Maths Book Part 1. By Abhishek Jain

Transcription

1 Maths Book Part 1 By Abhishek Jain

2 Topics: 1. Number System 2. HCF and LCM 3. Ratio & proportion 4. Average 5. Percentage 6. Profit & loss 7. Time, Speed & Distance 8. Time & Work

3 Number System Understanding Numbers A measurement carried out, of any quantity, leads to a meaningful value called the Number. This value may be positive or negative depending on the direction of the measurement and can be represented on the number line. Natural Numbers (N) The numbers 1, 2, 3, 4, 5 are known as natural numbers. The set of natural numbers is denoted by N. Hence, N = {1, 2, 3, 4 }. The natural numbers are further divided as even, odd, prime etc. Whole Numbers (W) All natural numbers together with 0 are collectively called whole numbers. The set of whole numbers is denoted by W, and W = {0, 1, 2, 3, } Integers (Z) The set including all whole numbers and their negatives is called a set of integers. It is denoted by Z, and Z = {, 3, 2, 1, 0, 1, 2, 3,. }. They are further classified into Negative integers, Neutral integers and positive integers. Negative integers (Z-) All integers lesser than Zero are called negative integers. Z- = { 1, 2, 3 }

4 Neutral integers (Z0) Zero is the only integer which is neither negative nor positive and it is called a neutral integer. Positive integers (Z+) All integers greater than Zero are called positive integers. Z + = {1, 2, 3,.., } Real number Real numbers are those numbers which can be easily identified & quantified or all numbers which can be represented on number line are called real number. E.g. 2, 5, 6, 7.88, 9.73, 17 etc. Real numbers are of two types: Rational numbers & Irrational numbers: Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or ). On the other hand, all those numbers that can be written as non-terminating, non-repeating decimals are non-rational, so they are called the "irrationals". Examples would be 2 ("the square root of two") or the number pi ( π = , from geometry). Types of rational numbers: There are two types of rational numbers: a. Integers b. Fractions Integers Integers are defined as: all negative natural numbers (..-4,-3,-2,-1 and 0 & all positive natural numbers (1,2,3,4,5..) Note that integers do not include decimals or fractions - just whole numbers. Fractions: All the rational numbers which are in form of " #, where p, q 0 and p is not a multiple of q are called fractions. The number on the bottom of fraction is called denominator and number on top of a fraction is called a numerator. Examples: 1.3, 1.5, 5.6, $ %, & % etc. In the fraction, $, 2 is the numerator and 5 is denominator. %

5 Irrational Numbers Any number which cannot be represented in the form " where p and q are integers and q 0 # is an irrational number. On the basis of non-terminating decimals, irrational numbers are nonterminating non recurring decimals. Such as is a non-terminating, nonrepeating number. Examples: π, 5, 7, etc. Imaginary number:- Imaginary numbers are those numbers which cannot be represented on number line. E.g. 2, 3, etc. Square roots of all negative numbers are imaginary numbers. Classification of Numbers i. Even Numbers: All numbers divisible by 2 are called even numbers. E.g., 2, 4, 6, 8, 10 Even numbers can be expressed in the form 2n, where n is an integer. Thus 0, 2, 6, etc. are also even numbers. ii. Odd Numbers: All numbers not divisible by 2 are called odd numbers. e.g. 1, 3, 5, 7, 9 Odd numbers can be expressed in the form (2n + 1) where n is any integer. Thus 1, 3, 9 etc. are all odd numbers. Addition / Subtraction even + even = even; even + odd = odd; odd + odd = even even even = even; even odd = odd; odd odd = even Multiplication even even = even; even odd = even; odd odd = odd. Multiplication of any number Positive positive = positive Positive negative = negative Negative negative = positive Division Positive / positive = positive Positive / negative = negative Negative / negative = positive iii. Prime Numbers: A natural number that has no other factors besides itself and unity is a prime number.

6 Examples: 2, 3, 5, 7, 11, 13, 17, 19 Important Observation about prime numbers: A prime number greater than 3, when divided by 6 leaves either 1 or 5 as the remainder. Hence, a prime number can be expressed in the form of 6K ± 1. But the converse of this observation is not true, that a number leaving a remainder of 1 or 5 when divided by 6 is not necessarily a prime number. For eg: 25, 35 etc Example 1:- If a, a + 2, a + 4 are consecutive prime numbers. Then how many solutions a can have? Solution:- No even value of a satisfies this. So a should be odd. But out of three consecutive odd numbers, atleast one number is a multiple of 3. So, only possibility is a = 3 and the numbers are 3, 5, 7. Verifying whether the number is prime number or not Can be done by trial division, that is to say dividing n by all prime numbers Smaller than n, thereby checking whether n is a multiple of m < n Example 2:- Find whether 161 is a prime number or not? Solution: 161 is little less than 13, from integers from 2 to 13, 161 is divisible by 7, hence 161 is not a prime number. iv. Composite Numbers: A composite number has other factors besides itself and unity. e.g. 8, 72, 39, etc. On the basis of this fact that a number with more than two factors is a composite we have only 34 composite from 1 to 50 and 40 composite from 51 to 100. v. Perfect Numbers: A number is said to be a perfect number if the sum of ALL its factors excluding itself (but including 1) is equal to the number itself. Or The sum of all the possible factors of the number is equal to twice the number.

7 vi. Co-Prime numbers: Two numbers are (prime or composite) said to be co-prime to one another, if they do not have any common factor other than 1. e.g. 35 & 12, since they both don t have a common factor among them other than 1. Conversion of recurring decimal into fractions: The p q form of any recurring = The whole number - non-recurring part As many 9's as number of digits in the recurring part followed by as many0's as digits in the non-recurring part Example:- Express as a fraction Solution: x = = = = Factors A divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. In general, it is said m is a factor of n for non-zero integers m and n if there exists an integer k such that n = k m 1 is divisor of every integer. Every integer is a divisor of itself. A positive divisor of n which is different from n is called a proper divisor. An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself. Any positive divisor of n is a product of prime divisors of n raised to some power. If a number equals the sum of its proper divisors, it is said to be a perfect number Finding the Number of Factors of an Integer p q r First make prime factorization of an integer n= a b c, where a, b and c are prime factors of n and p, q, and r are their powers. The number of factors of n will be expressed by formula (p+1) (q+1) (r+1) NOTE: This will include 1 and n itself. Example: Finding the number of all factors of 450? Solution: 450 = Total number of factors of 450 including 1 and 450 itself is.(1+1) (2+1 ) (2+1) = = 18 Finding the Sum of all the Factors of any natural number. p q r First make prime factorization of an integer n= a b c, where a, b and c are prime factors of n and p, q, and r are their powers. The sum of factors of n will be expressed by the formula:

8 Example: Finding the sum of all factors of Solution: 450 = The sum of all factors of 450 is Finding the product of all the factors of any natural number. First we need to find the number of factors of that number (N) Let s say number of factors of N = X then, product of all the factors of N = (N) 4 5 Example: Find product of all the factors of 36 Solution: 36 = number of factors are = (2+1) (2+1) = 9 so, product of factors = or = = answer Perfect Square A perfect square, is an integer that can be written as the square of some other integer. For example 16 is an perfect square of 4. There are some tips about the perfect square: The number of distinct factors of a perfect square is ALWAYS ODD. The sum of distinct factors of a perfect square is ALWAYS ODD. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Evenfactors. Perfect square always has even number of powers of prime factors. Division Division is the operation of determining how many times one quantity is contained in another. It is the inverse of multiplication. Formulae: Dividend = ( Divisor Quotient) + Remainder If 15 is divided by 6, then quotient will be 2 and remainder will be 3. Hence, we can write it as 15 = Here 15 is dividend, 6 is divisor, 2 is a quotient and 3 is a remainder If a number, when divided by 3 leaves remainder 1, the number can be written as 3x+1. Some important facts 1. The sum of five successive whole numbers is always divisible by The square of an odd number when divided by 8 leaves a remainder of The product of 3 consecutive natural numbers is divisible by The product of 3 consecutive natural numbers, the first of which is an even number is divisible by If a number is not a multiple of 3, then its square divided by 3 leaves a remainder of The last digit of the product of any nine consecutive numbers is always The sum of a two-digit number and a number formed by reversing its digits is divisible by 11.

9 E.g = 110, which is divisible by A three-digit number where all the digits are same is divisible by n - 7 is divisible by 3 for any natural n. Divisibility Rules Divisible By: If 2 The number is even. Or The last(units) digit of the number is 0, 2, 4, 6 or 8 3 The sum of the digits of number is divisible by 3. 4 The last two digits of the number is divisible by 4. And the numbers having two or more zeros at the end. Examples 84 is divisible by is not divisible by is divisible by 3. ( = 15) 346 is not divisible by 3. ( = 13) is divisible by 4. (56 is divisible by 4) is divisible by 4. (Two or more zeros at the end.) 5 The numbers having 0 or 5 at the end is divisible by 5. (5 is there at the end) 1234 is not divisible by 5. (0 or 5 is not there at the end) 6 The number is divisible by both 2 and is divisible by 6. (It is divisible by both 2 and 3) 7 The rule which holds good for numbers with more than 3 digits is as follows: (A) Group the numbers in three from units digit (B) Add the odd groups and even groups separately 6782 is not divisible by 6. (It is divisible by 2 but not divisible by 3) Let s take The groups are 85, 437, 954 Sum of odd groups = = 1039 Sum of even groups = 437 Difference = 602. Which is divisible by 7, hence the number is divisible by 7. (C) The difference of the odd and even groups should be 0 or divisible by

10 8 The number formed by last three digits is divisible by 8. And the numbers having three or more zeros at the end is divisible by 8. (056 is divisible by 8) is divisible by 8. (Three zeros at last) 9 The sum of all the digits is divisible by is divisible by 9. ( = 9 which is divisible by 9) 10 The number ends with zero is divisible by 10. (It ends with zero) 11 The difference between the sum of digits at even places and sum of digits at odd places is 0 or divisible by is divisible by 11. [( ) (0 + 3) = 11 which is divisible by 11] is divisible by 11. [( ) ( ) = 0] 12 The number is divisible by both 3 and is divisible by 12. ( The number is divisible by both 3 and 4) 13 Method: Multiply last digit of the number by 4 and add it to the remaining number. Continue this process until two digit number is achieved. If this two digit number is divisible by 13 then the number is divisible by is not divisible by 12. (The number is divisible by 3 but not divisible by 4) 182 is divisible by 13. (Multiply last digit by 4 i.e 2 4 = 8. Add it to the remaining number i.e = is divisible by 13 so 182 is divisible by 13) 2145 is divisible by 13 ( Multiply last digit by 4 i.e 5 4 = 20. Add it to the remaining number i.e = 234 which is not a two digit number so repeat the process. Multiply last digit of 234 by 4 i.e. 4 4= 16. Add it to the remaining number i.e = 39 which is divisible by 13 so 2145 is divisible by 13) 14 The number is divisible by both 2 and is divisible by 14. (The number is divisible by both 2 and 7)

11 15 The number is divisible by both 3 and is divisible by 15. (The number is divisible by both 3 and 5) 16 The number formed by last four digits is divisible by Method: Multiply last digit with 5 and subtract it from the remaining number. If the result is divisible by 17 then the original number is also divisible by 17. Repeat this process if required is divisible by 16. (The number formed by last four digits is divisible by 16) 4029 is divisible by 17 ( Multiply last digit by 5 i.e. 9 5= 45. Subtract it from the remaining number = 357 which is divisible by 17 so 4029 is divisible by 17) 18 The number is even and divisible by is divisible by 18. (It is even and divisible by 9) 19 Method: Multiply last digit with 2 and add it to the remaining number. If the result is divisible by 19 then original number is also divisible by 19. Repeat this process if required. Divisibility rule for any prime number greater than is divisible by 19. (Multiply last digit with 2 i.e 5 2 = 10. Add it to the remaining number 123 i.e = 133 which is divisible by 19 so 1235 is also divisible by 19) If any single digit written n times then it will be divisible by prime number(p) greater than 5 only if n is divisible by (P 1) Example:- What will be the remainder when times is divided by 7? Solution: If 4 is divided by 7, the remainder is 4. If 44 is divided by 7, the remainder is 2. If 444 is divided by 7, the remainder is 3 By checking like this, we come to know that is exactly divisible by 7. So if we take 6 fours, it is exactly divisible by 7. Similarly 12 fours is also exactly divisible by 7 and 42 fours will be exactly divisible by 7. So out of 44, the remaining two fours will give a remainder of 2. Now if we see all bold numbers are multiple of 6 and 6 is nothing but 1 less than 7(prime) FUNDA: How to calculate remainder, when a number is divided by 11, without division? Step 1: Add all the odd place numbers (O) and even place numbers (E) counted from right to left. Step 2: If O E is positive, remainder will be the difference less than 11. Step 3: If O E is negative, remainder should be (11 difference). Ex.1 what is the remainder when is divided by 11? Sol. Odd place digit sum (O) = = 19.

12 Even place digits sum (E) = = 27. Difference (D) = = 8 Remainder = 11 8 = 3. Ex.2 If 567P55Q is divisible by 88; Find the value of P + Q. Sol. The number is divisible by 8 means; the number formed by the last 3 digits should be divisible by 8 which are 55Q. Only Q = 2 satisfy this. From the divisibility rule of 11, ( ) (5 + P + 6) is divisible by 11. So 8-P is divisible by 11. if P= 8, then only it is possible. So P = 8 and Q = 2. So P + Q = 10. Ex.3 If the first 100 natural numbers are written side by side to form a big number and it is divided by 8. What will be the remainder? Sol. The number is According to the divisibility rule of 8, we will check only the last 3 digits. If 100 is divided by 8, the remainder is 4. Points to remember The number 1 is neither prime nor composite. 1) 2 is the only even number which is prime. 2) (x n + y n ) is divisible by (x + y), when n is an odd number. 3) (x n y n ) is divisible by (x + y), when n is an even number. 4) (x n y n ) is divisible by (x y), when n is an odd or an even number. 5) The difference between 2 numbers (xy) (yx) will always be divisible by 9. 6) The square of an odd number when divided by 8 will always give 1 as a remainder. 7) Every square number is a multiple of 3 or exceeds a multiple of 3 by unity. 8) Every square number is a multiple of 4 or exceeds a multiple of 4 by unity. 9) If a square number ends in 9, the preceding digit is even. 10) If m and n are two integers, then (m + n)! is divisible by m! n! 11) (a) n 1 if n is even / (a + 1) leaves a remainder of a if n is odd 12) Product of n consecutive numbers is always divisible by n!. Factorials Factorial of a positive integer n denoted by n!, is the product of all positive integers less than or equal to n. For instance. Trailing zeros Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow has 3 trailing zeros; The number of trailing zeros in the decimal representation of n!, the factorial of a non Negative integer, can be determined with this formula:

13 It's easier if you look at an example., where k must be chosen such that. Example: How many zeros are in the end (after which no other digits follow) of 32!? Solution: (Denominator must be less than 32, 5 $ = 25 is less) Hence, there are 7 zeros in the end of 32! The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. Finding the number of powers of a prime number, in the. The formula is :... till Example: What is the power of 2 in 25!? Solution: Unit digit Unit is same as finding remainder by 10 Determining the last digit of : 1. Last digit of is the same as that of ; 2. Determine the cyclicity number of ; 3. Find the remainder when divided by the cyclicity; 4. When, then last digit of is the same as that of and when, then last digit of is the same as that of, where is the cyclicity of number. 5. Integer ending with 0, 1, 5 or 6, in the integer power k > 0, has the same last digit as the base. 6. Integers ending with 2, 3, 7 and 8 have a cyclicity of Integers ending with 4 (e.g. ) have a cyclicity of 2. When n is odd will end with 4 and when n is even will end with Integers ending with 9 (e.g. ) have a cyclicity of 2. When n is odd will end with 9 and when n is even will end with 1. Example: What is the last digit of? Solution: Last digit of is the same as that of. Now we should determine the cyclicity

14 of : =7 (last digit is 7) =9 (last digit is 9) =3 (last digit is 3) =1 (last digit is 1) =7 (last digit is 7 again!) and so on.. So, the cyclicity of 7 is 4. Now divide 39 (power) by 4 (cyclicity), remainder is 3.So, the last digit of is the same as that of the last digit of, is the same as that of the last digit of, which is. Indices: 1. a m a n = a m + n 2. (a m ) n = a m n 3. a m /a n = a m n 4. (ab) n = a n b n 5. a 0 = 1 6. ( ) 1/m m a= a 7. If a m = a n then m = n 8. If a m = b then a = b 1/m Basic formulae for operations on numbers 1. (a + b) 2 = a 2 + b 2 + 2ab 2. (a b) 2 = a 2 + b 2 2ab 3. (a + b) 3 = a 3 + b 3 + 3ab(a + b) 4. (a b) 3 = a 3 b 3 3ab(a b) 5. (a 2 b 2 ) = (a + b) (a b) 6. (a 3 b 3 ) = (a b) (a 2 + ab + b 2 ) 7. (a 3 + b 3 ) = (a + b) (a 2 ab + b 2 ) 8. (a + b + c) 2 = a 2 + b 2 + c (ab + bc + ca) 9. a 3 + b 3 + c 3 3abc = (a + b + c) (a 2 + b 2 + c 2 ab bc ca) 10. If a + b + c = 0, then a 3 + b 3 + c 3 = 3abc 11. (a + b) 2 + (a b) 2 = 2(a 2 + b 2 ) 12. (a + b) 2 (a b) 2 = 4ab 13. (a 4 b 4 ) = (a b) (a + b) (a 2 + b 2 ) Some more properties on natural numbers Sum of 1 st n Natural number i.e n = {n (n + 1)} /2 Sum of squares of 1 st n natural number i.e n 2 ={ n(n+l)(2n+l) }/6

15 Sum of cubes of 1 st n Natural number i.e n 3 = [{n(n+i)}/2] 2 Sum of squares of 1 st n Odd natural number i.e (2n 1) 2 = {n (2n -1 )(2n + I)}/3 Sum of squares of 1 st n Even natural number i.e n 2 = {2n(n+1)(2n+1)}/3 Sum of cubes of 1 st n Odd natural number i.e (2n -1) 3 = n 2 (2n 2-1) Sum of cubes of 1 st n Even natural number i.e n 3 = 2 [n (n + 1)] 2 How many times a digit appears in a certain range (except 0)? Let s consider any digit (Say 2) and check how many times it appears in different ranges. From 1 9, the digit 2 is used 1 time From 1 99, the digit 2 is used 20 times. From 1 999, the digit 2 is used 300 times From , the digit 2 is used 4000 times and so on Some Other Properties of Numbers Squares: There is a definite relationship between the unit digits when square of a number is considered, we will see that one by one. 1. If unit digit of a perfect square is I then ten s digit has to be Even (e.g. 81, 121, 441 etc) 2. If unit digit of a number is 2 then it can t be a perfect square. 3. If unit digit of a number is 3 then it can t be a perfect square. 4. If unit digit of a perfect square is 4 then ten s digit has to be Even (e.g. 64,144 etc) 5. If unit digit of a perfect square is 5 then ten s digit has to be 2 (e.g. 25, 225, 625 etc) 6. If unit digit of a perfect square is 6 then ten s digit has to be Odd and multiple of If unit digit of a number is 7 & 8 then it can t be a perfect square 8. If unit digit of a perfect square is 9 then ten s digit has to be Even (e.g. 49, 169 etc) 9. If unit digit of a perfect square is 0 then ten s digit has to be 0 (e.g. 100,400,900 etc) Hence if a number end with 2, 3, 7 or 8 then it can t be perfect square.

16 Solved Examples Q1. What is the digit in the unit place of ? Solution: First we will find the unit digit of We know, the even power of 9 results in 1 as its unit digit. So, the last digit of will be 1. Last digit of By dividing 33 by 4, we get 1 as a remainder. So, the last digit of will be same as of 3 1 or 3. Last digit of = = 4 Q2. A is the smallest integer which when multiplied with 3 gives a number made of 5 s only. Sum of the digits of A is B. Sum of the digits of B is C. What is the value of C 3? Solution:- Smallest number made of 5 s and divisible by 3 is 555. Thus B = 5+5+5=15 and C =1+5 =6 3 So, C =216 Q is divisible by: Solution: x n y n is always divisible by (x y) x n y n is divisible by (x + y) when n is even. So, will be divisible by both (37 23) and ( ) So, will be divisible by (37 23) ( ) = = 840 Q4. Let N = The sum of the digits of N is: Solution:-Since N is written out as a sum of powers of 10, then N can be written as , and so the sum of the digits is 7. Q5. Let D be a decimal of the form D = 0. abcd abcd abcd, where digits a, b, c and d are integers lying between 0 and 9. At most three of these digits are zero. By what number D be multiplied so that the result is a natural number? abcd Solution:- D = 9999, Hence, once we multiply D with 9999 we will get a natural number. Q6. N = ( ) 2. What is the sum of the digits of N? Solution:- 1 2 = 1, 11 2 = 121, = so, = The sum of these numbers is 64. Q7. What is the remainder when is divided by 1924? Solution: = (1924 1) some oddpower = x or Remainder = ( 1) x = 1 or Remainder = =1923

17 Q8. N= How many zeroes are there at the end of N? Solution:- N = !. The highest power of 5 in 50! is 12, therefore there will be 12 zeroes at the end of N. Q9. What is the unit s digit of 3 45? Solution:-We see that last digits of the powers 3 1 = 3, 3 2 = 9, 3 3 = 27, 3 4 = 81, and then the digits start repeating. So the last digit of 3 5 will be 3, 3 6 will be 9, 3 7 will be 7 and so on. This is called cyclicity. In this case the cyclicity is 4, as the numbers start repeating after the 4th power. So we can find the last digits of any numbers raised to a high power. Since we have to find 3 45, we divide 45 by 4 and we get the remainder as 1. Hence the last digit of 345 will be same as 3 1, or 3. The numbers 2, 3, 7 and 8 have cyclicity 4 Q10. What is the last digit of 7 23 x 8 13? Solution:-Divide 23 by 4, we get remainder 3. Hence last digit of 7 23 will be same as 7 3 or 3. Divide 13 by 4, we get remainder as 1; hence last digit of 8 13 is same as 8 1 or 8. So the last digit of the product is 3 8 = 24, or 4 Q11. A man has a certain number of mangoes. He sells half of what he had and one more to A, half of the remainder and one more to B, half of the remainder and one more to C, half of the remainder and one more to D. Then he has only 1 left. How many did he have at the beginning? Solution:- In such sums, it is advisable to start from the back (try not to use x and form an equation, because you have four steps to solve). Before D, he had (1+1) x 2 = 4. Before C he had (4+1) x 2 = 10. Before B he had (10+1) x 2 = 22 and before A he had (22+1) x 2 =46. Q12.What is the minimum number by which I should divide 1250 so that I get a perfect square? Solution: 1250 = = = Therefore to get perfect square we should divide by 2. Hence, to get a perfect square you can also multiply by 2. Q13.What is the rightmost non-zero digit of the number ? Solution: As the given number is a multiple of 10, the last 2720 digits will be 0s. The right most non-zero digit of the number will be the unit digit of the number The unit digit of the powers of 3 follows a cyclicity of "4" In as 2720 is a multiple of "4", the number will have the same units digit as 3 4 which is "1". Hence, the rightmost non-zero digit of the number is "1". Q14.When is divided by 12, what would be the remainder? Solution: When 13 is divided by 12 the remainder is 1 any power of 13 will always give 1 as the remainder. But we add 3, so the remainder will be 4 Q is divisible by Solution: Take 2 32 common 2 32 ( ) 2 32 (15). Thus the number is divisible by 15.

18 Q16. What is the least number that must be subtracted from 2000 to get a number which is exactly divisible by 17? Solution: On dividing 2000 by 17, we get 11 as remainder. So, required number is 11. Q17. What is the least number that must be added to 3000 to obtain a number exactly divisible by 19? Solution: On dividing 3000 by 19, we get 17 as remainder So, number to be added = = 2 Q18. If the positive integer x leaves a remainder of 2 when divided by 8, what will the remainder be when x + 9 is divided by 8? Solution: The easiest way to get the answer here is to pick numbers. Pick a number for x that has a remainder of 2 when divided by 8, such as 10. Increase the number you picked by 9. In this case, = 19. Now divide 19 by 8, which gives a remainder 3. Q = Solution: = = 2 2 = 2 = Q20. A number is as much greater than 36 as is less than 86. Find the number. Solution: Let the number be x. Then, x 36 = 86 x OR 2x = = 122 OR x = 61. Hence, the required number is 61. Q21. Find a number such that when 15 is subtracted from 7 times the number, the Result is 10 more than twice the number. Solution: Let the number be x. Then, 7x - 15 = 2x + 10 or 5x = 25 or x = 5. Hence, the required number is 5. Q22. The sum of a rational number and its reciprocal is 13/6. Find the number. Solution: Let the number be x. Then, x + (1/x) = 13/6 OR (x 2 + 1)/x = 13/6 OR 6x 2 13x + 6 = 0 OR 6x 2 9x 4x + 6 = 0 OR (3x 2) (2x 3) = 0 It means x = 2/3 or x = 3/2 Hence the required number is 2/3 or 3/2. Q23. The sum of two numbers is 184. If one-third of the one exceeds one-seventh of the other by 8, find the smaller number. Solution: Let the numbers be x and (184 - x). Then, (x/3) - ((184 x)/7) = 8 OR 7x 3(184 x) = 168 OR 10x = 720 => x = 72. So, the numbers are 72 and 112. Hence, smaller number = 72. Q24. The difference of two numbers is 11 and one-fifth of their sum is 9. Find the numbers. Solution: Let the number be x and y. Then, x y = (i) and 1/5 (x + y) = 9 OR x + y = (ii) Adding (i) and (ii), we get: 2x = 56 or x = 28. Putting x = 28 in (i), we get: y = 17. Hence, the numbers are 28 and 17.

19 Q25. If the sum of two numbers is 42 and their product is 437, then find the absolute difference between the numbers. (S.S.C. 2003) Solution: Let the numbers be x and y. Then, x + y = 42 and xy = 437 x - y = [(x + y) $ 4xy] = [(42) $ 4 437] = [ ] = [16] = 4. Required difference = 4. Q26. The sum of two numbers is 16 and the sum of their squares is 113. Find the numbers. Solution: Let the numbers be x and (15 - x). Then, x 2 + (15 - x) 2 = 113 OR x x 2-30x = 113 OR 2x 2-30x = 0 or x 2-15x + 56 = 0 OR (x - 7) (x - 8) = 0 or x = 7 or x = 8. So, the numbers are 7 and 8. Q27. The average of four consecutive even numbers is 27. Find the largest of these numbers. Solution: Let the four consecutive even numbers be x, x + 2, x + 4 and x + 6. Then, sum of these numbers = (27 4) = 108. So, x + (x + 2) + (x + 4) + (x + 6) = 108 or 4x = 96 or x = 24. :. Largest number = (x + 6) = 30. Q28. The sum of the squares of three consecutive odd numbers is 2531.Find the numbers. Solution: Let the numbers be x, x + 2 and x + 4. Then, x 2 + (x + 2) 2 + (x + 4) 2 = 2531 => 3x x = 0 => x 2 + 4x = 0 => (x - 27) (x + 31) = 0 => x = 27. Hence, the required numbers are 27, 29 and 31. Q29. Of two numbers, 4 times the smaller one is less than 3 times the 1arger one by 5. If the sum of the numbers is larger than 6 times their difference by 6, find the two numbers. Solution: Let the numbers be x and y, such that x > y Then, 3x - 4y = 5...(i) and (x + y) - 6 (x - y) = 6 OR -5x + 7y = 6 (ii) Solving (i) and (ii), we get: x = 59 and y = 43. Hence, the required numbers are 59 and 43. Q30. The ratio between a two-digit number and the sum of the digits of that number is 4 : 1.If the digit in the unit's place is 3 more than the digit in the ten's place, what is the number? Solution: Let the ten's digit be x. Then, unit's digit = (x + 3). Sum of the digits = x + (x + 3) = 2x + 3. Number = l0x + (x + 3) = 11x x+3 / 2x + 3 = 4 / 1 OR 1lx + 3 = 4 (2x + 3) OR 3x = 9 => x = 3. Hence, required number = 11x + 3 = 36. Q31. A number consists of two digits. The sum of the digits is 9. If 63 is subtracted from the number, its digits are interchanged. Find the number. Solution: Let the ten's digit be x. Then, unit's digit = (9 - x).

20 Number = l0x + (9 - x) = 9x + 9. Number obtained by reversing the digits = 10 (9 - x) + x = 90-9x. therefore, (9x + 9) - 63 = 90-9x OR 18x = 144 => x = 8. So, ten's digit = 8 and unit's digit = 1. Hence, the required number is 81. Q32. A fraction becomes 2/3 when 1 is added to both, its numerator and denominator. and,it becomes 1/2 when 1 is subtracted from both the numerator and denominator. Find the fraction. Solution: Let the required fraction be x/y. Then, x+1 / y+1 = 2 / 3 => 3x 2y = - 1 (i) and, x 1 / y 1 = 1 / 2 or 2x y = 1 (ii) Solving (i) and (ii), we get : x = 3, y = 5 therefore, Required fraction= 3 / 5. Q is divided into two parts such that the sum of their reciprocals is 1/ 12.Find the two Parts. Solution: Let the two parts be x and (50 - x). Then, K L + K %MNL = 1 / 12 or (%M O P O) L(%MNL) = 1 / 12 or x 2 50x = 0 or (x 30) (x 20) = 0 or x = 30 or x = 20. So, the parts are 30 and 20. Q34. If three numbers are added in pairs, the sums equal 10, 19 and 21. Find the numbers. Solution: Let the numbers be x, y and z. Then, x+ y = 10...(i) y + z = 19...(ii) x + z = 21 (iii) Adding (i),(ii) and (iii), we get: 2 (x + y + z ) = 50 or (x + y + z) = 25. Thus, x = (25-19) = 6; y = (25-21) = 4; z = (25-10) = 15. Hence, the required numbers are 6, 4 and 15. Q35. What is the least number by which 825 must be multiplied in order to produce a multiple of 715? Solution: 825 = = In 825, 13 is missing. Rest all the factors of 715 are present in 825 Q36. (i) If X381 is divisible by 11, find the value of smallest natural number X. (ii) If 381Y is divisible by 9, find the value of smallest natural number Y. Solution: (i) X381 is divisible by 11. Hence, the divisibility rule for 11 must be satisfied i.e. (X + 8) (3 + 1) = 0 or a multiple of 11 i.e. X + 4 = 11 X = 7 (ii) 381Y is divisible by 9. Hence, the divisibility rule for 9 must be satisfied i.e. ( Y) must be divisible by 9 i.e. (12 + Y) = 18 Y = 6

21 Q37. Which of these is true? A. 3 3 is not a rational number. B. If a is rational number and n is an integer > 1, then a n is a rational number. C. 7 is not the cube of a rational number. Solution: Recall that rational numbers are those that can be expressed as ratios i.e., as a/b (b 0). 3 is irrational. If a is rational number, a 2, a 3 will be rational. Cube root of 7 is irrational. Hence all the statements are true Q38. What is the smallest number must be multiplied with, to make it a perfect square? Solution: = = We can make perfect square only if we have even power of all the prime numbers Now we have even power of 5 and 2 but we have 3 powers of 7 next even number after 3 is 4 So, to make the power 4 of 7 we have to multiply number with one more 7 So, answer is 7 Q39. If y is a positive non-zero integer, find the largest number using three y s and no operational sign: a.yyy b.yy y c.(y y ) y d. y QR Solution: Consider any positive non zero integer, say, y = 3. Now try all the given options, you will find that 3 &S = 3 27 is the largest number Q40. The number 6n 2 + 6n for natural n is always divisible by a.6 only b.6 and 12 c.12 only d.18 only e.24 only Solution: The above expression contains a 6, it is always divisible by 6. Also, n (n + 1) is divisible by 2. Hence, 6n (n + 1) is also divisible by 6 2 = 12. The correct answer is 12 and 6

22 Exercise Q1. If n is a positive integer, then the remainder when is divided by 7 is a)3 b)0 c)5 d) 6 e) None of these Q2. If a, a+2, and a+4 are prime numbers, then the number of possible solutions for a is: a)1 b)2 c)3 d) none e) None of these Q3. In a division sum, the divisor is 12 times the quotient and 5 times the remainder. if the remainder is 48, then dividend is? a) 4844 b)4488 c)4848 d) 4424 e) None of these Q4.What is the value of * in the number 17*24 so that it will be exactly divisible by 11? a)5 b)4 c)3 d)2 e) None of these Q5.What is the least number which is a perfect square but contains 847 as its factor? a) 5929 b)5595 c) 9595 d) 5599 e)5549 Q6. If N is an even integer, Which of the following is an odd integer? a) N +2 b) 3N 3 c) 2N(N) 2N d)n(n)+n+2 e) None of these Q7. x is odd while y is even. Which of these is odd (assume these are all integers)? a) xy b) xy +y c) (x+x)(y+y) d) y/x e) x+y Q8. What will be the remainder if is divided by 9? a)0 b) 8 c) 4 d) 7 e) 6 Q9. What s value for ? a) b) c) d) e) None of these Q10. What is the highest power of 7 in 5000!? (5000! means factorial ) a)4998 b)714 c)832 d)816 e) None of these Q11. What is the total number of different divisors including 1 and the number that can divide the number 6400? a)24 b)27 c)54 d)68 e) None of these Q12. A certain number when successfully divided by 8 and 11 leaves remainders of 3 and 7 respectively. What will be remainder when the number is divided by the product of 8 and 11, viz 88? a)3 b)21 c)59 d)68 e) None of these

23 Q13. What is the reminder when is divided by 6? a)0 b)3 c)4 d)5 e) None of these Q14. How many times will the digit '0' appear between 1 and 1000? a. 192 b. 292 c. 193 d.191 e. None of these Q15. What is the total number of different divisors of the number 7200? a.20 b.4 c.54 d.32 e. None of these Q16. The positive integer nearest to 1000 & divisible by 2, 3, 4, 5& 6 is (a) 1020 (b) 1040 (c) 960 (d) 1030 (e) None of these Q17. If the operation, ^ is defined by the equation x ^ y = 2x + y, what is the value of a in 2 ^ a = a ^ 3 (a) 0 (b) 1 (c) -1 (d) 4 (e) None of these Q18. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12? (a) 10 (b) 7 (c) 8 (d) 9 (e) None of these Q19. The number which is formed by writing any digit 6 times (e.g , , etc.) is divisible by (a) 7 (b) 11 (c) 13 (d) All of these (e) None of these Q20. How many times does digit 6 appear when you count from 11 to 100? (a) 18 (b) 19 (c) 20 (d) 21 (e) None of these Q21. Convert into a rational expression. a b c d e. None of these Q22. Rahul writes first hundred whole numbers. How many times does he write zero? Also find out how many times will number 9 occur in the set of first hundred whole numbers? a. 12, 19 b. 10, 20 c. 11, 19 d. 12, 20 e. None of these Q23. Which of following numbers is exactly divisible by 99? a b c d e. None of these Q24. Find the total number of factors of a number N = a. 64 b. 192 c. 24 d. 16 e. None of these Q25. Which of following numbers is not a prime number? a.71 b. 11 c. 119 d. 41 e. None of these

24 Q26. Let P be the set of even integers a such that 125 a 275, a is divisible by 7 but not by 11. How many elements does P contain? a.12 b.11 c.10 d.9 e. None of these Q27. Find the least number which when divided by 6, 7, 8, 9,10 leaves remainder 1. a b c d e. None of these Q28. Two numbers, x and y, are such that when divided by 6, they leave remainders 4 and 5 respectively Find the remainder when (X 2 + Y 2 ) is divided by 6. a. 2 b. 3 c. 4 d.5 e. None of these Q29. P is a prime number greater than 5. What is the remainder when P is divided by 6? a. 5 b.1 c.1 or 5 d. 4 e. None of these 71 Q30. What is the unit digit of317? a. 1 b.7 c.3 d.9 e. None of these Q31. A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor? a.13 b. 59 c.35 d. 37 e. None of these Q32. What is the remainder when is divided by 7? a. 3 b. 5 c.4 d.2 e. None of these Q33. What is the unit digit of ? a. 5 b. 3 c.4 d.2 e. None of these Q34. A number when divided by 5 and 7 successively leaves remainders 2 and 4 respectively. Find the remainder when same number is divided by 35. a. 20 b.8 c.22 d.27 e. None of these Q35. Which is the second smallest multiple of 7 that leaves a remainder of 9 when divided by 19? a. 171 b.259 c.218 d.161 e. None of these Q36. What will be the unit digit of 13? a. 7 b. 9 c.1 d.3 e. None of these 3430 Q37. What is the rightmost non-zero digit of the number 770? a. 7 b. 3 c. 9 d. 1 e. None of these Q38. Which of the following has most number of divisors? a. 99 b. 101 c. 176 d. 182 e. None of these

25 Q39. If each of the three nonzero numbers a, b, and c is divisible by 3, then abc must be divisible by which one of the following the numbers? a. 27 b. 81 c. 6 d. 12 e. None of these Q40. If we write down all the natural numbers from 259 to 492 side by side get a very large natural number How many 8's will be used to write this large natural number a. 32 b. 43 c. 52 d. 53 e. None of these Q41. What is the remainder when is divided by 800? a. 143 b. 243 c. 343 d. 443 e. None of these Q42. What is the value of M and N respectively if M N is divisible by 8 and 11, where M and N are single digit integers? a. 7,8 b. 8,6 c. 6,4 d. 5,4 e. None of these Q43. If n is a positive integer, which one of the following numbers must have a remainder of 3 when divided by any of the numbers 4, 5, and 6? a. 30n + 3 b. 90n + 3 c. 60n + 2 d. 60n + 3 e. None of these Q44. A certain number when divided by 222 leaves a remainder 35, another number when divided by 407 leaves a remainder 47. What is the remainder when the sum of these two numbers is divided by 37? a. 8 b. 9 c. 12 d. 17 e. None of these Q45. Find the remainder when 2 31 is divided by 5 a. 1 b. 4 c. 3 d. 2 e. None of these Q46. If N= , M= , then find the number of factors of N that are common with the factors of M. a. 8 b. 6 c. 18 d. 20 e. None of these Q47. HCF of two numbers is 2 and LCM is How many pairs of these 2 numbers are possible? a. 8 b. 16 c. 32 d. 4 e. None of these Q48. Find the remainder when 2 89 is divided by 89? a. 1 b. 2 c. 88 d. 87 e. None of these Q49. How many natural numbers below 660 are divisible by 5 and 11 but not by 3? a. 8 b. 9 c. 10 d. 11 e. None of these Q50. What is the remainder when is divided by 6? a. 3 b. 1 c. 0 d. 4 e. None of these

26 Q51. What is the unit's digit of the number a. 4 b. 6 c. 2 d. 8 e. None of these Q52. P = find the remainder when P is divided by 16 a. 4 b. 0 c. 8 d. 2 e. None of these Q53. The remainder when m + n is divided by 12 is 8, and the remainder when m n is divided by 12 is 6. If m > n, then what is the remainder when mn divided by 6? a. 1 b. 3 c. 5 d. 2 e. None of these Q54. If 0.0ababab. = $ %%, find ab. a. 18 b. 36 c. 76 d. 72 e. None of these Q55. Some birds were settled on the branches of a tree. First, they sat one to a branch and there was one bird left. Next they sat two to a branch and there was one branch left. How many branches were there? a. 3 b. 4 c. 5 d. 6 e. None of these Q56. The number 23x4534x01 is divisible by 11, how many values of x can be substituted? a. 1 b. 4 c. 9 d. 10 e. None of these Q57. Suppose x and y are the digits for which x01y is divisible by both 8 and 9. What is the sum of x and y? a.5 b. 6 c. 7 d. 8 e. None of these Q58. If the number A97B is to be divisible by 72, what are the respective values of A and B? a. 7 and 8 b. 1 and 6 c. 5 and 8 d. 0 and 6 e. None of these Q59. A number when divided by 195 leaves remainder 47. If the same number is divided by 15, then the remainder will be a. 1 b. 2 c. 3 d. 4 e. None of these Q60. The difference between a three digit number and a number formed by reversing the order of digits is always divisible by a. 3 only b. 9 only c. 11 only d. 9 and 11 e. None of these Q61. Seema weighs 5/9 of her weight plus 20 kg. What is the actual weight of Seema? a. 36 kg b. 72 kg c. 27 kg d. 45 kg e. None of these Q62. A number consists of three consecutive digits, the one in units place being the greatest of the three. The number formed by reversing the digits exceeds the original number by 22 times the sum of the digits. Find the number.

27 a. 231 b. 232 c. 233 d. 234 e. None of these Q63. Ram and Shyam each had got several numbers of pens. Total number of pens they had, were such that on attempting to distribute them among five students, they had two left. When they distributed them among 3 and 12, they again had two left each time. But when they distributed the pens among 7 students, they had none left. If they had pens in the ratio of 7:6, What is the number of pens did Ram and Shyam had? a. 91, 78 b. 98, 84 c. 35, 30 d. 84, 98 e. None of these Q64. Find the remainder when 7 63 is divided by 342. a. 0 b. 1 c. 2 d. 341 e. None of these Q65. Find the remainder obtained when is divided by 97. a. 10 b. 5 c. 2 d. 0 e. None of these Q 66. How many zeros are there at the end of the product ? a. 2 b. 4 c. 5 d. 6 e. None of these Q67. Find the no. of zeros in 752!. a. 186 b. 187 c. 192 d. 193 e. None of these Q68. A positive integer which when added to 999 gives a sum which is greater than when it is multiplied by 999. Which of the following could be the value of the positive integer? a. 2 b. 1 c. 3 d. 4 e. None of these Q69. What is the total number of prime number s less than 100? a. 24 b. 26 c. 25 d. 23 e. None of these Q70. If 43p is divisible by 3, then what is the largest value p can take? a. 2 b. 5 c. 9 d. 8 e. None of these Q71. If the number 6pq5 is divisible by both 3 and 5, which of the following digits can replace p and q? a. 9,7 b. 7,9 c. 6,5 d. both (a) and (b) e. None of these Q72. There are four prime numbers written in ascending order. The product of the first three is 1001 and that of the last three is Find the first prime number a. 6 b. 9 c.7 d. 5 e. None of these Q73. What is the remainder when is divided by 5 a. 4 b. 1 c. 2 d. 3 e. None of these

28 Q74. What is the remainder when 75 is divided by 4? a. 3 b. 2 c. 50 d. 6 e. None of these Q75. By what least number 250 must be multiplied to get a multiple of 15 a. 3 b. 5 c. 15 d. 25 e. None of these Answers 1. B 2. A 3. C 4. B 5. A 6. B 7. E 8. B 9. B 10. C 11. B 12. C 13. B 14. D 15. C 16. A 17. B 18. B 19. E 20. B 21. D 22. B 23. D 24. B 25. C 26. B 27. B 28. D 29. C 30. C 31. D 32. A 33. A 34. C 35. D 36. D 37. C 38. C 39. A 40. D 41. C 42. C 43. D 44. A 45. C 46. B 47. B 48. B 49. A 50. C 51. B 52. C 53. A 54. B 55. A 56. D 57. D 58. B 59. B 60. D 61. D 62. D 63. B 64. B 65. D 66. B 67. B 68. B 69. C 70. D 71. A 72. C 73. B 74. A 75. A

29 HCF and LCM What is highest common factor (HCF) and least common multiple (LCM)? How do you calculate HCF and LCM of two or more numbers? Are you looking for problems on HCF and LCM? This chapter will answer all these questions. HIGHEST COMMON FACTOR (HCF) The largest number that divides two or more given numbers is called the highest common factor (HCF) of those numbers. There are two methods to find HCF of the given numbers: Prime Factorization Method- When a number is written as the product of prime numbers, the factorization is called the prime factorization of that number. For example, 72 = = To find the HCF of given numbers by this method, we perform the prime factorization of all the numbers and then check for the common prime factors. For every prime factor common to all the numbers, we choose the least index of that prime factor among the given number. The HCF is product of all such prime factors with their respective least Indices. Example:- Find the HCF of 72, 288, and 1080 Solution: 72 = , 288 = , 1080 = The prime factors common to all the numbers are 2 and 3. The lowest indices of 2 and 3 in the given numbers are 3 and 2 respectively. Hence, HCF = = 72. Example:- Find the HCF of 36x 3 y 2 and 24x 4 y. Solution: 36x 3 y 2 = x 3 y 2 24x 4 y = x 4 y. The least index of 2, 3, x and y in the numbers are 2, 1, 3 and 1 respectively. Hence the HCF = x 2 y = 12x 2 y. Long division method- To find HCF of two numbers by division method, we divide the higher number by the lower number. Then we divide the lower number by the first remainder, the first remainder by the second remainder... and so on, till the remainder is 0. The last divisor is the required HCF.

30 Example:- Find the HCF of 288 and 1080 by the division method. Solution: Hence, the last divisor 72 is the HCF of 288 and CONCEPT OF CO-PRIME NUMBERS: Two numbers are co-prime to each other if they have no common factor except 1. For example, 15 and 32, 16 and 5, 8 and 27 are the pairs of co-prime numbers. If the HCF of two numbers N 1 and N 2 be H, then, the numbers left after dividing N 1 and N 2 by H are co-prime to each other. Therefore, if the HCF of two numbers be A, the numbers can be written as Ax and Ay, where x and y will be co-prime to each other. SOLVED PROBLEMS ON HCF Q1. Three company of soldiers containing 120, 192, and 144 soldiers are to be broken down into smaller groups such that each group contains soldiers from one company only and all the groups have equal number of soldiers. What is the least number of total groups formed? Solution: The least number of groups will be formed when each group has number of soldiers equal to the HCF. The HCF of 120, 192 and 144 is 24. Therefore, the numbers of groups formed for the three companies will be 5, 8, and 6, respectively. Therefore, the least number of total groups formed = = 19. Q2. The numbers 2604, 1020 and 4812 when divided by a number N give the same remainder of 12. Find the highest such number N. Solution: Since all the numbers give a remainder of 12 when divided by N, hence ( ), ( ) and ( ) are all divisible by N. Hence, N is the HCF of 2592, 1008 and Now 2592 = , 1008 = and 4800 = Hence, the number N = HCF = = 48. Q3. The numbers 400, 536 and 645, when divided by a number N, give the remainders of 22, 23 and 24 respectively. Find the greatest such number N. Solution: N will be the HCF of (400-22), (536-23) and (645-24). Hence, N will be the HCF of 378, 513 and > N = 27. Q4. The HCF of two numbers is 12 and their sum is 288. How many pairs of such numbers are possible? Solution: If the HCF if 12, the numbers can be written as 12x and 12y, where x and y are coprime to each other. Therefore, 12x + 12y = > x + y = 24. The pair of numbers that are co-prime to each other and sum up to 24 are (1, 23), (5, 19), (7,

31 17) and (11, 13). Hence, only four pairs of such numbers are possible. The numbers are (12, 276), (60, 228), (84, 204) and (132, 156). Q5. The HCF of two numbers is 12 and their product is How many such numbers are possible? Solution: Let the numbers be 12x and 12y, where x and y are co-prime to each other. Therefore, 12x 12y = > xy = 216. Now we need to find co-prime pairs whose product is = Therefore, the co-prime pairs will be (1, 216) and (8, 27). Therefore, only two such numbers are possible. LEAST COMMON MULTIPLE (LCM) The least common multiple (LCM) of two or more numbers is the lowest number which is divisible by all the given numbers. To calculate the LCM of two or more numbers, we use the following two methods: Prime Factorization Method: After performing the prime factorization of the numbers, i.e. breaking the numbers into product of prime numbers, we find the highest index, among the given numbers, of all the prime numbers. The LCM is the product of all these prime numbers with their respective highest indices. Example:- Find the LCM of 72, 288 and Solution: 72 = , 288 = , 1080 = The prime numbers present are 2, 3 and 5. The highest indices (powers) of 2, 3 and 5 are 5, 3 and 1, respectively. Hence the LCM = = Example:- Find the LCM of 36x 3 y 2 and 24x 4 y. Solution: 36x 3 y 2 = x 3 y 2 24x 4 y = x 4 y. The highest indices of 2, 3, x and y are 3, 2, 4 and 2 respectively. Hence, the LCM = x 4 y 2 = 72x 4 y 2. Division Method: To find the LCM of 72, 196 and 240, we use the division method in the following way: L.C.M. of the given numbers = product of divisors and the remaining numbers = = = SOLVED PROBLEMS ON LCM Example:- Find the highest four-digit number that is divisible by each of the numbers 24, 36, 45

32 and 60. Solution: 24 = 2 3 3, 36 = , 45 = and 60 = Hence, the LCM of 24, 36, 45 and 60 = = 360. The highest four-digit number is when divided by 360 gives the remainder 279. Hence, the number ( = 9720) will be divisible by 360. Hence the highest four-digit number divisible by 24, 36, 45 and 60 = Example:- Find the highest number less than 1800 that is divisible by each of the numbers 2, 3, 4, 5, 6 and 7. Solution: The LCM of 2, 3, 4, 5, 6 and 7 is 420. Hence 420, and every multiple of 420, is divisible by each of these numbers. Hence, the number 420, 840, 1260, and 1680 are all divisible by each of these numbers. We can see that 1680 is the highest number less than 1800 which is multiple of 420. Hence, the highest number divisible by each one of 2, 3, 4, 5, 6 and 7, and less than 1800 is Example:- Find the lowest number which gives a remainder of 5 when divided by any of the numbers 6, 7, and 8. Solution: The LCM of 6, 7 and 8 is 168. Hence, 168 is divisible by 6, 7 and 8. Therefore, = 173 will give a remainder of 5 when divided by these numbers. Example:- What is the smallest number which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively? Solution: The common difference between the divisor and the remainder is 4 (9-5 = 4, = 4, = 4). Now the LCM of 9, 18, and 24 is 72. Now 72-4 = = = Therefore, if we subtract 4 from 72, the resulting number will give remainders of 5, 14, and 20 with 9, 18, and 24. Hence, the number = 72-4 = 68. Example:- A number when divided by 3, 4, 5, and 6 always leaves a remainder of 2, but leaves no remainder when divided by 7. What is the lowest such number possible? Solution: the LCM of 3, 4, 5 and 6 is 60. Therefore, the number is of the form 60k + 2, i.e. 62, 122, 182, 242 etc. We can see that 182 is divisible by 7. Therefore, the lowest such number possible = 182. PROPERTIES OF HCF AND LCM 1) The HCF of two or more numbers is smaller than or equal to the smallest of those numbers.

33 2) The LCM of two or more numbers is greater than or equal to the largest of those numbers 3) If numbers N 1, N 2, N 3, N 4 etc. give remainders R 1, R 2, R 3, R 4, respectively, when divided by the same number P, then P is the HCF of (N 1 - R 1 ), (N 2 - R 2 ), (N 3 - R 3 ), (N 4 - R 4 ) etc. 4) If the HCF of numbers N 1, N 2, N 3... is H, then N 1, N 2, N 3... can be written as multiples of H (Hx, Hy, Hz.. ). Since the HCF divides all the numbers, every number will be a multiple of the HCF. 5) If the HCF of two numbers N 1 and N 2 is H, then, the numbers (N 1 + N 2 ) and (N 1 - N 2 ) are also divisible by H. Let N 1 = Hx and N 2 = Hy, since the numbers will be multiples of H. Then, N 1 + N 2 = Hx + Hy = H(x + y), and N 1 - N 2 = Hx - Hy = H(x - y). Hence both the sum and differences of the two numbers are divisible by the HCF. 6) If numbers N 1, N 2, N 3, N 4 etc. give an equal remainder when divided by the same number P, then P is a factor of (N 1 - N 2 ), (N 2 - N 3 ), (N 3 - N 4 )... 7) If L is the LCM of N 1, N 2, N 3, N 4.. all the multiples of L are divisible by these numbers. 8) If a number P always leaves a remainder R when divided by the numbers N 1, N 2, N 3, N 4 etc., then P = LCM (or a multiple of LCM) of N 1, N 2, N 3, N R. LCM and HCF of fractions HCF of fractions = LCM of fractions = TUV WX YZ[\]^_W]` aub WX c\yw[dy^_w]` aub WX YZ[\]^_W]` TUV WX c\yw[dy^_w]` Example:- Find the LCM and HCF of $% & &% K$ Ke Solution: LCM = HCF = aub WX $% & &% = Kg% TUV WX K$ & Ke h TUV WX $% & &% = % aub WX K$ & Ke &h Usage of HCF and LCM: I. Least number leaving remainder r in each case when divided by x, y and z. Thus, least number leaving remainder r in each case when divided by x, y and z = (LCM of x, y, z) + r The series of such numbers will be (LCM of x, y, z) n + r Example :- Find the least number which when divided by 5, 6 and 7 leaves a remainder 3 in each case? Solution: The smallest number which, when divided by 5, 6 and 7, leaves remainder 3 in each case is LCM (5, 6, 7) = =213 II. Least number leaving remainder (x-a), (y-a) and (z-a) when divided by x, y and z respectively.

34 Thus, least number leaving remainder (x-a), (y-a) and (z-a) when divided by x, y and z respectively = (LCM of x, y, z) a. The series of such numbers will be (LCM of x, y and z) n a. Example :- Find the smallest number which when divided by 4, 5 and 6 leaves remainder 2, 3 and 4? Solution: Here, we can see that (divisor remainder) =a, is same for all divisors, in this case a =2. HCF of 4, 5 and 6 is 60 so least number is LCM a = 60 2 =58 III. To find the largest measure that measures x, y and z exactly. Example :- A rectangular piece of cloth has dimension 18m 8m. What is least number of equal squares that can be cut out of this cloth such that no cloth is wasted? Solution: HCF (18, 8) = 2 So, minimum number of square that can be cut out of this piece = 18 8/2 2 = 36 Example :- In a school 437 boys and 342 girls have been divided into classes, so that each class has the same number of students and no class has boys and girls mixed. What is the least number of classes needed? Solution: We should have maximum number of students in a class. So, we have to find the HCF (437, 342) = 19 HCF is also factor of difference of the numbers So, number of classes = i&g + &i$ = = 41 classes Kj Kj Example :- The ratio between two numbers is 3 : 4. If their LCM is 192, what is the difference between the squares of the numbers? Solution: Let the two numbers be 3x and 4x. Therefore, the LCM of these two numbers will be 3 4 x = 12x. We know that the LCM of these two numbers is 192. Equating, we get 12x = 192 or x = 16. Hence, the two numbers are 3 16 and i.e., 48 and 64 We have to find out the difference between the squares of the numbers = = ( ) (64-18) = = 1792 Rapid Information List S. No Type of problem 1. Find the GREATEST NUMBER that will exactly divide given numbers. 2. Find the GREATEST NUMBER that will exactly divide x, y and z leaving remainders a, b and c respectively. 3. Find the LEAST NUMBER which is exactly divisible by x, y and y Approach to the problem Required number = HCF of given numbers (greatest divisor) Required number (greatest divisor) = HCF of (x-a), (y-b) and (z-c) Required number = LCM of x, y and z (least dividend)

35 Find the LEAST NUMBER which when divided by x, y and z leaves the remainders a, b and c respectively. 5. Find the LEAST NUMBER which when divided by x, y and z leaves the same remainder r in each case. 6. Find the GREATEST NUMBER that will exactly divide x, y and z leaves same remainder in each case. It is always observed that (x-a) = (y-b) = (z-c) = k (say) Required number = (LCM of x, y and z) - k Required number = (LCM of x, y and z) + r Required number = HCF of (x-a), (y-b) and (z-c) Find the HCF of HCF of fractions = Find the LCM of LCM of fractions = 9. Find the HCF of decimal numbers Step 1. Find HCF of given numbers without decimals Step 2. In the HCF make decimal point from right to left according to the maximum decimal places among the given numbers. 10. Find the LCM of decimal numbers Step 1. Find LCM of given numbers without decimals Step 2. In the HCF make decimal point from right to left according to the maximum decimal places among the given numbers. Student often get confused what to find in question HCF or LCM Whenever you see just DIVIDE or DIVIDES in question it means you need to find HCF And whenever you see DIVIDED BY or DIVISIBLE BY it means you need to find LCM Solved Examples: Ex. 1. Find the LCM of 70 and 90 [LIC 2005] Solution: To find LCM:

36 LCM = Ex. 2. Find the HCF of 22, 54, 108, 135, and 198 [Bank Clerk 2000] Solution: (i) To find HCF: Common factor is 1. So, HCF is 1 Ex. 3. Find the HCF of , and [CDS 2002] Solution: Let x = y = z = Common factors of x, y and z are 3, 5, 7. Highest Common factor = = 105 Ex. 4. Find the LCM of, and [Asst. Grade 2005] Solution: We know that LCM of fractions = To find LCM of fractions LCM = LCM of (4, 5, 7)

37 LCM = = 140 HCF of (5, 6, 15) 5 = = = HCF of (5, 6, 15) = 1 LCM = = 140 Ex. 5. Find the HCF of, [Asst. Grade 2004] Solution: We know that HCF of fractions = Here, and HCF = HCF of (4, 10, 20) 4 = = = HCF = 2 LCM of (9, 21, 63) LCM = = 63 HCF =

38 Ex. 6. Find the LCM of 0.24, 1.2 and 6 [SSC Grad, 2007] Solution: LCM of 0.24, 1.2 and 6 Make the same number of decimal places in all the given numbers by suffixing zeroes if necessary and then find the HCF/LCM without decimal point. Put the decimal point in the HCF/LCM results leaving as many digits on its right as there are in each of the numbers. The given numbers can be written as 1.20, 0.24, and 6.00 Now ignoring the decimals we find LCM of 120, 24, and 600 LCM = = = 600 Thus, required LCM = 600 i.e., 6. Ex. 7. Find the HCF and LCM of 0.72, 0.36 and 0.45 [SBI PO 2003] Solution: The given number can be written as 0.72, 3.6 and 0.45 Now ignoring number can be written as 72, 360 and 45 (i) To find HCF HCF of (0.72, 3.6 and 0.45) is 9 The required HCF is 0.09 (ii) To find LCM LCM = =360

39 Thus, required LCM is 3.6 Ex. 8. The HCF and LCM of two numbers are 63 and 1260 respectively. If one of the two numbers is 315, find the other number. [SSC Grad, 2000] Solution: We know that Given that, HCF = 63; LCM = 1260 One number (x) = 315 y = 252 Other number is 252. Ex. 9. The LCM of two numbers is 45 times their HCF. If one of the numbers is 125 and the sum of HCF and LCM is 1150, then the other number is? [SBI PO 2006] Solution: Given that, HCF + LCM = 1150; LCM = 45 (HCF) One number(x) = 125 Let, HCF + LCM = eqn. LCM = 45 (HCF) eqn. Substitute eqn. 2 in eqn. 1 HCF + 45HCF = HCF = 1150 HCF = => HCF = 25 LCM = 45 (25) => LCM = 1125 We know that, Another number is 225. Ex. 10. Find the greatest number which will divide 772 and 2778 so as to leave the remainder 5 in each case. [LIC ADO, 2005] Solution: We know that, To find the greatest number that will divide x, y and z leaving the same remainder in each case and when the value of the remainder is r is given, then Required number = H. C. F of

40 Given that x = 772; y = 2778; r = 5 Required number =HCF of (772-5) (2778-5) = HCF of (767, 2773) = 59 Ex. 11. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is: Solution: Required number = H.C.F. of (1657-6) and (2037-5) = H.C.F. of 1651 and 2032 = 127. Ex. 12. The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Then sum of the number is: Solution: Let the numbers be 2x and 3x. Then, their L.C.M. = 6x. So, 6x = 48 or x = 8. The numbers are 16 and 24. Hence, required sum = ( ) = 40. Ex. 13. If the sum of two numbers is 55 and the H.C.F. and L.C.M. of these numbers are 5 and 120 respectively, then the sum of the reciprocals of the numbers is equal to: Solution: Let the numbers be a and b. Then, a + b = 55 and ab = 5 x 120 = 600. The required sum = = a + b = 55 = 11 a b ab Ex. 14. Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is The sum of the three numbers is: Solution: Since the numbers are co-prime, they contain only 1 as the common factor. Also, the given two products have the middle number in common. So, middle number = H.C.F. of 551 and 1073 = 29; First number = = 19; Third number = = Required sum = ( ) = 85.

41 Exercise 1. Find the L.C.M. of 36, 48, 72 and 108. a.432 b. 428 c. 360 d. 106 e. None of these 2. The H.C.F and L.C.M. of any two numbers are 2079 and 27 respectively. If one of the two numbers is 189, find the other number. a. 309 b. 297 c. 180 d. 327 e. None of these 3. Find the greatest number that will divide 296, 492 and 1246 leaving remainders 8, 12 and 22 respectively a.12 b. 36 c. 24 d. 48 e. None of these 4. Which of following is a pair of co - prime number? a.32,52 b. 16,28 c. 18,35 d. 21,42 e. None of these 5. The L.C.M. of two numbers is 252 and their ratio is 6:7. Find the numbers. a. 32, 49 b. 16, 25 c. 36,42 d. 54,63 e. None of these 6. A room is 2.47m long and 2.09m broad. It is required to pave the floor with minimum square slabs. Find the number of slabs required for floor. a. 143 b. 361 c. 142 d. 165 e. None of these 7. Find the greatest number that will divide 378, 330 and 882 leaving remainders 3, 5 and 7 respectively. a. 25 b. 220 c. 300 d. 275 e. None of these 8. Which smallest number must be added to 9373 so that it exactly divisible by 4? a. 3 b. 2 c. 1 d. 4 e. None of these 9. Which smallest number must be added to so that it becomes divisible by 9? a. 3 b. 2 c. 5 d. 7 e. None of these 10. Six bells tolling at the interval of 2, 4, 6, 8, 10, and 12 seconds respectively. In 30 minutes how many times they toll? a. 4 b. 10 c. 15 d. 16 e. None of these 11. Find the L.C.M. of 6/7, 12/13, 7/15. a. 84 b. 1/84 c. 13 d. 1/13 e. None of these 12. The sum and the differences of the L.C.M. and the H.C.F. of two numbers are 312 and 264, respectively. Find the bigger number if their sum is 168. a. 108 b. 92 c. 96 d. 100 e. None of these 13. Find the H.C.F. of 120, 140, 180 and 420.

42 a. 15 b. 20 c. 25 d. 30 e. None of these 14. Find the least number which when divided by 36, 48 and 54 which will leaves the same remainder 7. a. 432 b. 436 c. 437 d. 439 e. None of these 15. Find the greatest number which on dividing 304, 554 and 854 leaves equal remainder. a. 35 b. 65 c. 50 d. 60 e. None of these 16. Find the greatest number which on dividing 277, 757 and 3307 leaves the same remainder 7 in each case. a. 30 b. 15 c. 60 d. 25 e. None of these 17. Find the L.C.M. of 9/10, 12/25, 18/35, 21/40. a. 5/252 b. 3/1400 c. 252/5 d. 1400/3 e. None of these 18. Find the greatest number that will divide 63, 45 and 69 so as to leave the same remainder. a. 6 b. 9 c. 8 d. 10 e. None of these 19. A wholesale sugar dealer has 780kilogram, 1650kilogram and 510kilogram sugar of three different qualities. He has to pack it all into boxes of equal size without mixing. Find the capacity of the largest possible box. a. 15 b. 30 c. 12 d. 25 e. None of these 20. Find the least perfect square number which is divisible by 2,3,4,5,6. a b. 900 c d e. None of these 21. Six bells commence tolling together and toll at intervals of 2, 4, 7, 8, 10 and 12 seconds respectively. In 30 minutes how many times do they toll together? a. 2 b. 3 c. 4 d. 5 e. None of these 22. Seven bells ring at intervals of 2, 3, 4, 6, 8, 9 and 12 minutes, respectively. They started ringing simultaneously at 6.00 in the morning. How many more times would all seven bells ring simultaneously till 5.00 in the evening of the same day? a. 6 b. 7 c. 8 d. 9 e. None of these 23. Find the LCM of 5/2, 8/9, 11/14 a. 400 b. 420 c. 440 d. 460 e. None of these 24. Find the largest number which divides 62, 132 and 237 to leave the same remainder in each case a. 35 b. 25 c. 45 d. 55 e. None of these 25. An electric wire is sold only in multiplier of 1 metre and a person required several lengths of wire, each 85 cm long. To avoid any wastage and to minimize labour, he should purchase

43 minimum length of a. 17m b. 17cm c. 170cm d. 1.7m e. None of these 26. Find the greatest number that will divide 55, 127 and 175, so as to leave the same remainder in each case a. 12 b. 18 c. 24 d. 30 e. None of these 27. Find the largest number of four digits exactly divisible by 12, 15, 18 and 27 a b c d e. None of these 28. The circumferences of the fore and hind wheels of a carriage are respectively. At any given moment, a chalk mark is put on the point of contact of each wheel with the ground. Find the distance travelled by the carriages so that both the chalk marks are again on the ground at the same time a m b. 2175m c cm d. 2175m e. None of these 29. Find the greatest number that will divide 478 and 719 leaving remainders 2 and 5 Respectively a. 236 b. 238 c. 240 d. 119 e. None of these 30. A certain card game has a special pack. If you deal out equally to five players there are two cards left undealt. The same is true if you deal to seven players. If the cards are dealt to three players there is one left undealt. Number of cards that cannot be there in the pack is a. 100 b. 150 c. 125 d. 175 e. None of these 31. The HCF of two numbers is 113 and their LCM is One of the numbers is Find the other a b c d e. None of these 32. Three numbers are in the ratio 1:2:3 and their H.C.F. is 12. The numbers are a. 12, 24, 36 b. 1, 2, 3 c. 6, 12, 18 d. 24, 48, 72 e. None of these 33. The least number of five digits which is exactly divisible by 12, 15 and 18 is a b c d e. None of these 34. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together? a. 10 b. 15 c. 16 d. 14 e. None of these 35. The traffic lights at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all chance simultaneously at 8: 20: 00 hours, then they will again change simultaneously at a. 8:27:12 hours b. 8:26:06 hours c. 8:25:12 hours d. 8:27:08 hours e. None of these

44 36. The greatest number that will divide 187, 233 and 279 leaving the same remainder in each case is a. 23 b. 46 c. 1 d. 92 e. None of these 37. The L.C.M of three different numbers is 120. Which of the following cannot be their H.C.F? a. 12 b. 24 c. 48 d. 8 e. Data inadequate 38. The sum of two numbers is 528 and their H.C.F is 33. The number of pairs of such numbers satisfying the above condition is a. 8 b. 4 c. 12 d. 16 e. None of these 39. Two numbers are in the ratio of 15: 11. If their H.C.F is 13, then the numbers are a. 195,143 b. 15, 11 c. 26, 13 d. Data inadequate e. None of these 40. Find the least number which when decreased by 4 is exactly divisible by 9, 12, 15 and 18 a. 180 b. 184 c. 176 d. Cannot be determined e. None of these 41. Find the least number which when divided by 12, 15, 20 and 24 leaves the same remainder 2 in each case a. 120 b. 122 c. 118 d. Cannot be determined e. None of these 42. The sum of two numbers is PQ and their difference is 1/7 of their sum. Find their HCF a. 7/6 (PQ) b.6/7 (PQ) c. 1/7 (PQ) d. 1/6 (PQ) e. None of these 43. The LCM of three numbers is 4752 and HCF is 6. If the two numbers are 48 and 66, then find the third number a. 36 b. 54 c. 42 d. 60 e. None of these 44. How many numbers between 200 and 300 which are exactly divisible by 6, 8, and 9? a. 2 b. 3 c. 4 d. 1 e. None of these 45. There are two electrical wires, one is 9 m 60 cm long aluminum wire and the other is 5 m 12 m long copper wire. Find the maximum length that can be equally cut from each wire in such a way that the total length of each wire is exactly divisible by it a. 64 cm b. 64 m c. 32 cm d. 128 cm e. None of these 46. Find the least number of girls, so that they can be arranged in the group of 15 or 20 or 30 a. 30 b. 60 c. 120 d. 180 e Find the least perfect square number which is exactly divisible by 3, 4, 5, 6 and 8 a b c d e. None of these 48. Find the greatest number less than 800, which is divisible by 8, 12, 28 a. 672 b. 660 c. 540 d. 720 e. None of these

45 49. Find the least number which when divided by 12, 15 and 20 leaves remaining 4, 7 and 12 respectively a. 50 b. 48 c. 52 d. 54 e. None of these 50. Find the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3 but when divided by 9 leaves no remainder a b. 187 c d e. None of these 51. Five balls begin to toll together and toll at intervals of 24, 40, 72 and 120 s. After what interval to time will they toll again together? a. 24 min. b. 48 min. c. 12 min. d. 72 min. e. None of these 52. Four wheels moving 12, 24, 20 and 30 revolutions in a minute starting at a certain point on the circumference downwards. After interval of time will they come together again in the same position? a. 20s b. 30s c. 1 min d. 45s e. None of these 53. The traffic lights at three different road crossings change after every and 108 s respectively. If they change simultaneously at 9 am, at what time will they change again simultaneously? a. 7 min 12 s b. 8 min c. 7 min 24 s d. 7 min e. None of these 54. Find the smallest four digit number that is exactly divisible by 8, 10 and 12 a b. 80 c d. 900 e. None of these 55. Find the greatest five digit number exactly divisible by 6, 8 and 12 a b c d e. None of these 56. Find the greatest number which will divide by 321, 428 and 535 exactly a. 107 b. 103 c. 101 d. 111 e. None of these 57. Find the greatest possible length which can be used to measure 4m 3cm, 4m 34cm and 4m 65cm exactly a. 31 m b. 31 cm c. 3.1 m d. 3.1 cm e. None of these 58. Find the least number of square tiles required to pave the floor of a room of 4 m 50 cm long and 5 m 25 cm broad a. 44 b. 42 c. 40 d. 80 e. None of these 59. Find the sum of the smallest four digit number and the largest five digit number a b c d e. None of these 60. Five balls begin to toll together and toll respectively at intervals of 6, 7, 8, 9 and 12. How many times they will toll together in one hour, excluding the one at the start?

46 a. 6 b. 7 c. 8 d. 9 e. None of these 61. Among how many students, 175 bananas and 105 oranges can be equally divided? a. 70 b. 25 c. 35 d. 45 e. None of these 62. Find out the LCM of 2 34, 5 32, 7 45, and 2 56, 3 32, 5 31, 7 24 a. 2 56, 3 32, 5 32, 7 45, b. 2 56, 3 32, 5 32, 7 24 c. 2 56, 5 32, 7 45, d. 2 34, 3 32, 5 32, 7 45, e. None of these 63. What is the smallest whole number that is exactly divisible by a. 128 b. 132 c. 130 d. 145 e. None of these 64. The least perfect square which is divisible by each of the numbers 12, 15, 20 and 24 is a b c d e. None of these 65. The least positive integer which leave a remainder 2, when divided by each of the numbers 4, 6, 8, 12 and 16 a. 98 b. 50 c. 48 d. 52 e. None of these 66. The number of integers between 200 and 600 that are divisible by 2, 3, and 7 is a. 6 b. 7 c. 9 d. 10 e. None of these 67. If three natural numbers, whose LCM is 360, are in the ratio 2:3:4, then the largest of them is a. 120 b. 60 c. 180 d. 240 e. None of these 68. The HCF of 168, 189 and 231 reduced by 8, gives a. 11 b. 13 c. 17 d. 21 e. None of these 69. The GCD and LCM of two numbers are 66 and 384 respectively. If the first number is divided by 2, the resulting answer is 66. The second number is a. 190 b. 192 c. 120 d. 90 e. None of these 70. The LCM and GCD of two numbers are 240 and 16 respectively. If the two numbers are in the ratio 3:5, the numbers are a. 16, 80 b. 48, 80 c. 12, 60 d. 6, 10 e. None of these 71. The LCM of two numbers is 280 and the ratio of the numbers of 7:8. Find the numbers a. 35 and 40 b. 21 and 24 c. 28 and 32 d. 32 and 48 e. None of these

47 Answers 1. A 2. B 3. C 4. C 5. C 6. A 7. A 8. A 9. B 10. D 11. A 12. C 13. B 14. D 15. C 16. A 17. C 18. A 19. B 20. B 21. A 22. D 23. A 24. A 25. A 26. C 27. C 28. A 29. B 30. B 31. A 32. A 33. D 34. C 35. A 36. B 37. C 38. B 39. A 40. B 41. B 42. B 43. B 44. A 45. A 46. B 47. D 48. A 49. C 50. C 51. B 52. B 53. A 54. C 55. A 56. A 57. A 58. B 59. A 60. B 61. C 62. A 63. B 64. C 65. B 66. D 67. A 68. B 69. B 70. B 71. A

48 Ratio & Proportion Ratio: If a and b are two quantities with the same units such that b is not equal to zero; then the quotient a/b is called the ratio between a and b. It is written as a:b. Ratio does not have any unit. The quantities a and b are called terms of the ratio. a' is called antecedent (first term) b' is called consequent (second term). Properties of Ratios: # If m = n = U, then A:B:C = p:q:r " # ] # If pa = qb = rc, then A:B: C = K " = K # = K ] # If m n = U o mpn then = UPo mnn UNo # If A:B = p: q and B:C = r: s Then, A: B: C = pr: qr: qs Proportion: Equality of two ratios is called proportion. If a : b = c : d, we say a, b, c, d are in proportion or a : b : : c : d. (1) Third proportion: For any two number a and b, third proportion (x) can be found ^ p = p O or x = p5 ^ (2) Fourth proportion: For any three numbers a, b and c, fourth proportion (x) can be found from ^ = q p q or x = p O ^ (3) Mean proportion: For any two numbers a and b, mean proportion (x) is found by ^ or x = ab O = O p Direct proportion:- Two values are called in direct proportion if one value changes accordingly with the second value.if one value increases, second also increases and if one decreases, second value automatically decreases. Examples of direct proportion:- 1. Time and work 2. Man and work 3. Number of articles and cost of an article 4. Work and efficiency 5. Speed and distance Let two quantities x and y are in directly proportion to each other Or, x y where is sign of proportionality.

49 Or, x = ky where k is a constant. Or, O Q = k Indirect or inverse proportion:- Two values are called in indirect proportion if one increases and second decreases automatically.and vice versa. Let x is inversely proportional to y then Or, x K Q Or, x = s Q Or, xy = k where k is a constant. Examples of indirect direction:- 1. Time and men 2. Men and time 3. Men and efficiency 4. Speed and time By using both concept of proportionality, we can say that Men Efficiency Time =Constant Work Solved examples Ex.1: The incomes of A and B are in the ratio 4 : 3 and their expenditures are in the ratio 3 : 2. If they save Rs.500 each, find the income of A? Solution: Let their incomes be Rs.4x and 3x respectively also their expenditures as 3y and 2y. Then 4x 3y = (1) & 3x 2y = (2) Solving equations (1) and (2) we get x = 500 and y = 500. the income of A is Rs Ex.2: A bag contains rupee, 50p, 25p coins in the ratio 5 : 3 : 2. If the total amount is Rs.70. Find the total number of coins? Solution: Let us suppose that number of coins are 5x, 3x and 2x respectively. Then total money will be 5x x K $ + 2x K i = 70 28x /4 = 70 x = 10 So, the total number of coins will be = (5x + 3x + 2x) = 10x = = 100

50 Ex.3: Two numbers are in the ratio 3 : 5, if 9 is added to each, then they are in the ratio 21 : 32. The second number is? Solution: Let the numbers be 3x and 5x. Then &OPj = $K or 32(3x + 9) = 21(5x + 9) %OPj &$ 9x = 99, x = 11 So the numbers are 33 and 55. Ex.4: The ratio of A s and B s salary is 9 : 4. If A s salary is increased by 15%, then his total salary becomes Rs What is the salary of B? Solution: Let the salaries of A and B be 9x and 4x So, 9x KK% KMM = Rs.5175 x = 500 B s salary is = Rs Ex.5: A can do a piece of work in 12 days; B is 60 % more efficient than A. Find the number of days that B takes to do the same piece of work? Solution: We know that time and efficiency are in inverse ratio so their product must be equal. Product of time taken by A and its efficiency = product of time taken by B and its efficiency Let A s efficiency is 100 so B s efficiency will be 160 K$ KMM Then, = y 160 or y = or y = 7.5 days KhM Ex.6: The monthly salaries of two persons are in the ratio of 4:7. If each receives an increase of Rs.25 in the salary, the ratio is altered to 3: 5. Find their respective salaries. Solution:Let the salaries be 4x and 7x Therefore, iop$% gop$% = & % 5(4x + 25) = 3(7x + 25) 20x = 21x + 75 x = 50 Therefore, their salaries are 4(50) & 7(50) i.e., 200 & 350 Ex.7: What is the value of 'x' if it is the mean proportional of x - 4 and x + 8? Solution:As x is the mean proportional of x - 4 and x + 8, we have x - 4 : x :: x : x + 8 Therefore, ONi = O O OPe or x 2 = (x-4) (x+8) or x 2 = x 2 + 4x 32, Therefore, x = 8 Ex.8: The ratio of A s and B s salary is 9 : 4. If A s salary is increased by 15%, then his total salary becomes Rs What is the salary of B? Solution: Let the salaries of A and B be 9x and 4x 9x KK% KMM = Rs.5175 x = 500 B s salary is = Rs.2000.

51 Ex.9. Which of the following proportions is correct if 12 = pq? a. 3:q : : 2:p b. 2: q : : 3:p c. 4: q :: p : 3 d. none of these Solution: 12 = pq or 4 3 = p q or 4:q = p:3 Ex.10. If it takes 3 painters 5 days to complete a job, how many days longer will it take if there are only 2 painters working? Solution: We know man is inversely proportional to time so men time = constant Let it will take x days when 2 painter are working Hence 3 5 = 2 x or x = 3 5/2 = 15/2 = 7.5 Ex.11. If 1 alpha = 2 betas and 1 beta = 3 gammas, how many alphas are equal to 36 gammas? Solution: If 1 beta = 3 gammas, Multiplying both sides by 12, we get 12 betas = 36 gammas We also know 2 betas = 1 alphas Hence 36 gammas = 12 betas = 6*2 betas = 6*1 alphas = 6alphas Hence 36 gammas = 6 alphas Ex.12. A quiz consists of true and false questions. The ratio of the number of true questions to the number of false questions is 4:3. About what percent of the questions are false? Solution: Percent of false questions = (3/7) 100 = = 43% approx. Ex.13. Jack is now 7 times as old as Jill. If 10 years from now, he will be 3 times as old as Jill, how old is Jack now? Solution: Let Jill be y years old. So, Jack is 7y years old now. Ten years from now, Jill will be (y + 10) and Jack will be (7y + 10). So, 7y + 10 = 3 (y + 10) = 3y Thus, 4y = 20 or y = 5. Jill is 5 years old and Jack is 35 years old now. Ex.14. Rs.432 is divided amongst three workers A, B and C such that 8 times A's share is equal to 12 times B's share which is equal to 6 times C's share. How much did A get? Solution: Given 8A = 12B =6C Dividing this by LCM of 8, 12 & 6 i.e. 24, we get 8A 12B 6C = = A B C = = or A:B:C = 3:2:4 So A s share = & 432 = 144 j

52 Ex.15. Two cogged wheels of which one has 32 cogs and other 54 cogs, work into each other. If the former turns 80 times in three quarters of a minute, how often does the other turn in 8 seconds? Solution: If 32 cogs turn 80 times in 45 second Then let 54 cogs turn x time in 8 second = 80 x x = 24 Ex.16. The ratio of A s and B s salary is 9 : 4. If A s salary is increased by 10%, then his total salary becomes Rs What is the salary of B? Solution: Let the salaries of A and B be 9x and 4x 9x 110/100 = Rs.4950 x = 500 B s salary is = Rs Ex.17: The ratio between larger and smaller numbers is twelve less than the product of these two numbers. The square of smaller number is half of the larger number. What is the total sum of these two numbers? Solution: Let the smaller and larger numbers be x and y respectively. According to the question, Q O + 12 = xy and x2 = Q $ Combining the above equations, we get x 3 x 6 = 0 or (x 2) (x2 + 2x + 3) = 0 Now x = 2 is the only positive value of x, whereas the other values are imaginary y = 8, Sum of the numbers = x + y = = 10. Problem on ages:- Problems on age are similar to those of Ratio and Proportion. However in certain questions, calculations can be shortened using the concepts of Singapore Math s. For example the age of the father 3 years ago was 7 times that of his son. At present the father's age is 5 times that of his son than. find the ages of father and son we proceed as follows: F: S 7:1 <-- 3 Years Ago 5:1 <-- Present Here if the ratios were the actual ages then Present ratio should be more than that of 3 years ago. Also the same years would be added to father's age and son's age. So what we do here is we make the ratio of present age to be more than that of 3 years ago and also make the difference of the ratio to be equal. To do that we find the difference of the ratios and cross multiply as:

53 F: S 7:1 <-- Difference is 6 5:1<--- Difference is 4 Thus the ratio of difference is 6:4 or 3:2, cross multiplying the upper ratio by 2 and lower by 3, we have: F: S 14:2 <-- three years ago 15:3 < --Present : Here we see that the difference is 1: 1 which is same and also the present age is more than that of 3 years ago. Now the calculation is very simple = 1 but we need difference is equal to 3 because the given age is 3 years ago So, we need to multiply both numerator and denominator by 3 to get the present ages of father and son Required answers, 15*3 = 45 years (father) and 3*3 = 9 years (son) The concept is very simple and gives the answer almost intuitively. Sometimes the questions are asked like this. The sum of ages of a mother and her daughter is 50 years. Also 5 years ago, the mother's age was 7 times the age of the daughter. What are the present ages of the mother and the daughter. Here 5 years ago, the sum of the ages will be 40 years. Divide 40 in the ratio of 7:1, Thus, 1/8 40 = 5 (daughter) 7/8 40 = 35 (mother) So mothers age today is = 40 years and daughter's age today is = 10 years. Shortcut Methods 1. If the age of A t years ago was n1 times the age of B, at present A s age is n2 times that of B then, A s present age is = B s present age is = 2. The present age of A is n1 times the present age of B. If t years hence, the age of A would be n2 times of B, then

54 A s present age = B s present age = 3. Age of A t years ago, was n1 times the age of B. if t years hence A s age would be n2 times of B then, A s present age = B s present age = 4. The sum of present ages of A and B is S years. If t years ago, the age of A was n times the age of B then, Present age of A = years Present age of B = 5. The sum of present ages of A and B is S years. If t years hence the age of A would be n times the age of B then, Present age of A = years Present age of B = 6. If the ratio of present ages of A and B is a:b and t years hence it will be c:d, then, A s present age is = B s present age is = If the difference is given, then Present age of A = Present age of B = years If age is given, then A = B = Ex. 1. The ratio between the present ages of M and N is 5: 3 respectively. The ratio between M s age 4 years ago an N s age after 4 years is 1: 1. What is the ratio between M s age after 4 years and N s age 4 years ago? [Bank PO 2000] Solution: M s age = 5x, N s age = 3x (5x -4) : (3x +4) = 1: 1 5x 4 = 3x + 4 2x =8 x =4

55 M s present age = 20 years, N s present age = 12 years. Required Ratio = 24: 8 = 3: 1 Ex. 2. Six years ago Jose was twice as old as Joseph. The ratio of their present age is 9:5 respectively, what is the difference between their present ages? [Bank PO 2000] Solution: Let their present ages be 9x and 5x. Six years ago, Difference between their present ages = Ex. 3. Sita s mother was 4 times of Sita s age ten years ago.after ten years, she will be twice as old as Sita. How old is Sita today? [SSC 2002 Solution: Take x as mother s present age, y as Sita s present age. Before 10 years (x - 10) = 4(y - 10) x - 10 = 4y - 40 x - 4y = (1) After 10 years (x + 10) = 2(y + 10) x + 10 = 2y + 20 x - 2y = (2) By solving (1) & (2) -2y = -40 y = 20 x = 50 Sita s age is 20 years. Ex. 4. A said to B I am twice as old as you were when I was as old you are now. Sum of their ages is 42. Find their present ages. [Delhi police, 2007] Solution: Clearly by choosing option (b) we see that A was equal to B s age, i.e. 18 years, 6 years ago and at that time B was 12 years old. So, A s present age, 24 years, is twice of B s that time of age, i.e. 12years. (Or) Let Present age of B=x and present age of A=42-x Then, 18 years ago A s age=x and B s age=x-(42+2x) =3x-42 By given condition, A s present age=2 (B s age some years ago) 42 x = 2(3x - 42) x = 18 B s present age = 18 years and A s present age = 24years. Ex. 5. The average age of 30 boys in a class is 15 years. One boy, aged 20 years, left the class, but two new boys came in his place whose ages differ by 5 years. If the average age of all the boys now in the class remains 15 years, the age of the younger newcomer is: [SSC Grad. 2002]

56 Solution: Let the ages of the two new boys be x and x +5. By given condition, Ex. 6. My grandfather was 8 times older than me 16 years ago. He would be 3 times my age, 8 years from now. Eight years ago, what was the ratio of my age to that of my grandfather? [SSC Grad. 2003] Solution: Let my age be x years. Then 16 years ago, my age = (x -16) years My grandfather s age = 8(x-16) years 8 years from now, my age = (x + 8) years My grandfather s age = 3 (x +8) years Also, 3(x -8)-8(x-16)= 24 5x = 128 Grandfather s present age = 8(x - 16) + 16 = 92.8 The ratio of their ages 8 years ago, Ex. 7. The present age of Mr. Sanyal is 3 times the age of his son. Six years hence the ratio of their ages will be 5:2 respectively. What is the present age of Mr. Sanyal? [Bank 2006] Solution: Let present age of son = x years Present age of Mr. Sanyal = 3x years 6x + 12 = 5x x 5x = x = 18 years Present age of Mr. Sanyal = 3 18 = 54 years Ex. 8. Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand's present age in years? Solution: Let the present ages of sameer and anand be 5x years and 4x years respectively. Then Anands's present age 5x+3/4x+3=11/9 9(5x+3)=11(4x+3) x=6. = 4x =24 years.

57 Ex. 9. The ratio between the present ages of P and Q is 5 : 7 respectively. If the difference between Q's present age and P's age afler 6 years is 2, what is the total of P's and Q's present ages? Solution: Let the present ages of P and Q be 5x years and 7x years respectively. Then 7x-(5x+6)=2 2x=8 x=4. Required sum = 5x+7x =12x =12 4 =48 years. Ex. 10. My brother is 3 years elder to me. My father was 28 years of age when my sister was born while my mother was 26 years of age when I was bom. If my sister was 4 years of age when my brother was born, then, what was the age of my father and mother respectively when my brother was born? Solution: Clearly,my mother was born 3 years before I was born and 4 years after my sister was born. So, father's age when =(28+4) brother was born = 32 years. mother's age when brother was born =(26-3) = 23 years. Ex. 11. Eighteen years ago, a father was three times as old as his son. Now the father is only twice as old as his son. Then the sum of the present ages of the son and the father is Solution: Let the present ages of Father and Son be 2x years and x years respectively. Then (2x-18)=3(x-18) 2x-18=3x-54 x=54-18 x=36. Required sum = (2x+x) =3x =3x36 =108 years. Ex. 12. The sum of the ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?

58 Solution: Let the ages of the children be x,(x+3),(x+6),(x+9) and (x+12) years. Then, x+(x+3)+(x+6)+(x+9)+(x+12)=50 5x=20 x=4. Age of the youngest child =4 years. Ex. 13. Ayesha's father was 38 years of age when she was born while her mother was 36 years old when her brother four years younger to her was born. What is the difference between the ages of her parents? Solution: Mother's age when Ayesha's brother was = 36 years. born Father's age when Ayesha's brother was born = (38+4) years. = 42 years. Required difference = (42-36) = 6 years. Ex. 14. Sneh's ago is 1/6 th of her father's age. Sneh's father's age will be twice of Vimal's age after 10 years. If Vimal's eighth birthday was celebrated two years before, then what is Sneh's present age? Solution: Vimal's age after 10 years = (8+2+10)years = 20 years. Sneh's father age after 10 years = 40 years Sneh's father Present age = 30 years Therefore Sneh's age = (1/6 x 30)years = 5 years. Ex. 15. A father said to his son, "I was as old as you are at the present at the time of your birth". If the father's age is 38 years now, the son's age five years back was: Solution: Let the son's present age be x years. Then, (38 - x) = x 2x = 38. x = 19. Son's age 5 years back (19-5) = 14 years. Ex. 16. Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand's present age in years? Solution: Let the present ages of Sameer and Anand be 5x years and 4x years respectively. Then, 5x + 3 = 11 4x + 3 9

59 9(5x + 3) = 11(4x + 3) 45x + 27 = 44x x - 44x = x = 6. Anand's present age = 4x = 24 years. Ex. 17. Six years ago, the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar's age at present? Solution: Let the ages of Kunal and Sagar 6 years ago be 6x and 5x years respectively. Then, (6x + 6) + 4 = 11 (5x + 6) (6x + 10) = 11(5x + 10) 5x = 10 x = 2. Sagar's present age = (5x + 6) = 16 years. Partnership:- When two or more persons agree to run a business jointly, they are called partners and this deal is called partnership. Simple Partnership When all the partners invest their money for the same time, it is called simple partnership. Compound Partnership When all the partners invest their money for the different times, it is called compound partnership. 1. If the capitals of three partners be Rs. C1, Rs. C2 and Rs. C3 for the same period and the total profit be Rs. P, then shares of the partners in the profits are, and 2. If the capitals of three partners be Rs. C1, C2 and RS. C3 for the periods t1, t2 and t3 respectively and the total profit be Rs.P, then shares of the partners in the profits are:, and 3. If in a partnership the investments made by first, second and third partners are x1, x2, x3 respectively, the time period be t1, t2, t3 then ratio of profits is given by x1t1 : x2t2 : x3t3. 4. If is the ratio of investments and p1:p2:p3 be the ratio of profits, the ratio of time periods is given by

60 5. If p1 : p2 : p3 is the ratio of profits and t1 : t2 : t3 be the ratio of time periods, the ratio of investments is given by Ex. 1. Mr. Burkit started a business investing Rs After 3 months Peter joined him with a capital of Rs After another 6 months, Ronald joined them with a capital of Rs At the end of the year, they made a profit of Rs Find the ratio of their capitals? [Bank Po, 2007] Solution: Burkit invested his capital for 12 months, Peter for 9 months and Ronald for 3 months. So, ratio of their capitals = ( ) : ( ) : ( ) = : : = 2 : 2 : 1 ( Total investment = Investment time) Ratio of their capitals = 2 : 2 : 1 Ex. 2. Mr. Sanjay started a business investing Rs in In 1997 he invested an additional amount of Rs. 10,000 and Mr. Suresh joined him with an amount of Rs In 1998 Mr. Sanjay invested another additional amount of Rs. 10,000 and Mr. Ram joined them with an amount of Rs. 35,000. What will be Suresh s share in the profit of Rs earned at the end of three years from the start of the business in 1966? [Bank PO 2001] Solution: Sanjay Suresh Ram , ,000 35, ,000 35,000 35,000 Suresh s share = $ 1,50,000 = Rs. 50,000 h Ex. 3. The ratio of investments of two partners A and B is 11:12 and the ratio of their profit is 2:3.If A invested the money for 8 months, then the time for which B invested the money is: [SSC Grad 2005] Solution: Let A s capital be Rs. 11x and B s capital be Rs. 12x. Ex. 4. Rs.1290 is divided between A, B, and C. So that A s share is 1 K $ time C s. What is C s share? [CDS 2004] 1 & i Solution: times B s and B s share is

61 C s share Ex. 5. A, B and C invest Rs.4000, Rs.5000 and Rs.6000 respectively in a business and A gets 25% of profit for managing the business, the rest of the profit is divided by A,B and C in proportion to their investment. If in a year, A gets Rs.200 less than B and C together, what was the total profit of that year? [RRB, 2000] Solution: After giving 25% of the total profit amount to A for managing the business, the rest 75% of total profit is divided among A, B, and C in proportion to their investments. In 75% of total profit 75% of total profit = 4x+5x+6x Total profit = K%O g%% = 20x Share of A = 4x + 25% of 20x = 9x Share of B = 5x = 5x Share of C = 6x = 6x Given, (5x + 6x) - 9x = 200 x = 100 Total profit = 20x = = Rs.2000 Ex. 6. In a partnership, A invests K of the capital for K of the time, B invests K of the capital for h h & K of the time and C, the rest of the capital for whole time. Find A s share of the total profit of & Rs.2300 [Asst. Grade 2003] Solution: Capital of C = Let the total time be 12 months Share of Ex. 7. Sumit and Ravi started a business by investing Rs and respectively. In what ratio the profit earned after 2 years be divided between Sumit and Ravi respectively. Solution: P:Q = 85000:15000 = 17:3 Important to note there that if both have invested for different period of times then we had to

62 multiply with number of months to get the desired ratio. Ex. 8. A and B invest in a business in the ratio 3 : 2. If 5% of the total profit goes to charity and A's share is Rs. 855, the total profit is: Solution: Let the total profit be Rs After paying to charity, A's share = Rs. 95 x 3 = Rs If A's share is Rs. 57, total profit = Rs If A's share Rs. 855, total profit = 100 x 855 = Ex. 9. A, B and C jointly thought of engaging themselves in a business venture. It was agreed that A would invest Rs for 6 months, B, Rs for 5 months and C, Rs. 10,000 for 3 months. A wants to be the working member for which, he was to receive 5% of the profits. The profit earned was Rs Calculate the share of B in the profit. Solution: For managing, A received = 5% of Rs = Rs Balance = Rs. ( ) = Rs Ratio of their investments = (6500 x 6) : (8400 x 5) : (10000 x 3) = : : = 13 : 14 : 10 B's share = Rs x 14 = Rs Ex. 10. A and B started a partnership business investing some amount in the ratio of 3 : 5. C joined then after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C? Solution: Let the initial investments of A and B be 3x and 5x. A : B : C = (3x x 12) : (5x x 12) : (5x x 6) = 36 : 60 : 30 = 6 : 10 : 5. Ex. 11. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B's share in total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business? Solution: A : B : C = (20,000 x 24) : (15,000 x 24) : (20,000 x 18) = 4 : 3 : 3. B's share = Rs x 3 = Rs. 7, Ex. 12. Aman started a business investing Rs. 70,000. Rakhi joined him after six months with an amount of Rs.. 1,05,000 and Sagar joined them with Rs. 1.4 lakhs after another six months. The amount of profit earned should be distributed in what ratio among Aman, Rakhi and Sagar respectively, 3 years after Aman started the business? Solution: Aman : Rakhi : Sagar = (70,000 x 36) : (1,05,000 x 30) : (1,40,000 x 24) = 12 : 15 : 16.

63 Exercise 1. If A: B=3:4 and B: C=8:9, then find the value of A: B: C a. 3:4:9 b. 3:8:9 c. 6:8:9 d. 2:4:7 e. None of these 2. If a: b=b:c then ratio a 4 :b 4 =? a. a 3 :c 2 b. a 4 :c 2 c. a 4 :c 4 d. a 2 :c 2 e. None of these 3. If ^ = p = q then find a: b: c g j KK a. 3:4:9 b. 3:8:9 c. 6:8:9 d. 7:9:11 e. None of these 4. If A = K B and B = K C, then find the value of A: B: C. i $ a. 3:4:9 b. 1:4:8 c. 6:8:9 d. 2:4:7 e. None of these 5. If (a+b) : (a-b)=5:3, then (a 2 +b 2 ) : (a 2 -b 2 )=? a. 13:15 b. 15:13 c. 17:15 d. 15:17 e. None of these 6. The total number of students in a school is If the number of girls in the school is 1200, what is the ratio of the total number of boys to the total number of girls in the school? a. 47:60 b. 15:13 c. 60:47 d. 15:17 e. None of these 7. The average age of a person and his son is 30 yr. The ratio of their ages is 4:1 respectively. Find the age of son a. 20yr b. 16yr c. 14yr d. 12yr e. None of these 8. Rahul obtained 12 marks more than that of Tanvi. If the ratio of their marks is 3:4, find the sum of their marks a. 96 b. 84 c. 7 d. 12 e. None of these 9. One-half of a certain number is equal to 65% of the 2nd number. Find the ratio of 1st to 2nd number a. 13:10 b. 15:13 c. 10:13 d. 15:17 e. None of these 10. The total weight of Sanjay and Suresh is 120 kg. If Sanjay weights 30 kg more than Suresh, what is the ratio of the weights of Suresh to that of Sanjay? a. 3:5 b. 5:3 c. 7:5 d. 5:7 e. None of these 11. Present age of Meena is 8 times of her daughter s age. 8 yr hence, the ratio of the ages of Meena and her daughter will be 10:3. Find the present age of Meena a. 20yr b. 32yr c. 14yr d. 28yr e. None of these 12. Amit and Suresh have invested in the ratio of 4:7. If both invested a total amount of Rs , then find the investment of Suresh a b c d e. None of these

64 Ans: Rs The present ratio of ages of A and B is 4:5. 18 yr ago, this ratio was 11:16. Find the sum of the present ages of A and B a. 90yr b. 75yr c. 60yr d. 54yr e. None of these 14. The ratio of the ages of a father to that of his son is 5:2. If the product of their ages in yr is 1000, then find the father s age after 10 yr a. 90yr b. 75yr c. 60yr d. 54yr e. None of these 15. If 70% of a number is equal to three-fifth of another number, what is the ratio between the 1st and the 2nd number, respectively? a. 6:7 b. 5:3 c. 7:5 d. 5:7 e. None of these 16. In a class, the number of boys and girls is in the ratio of 4:5. If 10 more boys join the class, the ratio of numbers of boys and girls becomes 6:5. How many girls are there in the class? a. 20 b. 25 c. 35 d. 45 e. None of these 17. The sum of the ages of a father and his son is 100 yr now. 5 yr ago, their ages were in the ratio of 2:1. Find the ratio of the ages of father and son after 10 yr a. 6:7 b. 5:3 c. 7:5 d. 5:7 e. None of these 18. The total number of students in a school is If the number of boys in the school is 4545, what will be the ratio of the total number of girls in the school? a. 303:275 b. 275:303 c. 7:5 d. 5:7 e. None of these 19. The average age of a woman and her daughter is 16 yr. The ratio of their ages is 7:1. What is the woman s age? a. 28yr b. 35yr c. 21yr d. 14yr e. None of these 20. The ages of Vaibhav and Jagat are in the ratio of 12:7. After 6 yr, the ratio of their ages will be 3:2. What is the difference in their ages? a. 10yr b. 25yr c. 30yr d. 49yr e. None of these 21. The average age of a man and his two twin sons born on the same day is 30 yr. The ratio of the ages of father to one of his sons is 5:2. What is the father s age? a. 90yr b. 75yr c. 60yr d. 50yr e. None of these 22. A sum of money is divided amongst A, B, C and D in the ratio of 3:7:9:13. If the share of B is Rs. 4872, what will be the total amount of money of A and C together? a b c d e. None of these 23. The speeds of three cars are in the ratio of 2:3:4. Find the ratio between the time taken by these cars to cover the same distance a. 6:4:3 b. 3:8:9 c. 6:8:9 d. 2:4:7 e. None of these

65 24. The ratio of angles of a triangle is 3:2:1 and the largest angle is 90. Find the difference between the largest and smallest angles a. 60 b. 90 c. 45 d. 75 e. None of these 25. The ratio between the present ages of Punit and Suresh is 4:5 respectively. After 7 yr Punit s age will be 31 yr. What was Suresh s age four years ago? a. 26yr b. 24yr c. 18yr d. 16yr e. None of these 26. In a town, 80% of the populations are adults of which the men and women are in the ratio of 9:7, respectively. If the number of adult women is 4:2 lakhs, what is the total population of the town? a. 24yr b. 36yr c. 48yr d. 12yr e. None of these 27. Mr. Shrimant inherits 2505 gold coins and divides them amongst his three sons. Bharat, Parat and Marat, in a certain ratio out of the total coins received by each of them, Bharat Sells 30 coins, Parat donates his 30 coins and Marat losses his 25 coins. Now, the ratio of gold coins with them is 46:41:34 respectively. How many coins did Parat receive from his father? a. 800 b. 600 c. 850 d. 900 e. None of these 28. Rs. 600 are divided among A, B and C such that Rs 40 more than $ th of A s share, Rs 20 more % than $ j th of B s share and Rs 10 more than th of C s share, all are equal. What is B s share? g Kg a. 250 b. 300 c. 270 d. 280 e. None of these 29. Perimeter of a rectangle is 40 cm and the length and the breadth are in the ratio of 3:2 respectively. What is the area of rectangle is sq cm? a. 84 b. 75 c. 96 d. 48 e. None of these 30. The expensive on rice, fish and oil of a family are in the ratio of 12:17:3. The prices of these articles are increased by 20%, 30% and 50%, respectively. Find the total percentage increase in the expenses of the family a. 28 K % b. 28 K % c. 28 % % d. 28 & % e. None of these e j e e 31. Salary of Mr. X is 80% of the salary of Mr. Y and the salary of Mr. Z is 120% of the salary of Mr. X. What is the ratio between the salaries of X,Y,Z respectively? a. 33:14:29 b. 30:28:19 c. 16:38:19 d. 20:25:24 e. None of these 32. If the positions of the digits of a two-digit number are interchanged, the number newly formed is smaller than the original number by 45. Also, the ratio of the new number to the original number is 3:8. What is the original number? a. 72 b. 75 c. 96 d. 48 e. None of these 33. The ratio of water and spirit in the mixture is 1:3. If the volume of the solution is increased by 25% by adding spirit only, what is the resultant ratio of water and spirit? a. 1:3 b. 4:1 c. 1:4 d. 5:7 e. None of these

66 34. A cat takes 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speeds of the cat to that to the dog? a. 15:34 b. 14:15 c. 15:16 d. 15:17 e. None of these 35. In the month of January, Arun s income and expenses were Rs and Rs 9000 respectively. His monthly expenses vary directly as the square of his monthly income. What is his income when it just equals his expenses? a b c d e. None of these 36. Two vessels P and Q contain milk and water mixed in the ratios of 5:3 and 2:3, respectively. When these mixtures are mixed to form a new mixture containing half milk and half water, then find the ratio in which P and Q are mixed? a. 1:3 b. 4:5 c. 1:4 d. 5:7 e. None of these 37. Rs 5625 are divided amongst A, B and C so that A receives K as much as B and C together $ receive and B receive K as much as A and C together receives. Find the sum of shares of A and B i a b c d e. None of these 38. Rs 710 were dividing amongst A, B and C in such a way that A had Rs 40 more than B and C had Rs 30 more than A. How much was C s share? a. 200 b. 250 c. 270 d. 300 e. None of these 39. The ratio of 1st and 2nd classes train fairs between two stations is 3:1 and that of the number of passengers travelling between these stations by 1st and 2nd classes is 1:50. If on a particular day, Rs 2650 be collected from the passengers travelling between these stations, then find the amount collected from 2nd class passengers a b c d e. None of these 40. The ratio of the number of students studying in schools A, B and C is 5:6:8. If the number of students in each of the schools is increased by 30%, 25% and 25%, respectively, what will be the new ratio of the numbers of students in schools A, B and C? a. 13:14:29 b. 13:15:20 c. 16:38:19 d. 20:25:24 e. None of these 41. In a college, the number of students studying History, Economics and Mathematics are in the ratio of 4:7:9. If the number of students a History, Economics and Mathematics be increased by 30%, 20% and 40% respectively, what will be the new ratio? a. 13:14:29 b. 13:15:20 c. 16:38:19 d. 26:42:63 e. None of these 42. Ruchi s father was 38 yr old when she was born while her mother was 36 yr old when her brother was born who is 4 yr younger to her. What is the difference between the ages of her parents? a. 4yr b. 6yr c. 8yr d. 2yr e. None of these

67 43. The difference between the two adjacent angles of a parallelogram is 20. What would be the ratio between the smaller and the larger angles of the parallelogram respectively? a. 1:3 b. 4:5 c. 1:4 d. 5:7 e. None of these 44. There are two triangles A and B. The angles of triangle A are in the ratio of 3:4:5 and the angles of triangle B are in the ratio of 5:6:7. What is the difference between largest angle of triangle A and the smallest angle of triangle B? a. 25 b. 20 c. 40 d. 15 e. None of these 45. The capacities of two glasses are same. They are filled with $ rd part and $ th part of water, & % respectively. Remaining part of both the glassed are filled with milk. If mixtures of both the glasses are mixed in a big glass, what is the ratio of water and milk in it? a. 8:7 b. 4:5 c. 1:4 d. 5:7 e. None of these 46. Rs are distributed amongst A, B and C. If money given to them is decreased by Rs 26, Rs 28 and Rs 32 respectively, then they have money in the ratio of 9:13:8. What is the amount given to A? a. 566 b. 665 c. 656 d. 156 e. None of these 47. A person gave $ th part of his income to his elder son and 30% part to his younger son. He % saved his remaining money in three trusts A, B and C in the ratio of 3:5:2. If difference between the amounts got by his both sons is Rs How much amount he saved in trust C? a b. 800 c d e. None of these 48. In a factory, the ratio of the numbers of employees of three types A, B and C is 9:13:18 and their wages are in the ratio of 10:7:4. If number of employees of type C is 54 and wages of every employees of type B is Rs 1400, then find the total wages of all the employees a b c d e. None of these 49. If (a + b) : (b + c) : (c + a) = 6 : 7 : 8 and a + b + c = 14. Find C a. 8 b. 6 c. 4 d. 2 e. None of these 50. Number of employees in a factory decreases in the ratio of 8:7 and salary of employees increases in the ratio of 5:6. Find weather the total salary given to the employees is increased or decreased and in what ratio? a. 21:20 deacreased b. 20:21 deacreased c. 20:21 deacreased d. 20:21 increased e. None of these 51. If ^ = p = q ^PpPq, then find the value of & % g p a. 3 b. 6 c. 4 d. 2 e. None of these 52. A profit of Rs. 7,200 is to be distributed among A, B and C in the ratio of 3:2:1. The difference between the amounts received by A and C is a b c d e. None of these

68 53. The ratio of the prices of three different types of cars is 4:5:7. If the difference types the prices of costliest and the cheapest car is Rs. 6000, what is the price of the car of moderate price? a. 1,00,000 b. 1,20,000 c. 2,40,000 d. 1,50,000 e. None of these 54. A certain amount was divided between Salim and Rahim in the ratio of 4:3. If Rahim s share was Rs , find out the amount a b c d e. None of these 55. karan got twice as many marks in English as in Science. His Total marks in English, Science and Maths are 180. If the proportion of his marks in English and Maths is 2:3, what are his marks is science? a. 30 b. 60 c. 40 d. 20 e. None of these 56. An amount is to be divided among a, B and C in the ratio 3:5:6 respectively. If the difference between A and B s share is Rs What is C s share? a b c d e. None of these 57. A s share in a business is & th of B s share. B got a profile of Rs. 1,200 by investing Rs. 40,000 i in the business. What will be the ratio of A s profit to his investment in the business? a. 18:23 b. 9:400 c. 400:9 d. 23:18 e. None of these 58. Rs is distributed among A, B and C in the ratio of 6:5:4. The difference between shares of A and C is a. 600 b. 640 c. 300 d. 320 e. None of these 59. The price of a scooter and a moped are in the ratio of 7:4. If a scooter costs Rs more than a moped, what is the price of a moped? a b c d e. None of these 60. A started with Rs and B joined afterwards with Rs If the profits at the end of one year were divided in the ratio 2:1 respectively, then B would be a. 3 months b. 6 months c. 4 months d. 2 months e. None of these 61. Two numbers are in the ratio 5:9 with the addition of 9 their ratio becomes 16:27. Then sum of the two numbers would be a. 144 b. 99 c. 55 d. 199 e. None of these 62. The price of a scooter and a T.V set are in the ratio of 3:2. If the scooter costs Rs. 6,000 more than the T.V. set, what is the price of the T.V. set? a b c d e. None of these 63. An amount of money is to be distributed among A, B and C in the ratio of 2:5:7. If the total of A s and B s share is Rs what is the difference between B s and C s share? a b c d e. None of these

69 64. An amount is to be divided among A, B and C in the ratio 3:5:6 respectively. If the difference between A and B s share is Rs What is C s share? a b c d e. None of these 65. Find out the ratio whose value is 2/3 and the antecedent is 18 a. 18:27 b. 9:400 c. 400:9 d. 23:18 e. None of these 66. $ : % e : What number? & h g a. KK g b. KM & c. KM g d. K$ g e. None of these 67. A : B = 2 : 3, B : C = 4 : 5, C : D = 6 : 7 find the ratio of A and D? a. 16:27 b. 16:35 c. 35:16 d. 23:18 e. None of these 68. Two equal glasses are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a tumbler. Find the ratio of milk and water in the tumbler a. 18:27 b. 9:400 c. 7:17 d. 23:18 e. None of these 69. If in 30 litre mixture of milk and water the ratio of milk and water is 7:3, find out the quantity of water to mix in order to make this ratio 3:7 a. 60 litres b. 15 litres c. 10 litres d. 40 litres e. None of these 70. In a big there are coins of 25 paisa, 10 paisa and 5 paisa in the ratio 1:2:3. If there are in all Rs. 30, how many 5 paisa coins are there? a. 140 b. 150 c. 120 d. 240 e. None of these 71. Shyam has a sister who is half of his age. When Shyam double, what will be the ratio of his age to his sister s age? a. 4/3 b. 5/3 c. 3/5 d. 3/4 e. None of these 72. On a certain railway the first and second class fares are 7 paisa and e paisa per kilometer respectively. A man who travelled 100 km spent Rs in going part of distance by first class and the in second class. How many kilometers did he travel in first class? a. 14 b. 15 c. 12 d. 10 e. None of these 73. Ratio of A s age to B s first equal to 4:3. A will be 26 years old after 6 years, how old is B now? a. 14 yrs b. 15 yrs c. 12 yrs d. 10 yrs e. None of these men with 10 boys can do in 6 days as much work as 21 men with 30 boys can do in 5 days. How many boys must help 40 men to do the same work in 4 days? a. 14 b. 15 c. 12 d. 10 e. None of these 75. A contractor took a contract for building 12 kilometer road in 15 days and employed 100 laboures on the work. After 9 days he found that only 5 kilometer road had been constructed.

70 How many more laboures should be employed to ensure that the work may be completed within the given time? a. 144 b. 105 c. 120 d. 110 e. None of these 76. What numbers has 5 to 1 ratio to the number 10? a. 40 b. 50 c. 20 d. 10 e. None of these men do a work in 20 days. In how many days will 20 men do the full work? a. 14 b. 15 c. 12 d. 10 e. None of these 78. Three persons start business and make profit of Rs If their capitals are as K % : K h : K e how should the profit be divided? a. 480,400,350 b. 480,450,350 c. 480,400,300 d. 450,400,350 e. None of these 79. A, B and C purchased the mangoes in a ratio 5:3:2. If the difference of mangoes of A and C is 60, find out the total number of mangoes purchased by them a. 400 b. 500 c. 200 d. 100 e. None of these 80. A company makes a profit of Rs Out of this 20% is paid for taxes and the rest be divided among its partners A, B and C in proportion of 1 : 1 K : 2 Find the share of each $ a. 480,400,350 b. 80,120,160 c. 480,400,300 d. 450,400,350 e. None of these 81. A garrison of 2,200 men has provision for 16 weeks at the rate of 45 gm per day per man. How many men must leave so that the same provision may last for 24 weeks at 33 gm per day per man? a. 400 b. 500 c. 200 d. 100 e. None of these 82. If 5 men can do a piece of work in 20 days, in how many days will 10 men and 5 boys do the same work if 1 man does as much work as 2 boys? a. 4 b. 5 c. 2 d. 8 e. None of these 83. In 21 cows eat as much as 15 oxen, how many cows will eat as much as 25 oxen? a. 34 b. 35 c. 32 d. 30 e. None of these 84. There are three containers of equal capacity. First container is half full, the second is one third full and the third is empty. If all the water in the containers is divided equally among the containers, what part of the third container will be full? a. 1/4 b. 5/8 c. 5/18 d. 18/5 e. None of these 85. The annual incomes of two persons are in the ratio 9:7 and their expenses are in the ratio 4:3. If each of them saves Rs per year, then their annual incomes are a and b and c and 15000

71 d and e. None of these 86. A person divider Rs. 10,800 among his 3 sons in the ratio 3:4:5. Second son kept Rs for himself, gave Rs. 600 to his wife, and divided the remaining money among his 2 daughters received a b c d e. None of these 87. In a school the number of boys and that of the girls are in the respective ratio of 2:3. If the number of boys is increased by 20% and that of girls is increased by 10%, what will be the new ratio of number of boys to that of the girls? a. 8:11 b. 9:400 c. 400:9 d. 23:18 e. None of these 88. Income of two companies A and B are in the ratio of 5:8. Had the income of company A been more by Rs. 25 lakhs, the ratio of their incomes would have been 5:4 respectively. What is the income of company B? a. 34 lakhs b. 35 lakhs c. 40 lakhs d. 34 lakhs e. None of these 89. A total of 91 boys are seated in three rows. The ratio between the number of boys seated in the first and the second row is 5:2 respectively and the ratio between the number of boys seated in the second and third row is 1:3 respectively. How many boys were there in the second row? a. 14 b. 15 c. 12 d. 10 e. None of these 90. The ratio of the income of A and B as also those of B and C is 3:2 If a third of A s income exceeds a fourth of C s income by Rs. 1000, What is B s income? a b c d e. None of these 91. A sum of money is divided among 160 males and some females in the ratio 16:21. Individually each male gets Rs. 4 and each female Rs. 3. The number of females is a. 280 b. 150 c. 250 d. 270 e. None of these 92. The monthly incomes of two persons are in the ratio 4:7, and their expenses are in the ratio 11:20. If each of them saves Rs.400 per month, then their monthly income must be respectively a and 4200 b and 8400 c and d and e. None of these 93. Divide Rs among A, B and C so that A gets 2/9 of B s share and C gets 3/4 of A s share Find A s, B s and C s share a. Rs. 200, Rs. 950, Rs. 150 b. Rs. 200, Rs. 900, Rs. 150 c. Rs. 250, Rs. 950, Rs. 150 d. Rs. 200, Rs. 950, Rs. 100 e. None of these

72 94. Rs. 180 contained in a box consists of one-rupee, 50-paisa and 25 paisa coins in the ratio 2:3:4. What is the number of 50-paise coins? a. 280 b. 150 c. 250 d. 120 e. None of these 95. The students in three classes are in the ratio 2:3:5. If 20 students are increased in each class, the ratio changes to 4:5:7. The total numbers of students before the increase were a. 100 b. 150 c. 250 d. 270 e. None of these 96. If 378 coins of one rupee, 50-paisa and 25-paisa coins whose values are in the ratio of 13:11:7, the number of 50-paisa coins will be a. 280 b. 150 c. 132 d. 270 e. None of these 97. The ratio of the number of boys to that of girls in a school is 4:1. If 75% of boys and 70% of the girls are scholarship-holders, then the percentage of students who do not get scholarship is: a. 28 b. 26 c. 25 d. 27 e. None of these 98. A total profit of Rs. 3,600 is to be distributed amongst A, B and C such that A:B::5:4 and B:C::8:9. What is the share of C? a b c d e. None of these 99. The ratio of number boys & girls in a school of 720 students is 7:5. How many more girls should be admitted to make the ratio 1:1? a. 100 b. 150 c. 120 d. 270 e. None of these 100. Annual incomes of A & B are as 4:3 & their expenses are as 3:2. If each saves Rs. 600 per annum, then annual income of A is a b c d e. None of these Ans: 2, A man s coat costs seven times as much his shirt, but the bill for the two together was Rs The price of the shirt is a. 28 b. 52 c. 25 d. 26 e. None of these 102. The speeds of three cars in the ratio 2:3:4. The ratio between the times taken by three cars to travel the same distance is a. 3:4:6 b. 4:3:6 c. 6:4:3 d. 4:6:3 e. None of these 103. Four numbers in the ratio 1:3:4:7 add up to give a sum of 105. Find the value of the biggest number a. 28 b. 49 c. 30 d. 14 e. None of these 104. If the ratio of the ages of Maya and Chhaya is 6:5 at present, and 15 years from now, the ratio will get changed to 9:8, then find Maya s present age a. 28 yrs b. 49 yrs c. 30 yrs d. 14 yrs e. None of these

73 105. If Rs. 58 is divided among 150 children such that each girl and each boy gets 25p and 50p respectively. Then how many girls are there? a. 28 b. 68 c. 60 d. 62 e. None of these 106. If 391 bananas were distributed among three monkeys in the ratio 1/2:2/3:3/4, how many bananas did the first monkey get? a. 128 b. 149 c. 103 d. 104 e. None of these 107. If two numbers are in the ratio of 5:8 and if 9 be added to each, the ratio becomes 8:11. Find the lower number a. 15 b. 49 c. 30 d. 14 e. None of these 108. What number must be taken from each term of the fraction $g that it become 2:3? a. 28 b. 11 c. 30 d. 14 e. None of these 109. Twenty litres of a mixture contains milk and water in the ratio of 5:3. If four litres of this mixture is replaced by four litres of milk, then the ratio of milk to water in the new mixture will be a. 3:7 b. 3:5 c. 7:3 d. 5:3 e. None of these 110. The ratio of Vimal s age and Aruna s age is 3:5 and the sum of their ages is 80 years. The ratio of their ages after 10 years will be a. 1:2 b. 3:5 c. 7:3 d. 5:3 e. None of these 111. The average age of 12 students is 20 years. If the age of one more student is included, the average decreases by 1, what is the age of the new student? a. 5 b. 9 c. 6 d. 7 e. None of these 112. Pushpa is twice as old as Rita was 2 years ago. If the difference of their ages be 2 years, how old is Pushpa today? a. 10 b. 9 c. 6 d. 7 e. None of these 113. Jayesh is as much younger to Anil as he is older to Prashant. If the sum of the ages of Anil and Prasanth is 48 years, what is the age of Jayesh age? a. 25 b. 29 c. 26 d. 27 e. None of these 114. The ratio between the present ages of M and N is 5:3 respectively. The ratio between M s age 4 years ago and N s age after 4 years is 1:1. What is the ratio between M s age after 4 years and N s age 4 years ago? a. 1:2 b. 3:5 c. 3:1 d. 5:3 e. None of these 115. When the son will be as old as the father today, their ages will add up to 136 years. When the father was as old as the son is today, their ages added upto 40 years. Find the present ages of the son and father in years? a. 32,57 b. 32,56 c. 15,35 d. 20,45 e. None of these &%

74 116. In a Cricket Eleven, the average age of eleven players is 28 years. Out of these, the average ages of three groups of three players each are 25 years, 28 years and 30 years respectively. If in these groups, the captain and the youngest player are not included, and the captain is eleven years older than the youngest player, what is the age of the captain? a. 25 b. 35 c. 26 d. 27 e. None of these 117. Average age of the students in a class of 50 is 13. The weight of each is directly proportional to the height of each and the height is also found to be in direct proportion with the age. A student of age 11 has 165 cms of height, with weight being 33 kgs. Find the average weight of the class a. 25 b. 29 c. 39 d. 27 e. None of these Directions for (Q. No 118 & 119): Refer to the data below and answer the questions that follow: The average age of A and B is 20 years. If C is to replace A, the average age would be 19 and if he were to replace B, the average would be 21. Also, D is half the age of C 118. Average age of B and D is a. 25 b. 14 c. 26 d. 15 e. None of these 119. Average age of A, B, C and D is a b c d e. None of these friends A, B, C, D, E and F whose ages are in integers have following relation between their ages. B s age is % j times that of A, whereas age of C and D is times that of B. F s age are h KM between that of A and B and E is & times old as B and 3 years older to A. What is the average $ weight of all the 6 friends? a. 13 b. 10 c. 12 d. 15 e. None of these 121. The sum of the ages of the four members of the Rathore family is 140 years. Five years ago the ages of the four members Inder, Sunder, Mrs. Rathore and Mr. Rathore were in the ratio of 2:3:7:8. After how many years would Sunder be as old as his mother is today? a. 25 b. 24 c. 26 d. 27 e. None of these 122. Many years ago when Mr. Waugh was asked as to who among Steve and Mark is elder, he said, In two years from now Steve will be twice as old as he was three years ago and in three years from now Mark will be three times as old as he was three years ago. How old is Mark today, if Mr. Waugh was asked the question 27 years back and who is older Mark or Steve? a. 25 yrs, mark is older b. 33 yrs, mark is older c. 32 yrs, mark is older d. 33 yrs, both are of the same age e. None of these Directions for (Q. No ): These questions are based on the following data. I was visiting my friend s house at Tuticorin for the first time. I knew that he has five children, two daughters and three sons. The daughters were Sonali and Monali and the sons were Surya, Sagar and Prithvi. When I stepped into the house I was greeted by Sagar and Sonali. Sagar introduced

75 himself and told that he was twice as old as Sonali. The next to meet me was Monali who said that she and Sonali together were twice as old as Sagar. However Sagar felt relieved when Surya walked in and introduced himself and said that he and Sagar were twice as old as their sisters together. In the afternoon Prithvi came to my room when was relaxing and said Sorry Uncle, I was busy with my friends, since it is my 21st birthday today and hence could not meet you in the morning. When I greeted him and said that it was fine, he looked relieved and said, You know uncle, that the three of us Sugar, Surya and me are five times as old as our sisters together. Saying this he went away. Can you help me find out the solution for the following questions? 123. What is the age of Surya? a. 13 years 10 months b. 10 years 6 months c. 12 yrs d. 15 yrs 6 months e. None of these 124. What is the age of Monali? a. 5 years 10 months b. 5 years 3 months c. 12 yrs d. 5 yrs 6 months e. None of these 125. Who is the youngest of the siblings? a. sonali b. sagar c. surya d. prithvi e. None of these 126. How many years older than Sonali is Monali? a. 3 years 6 months b. 3 years 3 months c. 2 yrs d. 5 yrs 6 months e. None of these 127. After how many years would Surya be twice as old as Sagar? a. 3 years 6 months b. 3 years 3 months c. 2 yrs d. 5 yrs 6 months e. None of these 128. Average age of a father and his two sons is 27 years. Five years ago, the average of age of the two sons was 12 years. If the difference between the ages of the two sons is 4 years, then the present age of the father is a. 25 b. 24 c. 26 d. 47 e. None of these 129. The average age of 30 boys in a class is 15 years. One boy, aged 20 years, left the class, but two new boys came in his place whose ages differ by 5 years. If the average age of all the boys now in the class remains 15 years, the age of the younger newcomer is a. 15 b. 24 c. 26 d. 27 e. None of these 130. A father said to his son I was as old as you are at present at the time of your birth. If the father s age is 38 years now, the son s age five years back was a. 15 b. 14 c. 26 d. 27 e. None of these 131. The average age of 8 persons is increased by 2 years, when one of them, whose age is 24 years is replaced by a new person. The age of the new person is

76 a. 25 b. 24 c. 26 d. 27 e. None of these 132. Harsha is 40 years old and Rith is 60 years old. How many years ago was the ratio of their ages 3:5? a. 20 b. 24 c. 10 d. 27 e. None of these 133. The average age of 8 men is increased by 2 years when two of them whose ages are 21 and 23 years are replaced by two new men. The average age of the two new men is a. 25 b. 24 c. 26 d. 27 e. None of these 134. My grandfather was 8 times older than me 16 years ago. He would be 3 times my age, 8 years from now. Eight years ago, what was the ratio of my age to that of my grandfather? a. 1:2 b. 1:5 c. 7:3 d. 5:3 e. None of these 135. Ages of A and B are in the ratio of 2:3 respectively. Six years hence the ratio of their ages will become 8:11 respectively. What is B s present age? a. 20 b. 24 c. 10 d. 27 e. None of these 136. Jayesh is twice as old as Vijay and half as old as Suresh. If sum of Suresh s and Vijay s ages is 85 years what is Jayesh s age in years? a. 20 b. 34 c. 10 d. 27 e. None of these 137. Six years ago Jagannath was twice as old as Badri if the ratio of their present age is 9:5 respectively, what is the difference between their present ages? a. 20 b. 24 c. 10 d. 27 e. None of these 138. Jayesh is as much younger than Anil as he is older than Prasanth. If the sum of the ages of Anil and Prasanth is 48 years, what is Jayesh s age in years? a. 20 b. 24 c. 10 d. 27 e. None of these 139. The average age of 30 students in a class is 15 years. If 6 students of this class have the average age of 16 years, then the average of the remaining 24 students would be a. 13 years 10 months b. 10 years 6 months c. 14 yrs 9 months d. 15 yrs 6 months e. None of these 140. The sum of the ages of two men is 85. Five years ago the difference between their ages was 1/9 of the present age of the elder. The elder s present age is a. 45 b. 24 c. 10 d. 27 e. None of these 141. The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member than the old member a. 20 b. 24 c. 10 d. 40 e. None of these

77 142. Father is 5 years older than the mother and the mother s age is now thrice the daughter s age. If the daughter is 10 years old now, what was father s age, when the daughter was born? a. 25 b. 24 c. 10 d. 27 e. None of these 143. The age of Abdul is five times of his daughter and 5 years more than his wife. If the age of his daughter is 7 years, the age of his wife is a. 20 b. 30 c. 10 d. 27 e. None of these years ago, the average age of family of 3 members was 19 years. 2 children Sonu and Monu having been born, average age of the family is same to date. What is the present age of the youngest member of the family assuring that Sonu and Monu s ages differ by 2 years? a. 2 yrs b. 4 yrs c. 1 yr d. 3 yrs e. None of these 145. Present age of Anand is 47 years and his wife s age is just 38 years. When they were married, Anand was exactly one and half as old as his wife. How many years ago they were married? a. 20 b. 24 c. 10 d. 27 e. None of these 146. A said to B I am twice as old as you were when I was old as you are now. Sum of their ages is 42. Find their present ages a. 20, 24 b. 24, 18 c. 10, 15 d. 27, 35 e. None of these 147. Mr. Singh have five children Dolly, Polly, Molly, Solly and Lolly. Dolly was the eldest and the children were born in the same sequence such that the age difference between any two consecutive children is the same. If Dolly is 14 years old, what are the possible ages of Molly, if the difference in the ages of the children has integral values? I. 13 years II. 2 years III. 11 years a. only I b. only II c. only III d. I & II e. None of these 148. The captain of a cricket team of 11 players is 25 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is 1 year less than the average of the whole team? a. 20 b. 24 c. 10 d. 27 e. None of these 149. The difference in the age, of a mother and her son is 24 years. Four years hence the mother s age will be 4 times that of the son s age. What is the mother s age? a. 32 b. 36 c. 40 d. 28 e. None of these 150. Supriya is 25 years younger than her mother today. Ten years ago, Supriya was one-sixth as old as her mother. How old is Supriya s mother now? a. 32 b. 36 c. 40 d. 28 e. None of these 151. Ten years ago the average age of a family of 4 members was 24 years. Three children

78 having been born, the average age of the family is same today. What are the present ages of children, if two children are identical twins and differ by two years from the elder one? a. 32,20,20 b. 36,30,30 c. 12,10,10 d. 12,40,40 e. None of these 152. Utpal and Nikil undertook a contract, investing Rs and Rs respectively. They earned a profit of Rs in the business. What was the share of Nikhil? a b c d e. None of these 153. A and B started a business investing Rs. 20,000 and Rs. 30,000 respectively. A withdrew his capital after 6 months. At the end of the year A received Rs as his share in the profit. How much was B s profit? a b c d e. None of these 154. Ajay, Keshav and Vijay invested Rs. 8000, Rs and Rs. 8,000 respectively in a business. Ajay left after six months. If after 8 months there was a profit of Rs. 4005, then what will be the share of Keshav? a. 890 b. 360 c. 400 d. 280 e. None of these 155. Ramesh started a business investing Rs. 30,000. Six months later, Yogesh joined him investing Rs. 15,000. If they make the profit of Rs. 10,000 at the end of the year, how much should the share of Ramesh be? a b c d e. None of these 156. In a factory the ratio of male workers to female workers was 5:3. If the number of female workers was less by 40, what was the total number of workers in the factory? a. 320 b. 360 c. 160 d. 180 e. None of these 157. Dilip, Ram and Avtar started a shop by investing Rs , Rs. 81,000 and Rs. 72,000 respectively. At the end of one year, the profit was distributed. If Ram s share was Rs , what was the total profit? a b c d e. None of these 158. Anuj invested Rs. 80,000 and started a business. After four months, Gopal joined him what with a capital of Rs. 90,000. If at the end of the year, the total profit earned by them was Rs. 14,000. What was Gopal s share in it? a b c d e. None of these 159. A started a business investing Rs. 18,000. After three months B joined him investing Rs. 27,000. If at the end of the year, the total profit is Rs. 3,400, what will be A s share in it? a b c d e. None of these 160. Jeewan started a business investing Rs. 50, months after he started. Anil joined him with a capital of Rs. 90,000. If at the end of the year, the total profit in the business is Rs. 22,000. What would be Anil s share it? a b c d e. None of these

79 161. Amal started a business investing Rs. 20,000. Three months later, Kapil joined him investing Rs. 30,000. If they make a profit of Rs. 6,800 at the end of the year, what would be kapil s share? a b c d e. None of these 162. Subodh started a business investing Rs After 4 months Nepal joined him investing Rs. 30,000. If the total profit earned by them at the end of the year is Rs. 13,000. What will be the difference between their shares? a b c d e. None of these 163. A started with Rs and B joined afterwards with Rs If the profits at the end of one year were divided in the ratio 2:1 respectively, then B would have joined A for business after a. 3 months b. 6 months c. 4 months d. 8 months e. None of these 164. Srikanth and Vividh started a business investing amounts of Rs and Rs , respectively. If Vividh s share in the profit earned by them is Rs 9000, what is the total profit earned by them together? a b c d e. None of these 165. A starts a business with Rs and B joins him after 6 months with an investment of Rs What will be the ratio of the profits of A and B at the end of year? a. 1:2 b. 2:5 c. 7:3 d. 5:3 e. None of these 166. Rajan and Sajan started a business initially with Rs and Rs respectively. If total profits at the end of the year are Rs , what is the Rajan s share in the profit? a b c d e. None of these 167. Ram, Shyam and Kamal together start a business in partnership. The ratio of their capitals is 3:4:7. If their annual profit be Rs , what will be kamal s share in this profit? a b c d e. None of these 168. Reena, Reema and Meera take a car on rent and they use that car for 14h, 16h and 22h, respectively. What amount will be paid by Meera, if total rent of the cat is Rs. 2080? a. 300 b. 840 c. 820 d. 880 e. None of these 169. Pinku, Rinku and Tinku divide an amount of Rs. 4,200 amongst themselves in the ratio of 7:8:6 respectively. If an amount of Rs. 200 is added to each of their shares, what will be the new respective ratio of their shares of amount? a. 7:8:9 b. 8:9:7 c. 9:8:7 d. 9:7:8 e. None of these 170. Ramesh and Priya started a business initially with Rs and Rs. 6600, respectively investments done by both the persons are for different time periods. If the total profit is Rs. 5460, what is the profit of Ramesh?

80 a b c d. data inadequate e. None of these 171. A, B and C together start a business. B invests 1/6 of the total capital while investments of A and C are equal. If the annual profit on this investment is Rs , find the difference between the profits of B and C a b c d e. None of these 172. Avinash invested an amount of Rs. 25,000 and started a business. Jitendra joined him after one year with an amount of Rs. 30,000. After two years from starting the business they earned the profit of Rs, 46,000. What will be Jitendra s share in the profit? a b c d e. None of these 173. Mr. Nilesh Agarwal opened a workshop investing Rs. 40,000. He invested additional amount of Rs. 10,000 every year. After two years his brother Suresh joined him with an amount of Rs. 85,000. Thereafter Suresh did not invest any additional amount. On completion of four years from the opening of workshop they earned an amount of Rs. 1, 95,000. What will be Nilesh s share in the earning? a b c d e. None of these 174. A invests Rs for one year in a business. How many B should invest in order that the profit after 1 year may be divided into ratio of 2:3? a b c d e. None of these 175. Three friends A, B and C started a business by investing amount in the ratio of 5:7:6 respectively. After a period of six months C withdrew half of the amount invested by him. If the amount invested by A is Rs. 40,000 and the total profit earned at the end of one year is Rs. 33,000. What is C s share in profit? a b c d e. None of these 176. Mr. Shivkumar started a business investing Rs. 25,000 in In 1997 he invested an additional amount of Rs. 10,000 and Mr. Rakesh joined him with an amount of Rs. 35,000. In 1998 Mr. Shivkumar invested another additional amount of Rs. 10,000 and Mr. Suresh joined them with an amount of Rs. 35,000. What will be Rakesh s share in the profit of Rs. 1,50,000 earned at the end of three years from the start of the business in 1996? a b c d e. None of these 177. A and B started business with Rs and Rs respectively. After 8 months a withdraw Rs and B advances Rs more. At the end of the year, their profits amounted to Rs Then the share of A is a. 200 b. 220 c. 240 d. 300 e. None of these 178. A and B started a business jointly. A s investment was thrice the investment of B and the period of his investment was two times the period of investment was two times the period of investment of B. If B received Rs as profit, then their total profit is a b c d e. None of these

81 179. A and B are partners in a business. A contributes K of the capital for 15 months and B i received $ of the profit. Find how long B s money was used? & a. 3 months b. 6 months c. 4 months d. 8 months e. None of these 180. A, B and C are partners of a company. During a particular year A received one-third of the profit, B received one-fourth of the profit and C received the remaining Rs. 5,000. How much did A received? a b c d e. None of these 181. A began a business with Rs and was joined afterwards by B with Rs When did B join if the profits at the end of the year were divided in the ratio 2:1? a. after 3 months b. after 6 months c. after 4 months d. after 8 months e. None of these 182. A and B entered into a partnership investing Rs. 16,000 and Rs. 12,000 respectively. After 3 months, A withdrew Rs while B invested Rs more. After 3 more months, C joins the business with a capital of Rs. 21,000. The share of B exceeds that of C, out of a total profit of Rs. 26,400 after one year by a b c d e. None of these 183. A and B started a business jointly. A s investment was thrice the investment of B and the period of his investment was two times the period of investment of B. If B received Rs as profit, then their total profit is a b c d e. None of these 184. Rs. 700 is divided among A, B and C so that A receives half as much as B and B half as much as C. Then C s share is a. 200 b. 220 c. 400 d. 300 e. None of these 185. Manoj received Rs as his share out of the total profit of Rs which he and Ramesh earned at the end of one year. If Manoj invested Rs. 20,000 for 6 months, whereas Ramesh invested his amount for the whole year, what was the amount invested by Ramesh? a b c d e. None of these 186. Yogesh started a business investing Rs. 45,000. After 3 months, Pranab joined him with a capital of Rs. 60,000. After another 6 months, Atul joined them with a capital of Rs. 90,000. At the end of the year, they made a profit of Rs. 20,000. What would be Atul s shate it? a b c d e. None of these 187. Nirmal and Kapil started a business investing Rs and Rs respectively. After 6 months Kapil withdrew half of his investment. If after a year, the total profit was Rs. 4,600, what was Kapil s share in it? a b c d e. None of these

82 188. Jayant opened a shop investing Rs. 30,000. Madhu joined him 2 months later, investing Rs. 45,000. They earned a profit of Rs. 54,000 after completion of one year. What will be be Madhu s share of profit? a b c d e. None of these 189. A, B, C enters into a partnership investing Rs. 35,000, Rs. 45,000 and Rs. 55,000 respectively. The respectively shares of A, B, C in an annual profit of Rs. 40,500 are a. Rs. 10,500, Rs. 13,500, Rs. 16, 500 b. Rs. 10,500, Rs. 13,500, Rs. 16, 000 c. Rs. 10,500, Rs. 12,500, Rs. 16, 500 d. Rs. 10,000, Rs. 13,500, Rs. 16, 500 e. None of these 190. Arun, Kamal and Vinay invested Rs. 8000, Rs and Rs respectively in a business. Arun left after six months, If after eight months, there was a gain of Rs. 4005, then will be the share of Kamal? a. 800 b. 890 c. 850 d. 820 e. None of these 191. Two friends P and Q started a business investing in the ratio of 5:6. R joined them after six months investing an amount equal to that of Q s. At the end of the year 20% profit was earned which was equal to Rs. 98,000. What was the amount invested by R? a b c d e. None of these 192. In a business A and C invested amounts in the ratio 2:1 whereas the ratio between amounts invested by A and B was 3:2. If Rs. 1,57,300 was their profit, how much amount did B receive? a b c d e. None of these 193. A and B started a business in partnership investing Rs. 20,000 and Rs. 15,000 respectively. After six months, C joined them with Rs. 20,000. What will be B s share in the total profit of Rs. 25,000 earned at the end of 2 years from the starting of the business? a b c d e. None of these 194. A, B and C rent a pasture. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months and C puts 15 oxen for 3 months for grazing. If the rent of the pasture is Rs. 175, how much C pays as his share of rent? a. 25 b. 35 c. 45 d. 50 e. None of these 195. A and B started a partnership business investing some amount in the ratio of 3:5. C joined them after six months with an amount equal to that of B. In what proportion should the profit at the end of one year be distributed among A, B and C? a. 5:10:6 b. 10:6:5 c. 6:10:5 d. 5:6:10 e. None of these 196. Mr. Nilesh Agarwal opened a workshop investing Rs. 40,000. He invested additional amount of Rs. 10,000 every year. After two years his brother Suresh joined him with an amount

83 of Rs. 85,000. Thereafter Suresh did not invest any additional amount. On completion of four years from the opening of workshop they earned an amount of Rs. 1, 95,000. What will be Nilesh s share in the earning? a b c d e. None of these 197. A invests Rs for one year in a business. How much B should invest in order that the profit after may be divided into ratio of 2:3? a b c d e. None of these 198. Three friends A, B and C started a business by investing amount in the ratio of 5:7:6 respectively. After a period of six months C withdrew half of the amount invested by him. If the amount invested by A is Rs. 40,000 and the total profit earned at the end of one year is Rs. 33,000, what is C s share in profit? a b c d e. None of these 199. Mr. Shivkumar started a business investing Rs. 25,000 in In 1997 he invested an additional amount of Rs. 10,000 and Mr. Rakesh joined him with an amout of Rs. 35,000. In 1998 Mr. Suresh joined them with an amount of Rs. 35,000. What will be Rakesh s share in the profit of Rs. 1,50,000 earned at the end of three years from the start of the business in 1996? a b c d e. None of these 200. A and B started business with Rs and Rs respectively. After 8 months A withdraws Rs and B advances Rs more. At the end of the year, their profits amounted to Rs Then the share of A is a. 480 b. 440 c. 420 d. 240 e. None of these

84 Answers 1. C 2. D 3. D 4. B 5. C 6. A 7. D 8. B 9. A 10. A 11. B 12. B 13. A 14. C 15. A. 16. B 17. B 18. A 19. A 20. A 21. D 22. A 23. A 24. A 25. A 26. D 27. C 28. D 29. C 30. A 31. D 32. A 33. C 34. C 35. B 36. B 37. D 38. C 39. A 40. B 41. D 42. B 43. B 44. A 45. A 46. C 47. D 48. C 49. B 50. D 51. A 52. D 53. A 54. B 55. A 56. A 57. B 58. B 59. B 60. A 61. A 62. C 63. A 64. B 65. A 66. C 67. B 68. C 69. D 70. B 71. A 72. D 73. B 74. D 75. D 76. B 77. B 78. C 79. C 80. B 81. C 82. D 83. B 84. C 85. E 86. D 87. A 88. C 89. A 90. D 91. A 92. B 93. B 94. D 95. A 96. C 97. B 98. A 99. C 100. A 101. B 102. C 103. B 104. C 105. B 106. E 107. A 108. B 109. C 110. A 111. D 112. A 113. E 114. C 115. B 116. B 117. C 118. B 119. D 120. C 121. B 122. D 123. B 124. B 125. A 126. A 127. A 128. D 129. A 130. B 131. E 132. C 133. E 134. B 135. D 136. B 137. B 138. B 139. C 140. A 141. D 142. A 143. B 144. D 145. A 146. B 147. B 148. E 149. D 150. C 151. C 152. E 153. E 154. A 155. D 156. C 157. D 158. B 159. B 160. A 161. A 162. A 163. A 164. C 165. B 166. D 167. D 168. D 169. B 170. D 171. D 172. D 173. C 174. C 175. A 176. C 177. C 178. B 179. E 180. A 181. E 182. A 183. B 184. C 185. B 186. D 187. B 188. C 189. A 190. B 191. B 192. D 193. A 194. C 195. C 196. B 197. B 198. E 199. B 200. D

85 Average Whenever we are asked the marks scored by us in any examination, we usually tell the marks in percentage, taking the percentage of total marks of all subjects. This percentage is called average percentage. Generally speaking averages can be broadly classified into two categories: 1. Mathematical Averages 2. Averages of position 1. Mathematical Averages Arithmetic average or mean: The average or mean or arithmetic mean of a number of quantities of the same kind is equal to their sum divided by the number of those quantities. Ex: The average of 3, 10, 11, 17, and 24 is Geometric mean: Geometric mean of Geometric mean is useful in calculating averages of ratios such as average population growth rate, average percentage increase etc. Harmonic mean: Harmonic means of is denoted by Harmonic mean is useful for finding out average speed of a vehicle, average production per day etc. 2. Averages of position Median: If we arrange N numbers in order, increasing or decreasing, the median is the middle number if N is odd; and the average of the two middle if N is even. Mode: The mode of a set of numbers is that value which occurs with the greatest frequency. In other words, it is the most common value. In general, then mode may not exist or is it exists it may not be unique. Formulae 1. Average = 2. Suppose a man covers a certain distance at kmph and an equal distance at kmph. Then, the average speed during the whole journey is

86 3. Sum of first n natural number n = n( n + 1) / 2 4. Sum of square of first n natural number n 2 = n( n + 1) (2n+ 1 )/6 5. Sum of cube of first n natural number n 3 = [n(n + 1) / 2] 2 6. Sum of first n even number n = n(n + 1) 7. Sum of first n odd number (2n 1) = n 2 Some properties of Average: i) If each observation is increased by a, the average of these observations will also increase by a. ii) If each observation is decreased by a, the average of these observations will also decrease by a. iii) If each observation is multiplied by a, the average of these observations will also multiply by a. iv) If each observation is divided by a, the average of these observations will also divided by a. v) If there are n consecutive odd numbers and we want to find the difference between highest and lowest number then it will be (n-1) 2 vi) If there are n consecutive even numbers and we want to find the difference between highest and lowest number then it will be (n-1) 2 The average score of three students A, B, and C is 50. When the score of another student D is added to the group, the average score become 47. What is the score of student D? Answer: for most of you, the score of student D would be = 38. For me, the calculation would just be 47-9 = 38. Some of you might have understood what I did. Let me start explaining through a simple example. Then we shall extend our explorations to more complex problems. The two images above show two beam balances, one with equal arms and the other with the arms lengths in the ratio 1: 3. The dotted line in both cases shows the average value. In the first beam balance, if you move any of the pan one unit towards the average value, the other pan would also move one unit in the direction of the average value to keep the average constant. For example, let the weights in the two pans be 50 kg and 60 kg. The average is 55 kg. If you increase the weight in 50 kg pan by one unit (i.e. 51) bringing it nearer to the average, you will have to decrease the weight in the other pan by one unit (i.e. 59), bringing it one unit

87 nearer to the average, to keep the average constant. In the second beam balance, if the arm of length 3 moves 1 unit towards the average, the arm of length 1 will have to move 3 units towards the average to keep the average constant. For example, let the weight in pan of 3 unit arm length be 40 kg and the weight in the pan of 1 unit arm length be 60 kg. Now if I increase the weight in the first pan by 1 unit, I shall have to decrease the weight in the second pan by 3 units to keep the average constant. The same rule applies if I am moving a pan away from the average. Now, understand this- the second beam balance can also be represented with a beam balance of equal arms but one arm having thrice the weight on the other arm. Savvy? Now let s see the problem once again. Now there are three weights on one arm. The average of the four weights is 47. To move each of the three weight 3 units away from the average (50-47) I shall have to move the weight D 3 3 nine units away from the average. Therefore, weight D = 47-9 = 38. Q) Four friends have an average weight of 68. If Rahim is also included in the group, the average weight becomes 72. what is Rahim s weight? Answer: Same process: there are four people at a distance of 4 units from the average weight of 72. To balance them, we will have to place a person at 4 4 = 16 units from 72 on the other side. Therefore, Rahim s weight = = 88. Q) A batsman in his 20 th innings makes a score of 93 and thereby increases his average by 3. What is the average after 20 innings? Answer: If you have understood what I have said so far, the new average is nothing but = 36. Let the new average be A. Therefore, there are 19 scores at a distance of 3 units from A. To balance these scores we need one score (which is 93) 19 3 = 57 units from A on the other side. Therefore A = = 36. Ex.1: Find the average of all prime numbers between 30 and 50? Sol: There are five prime numbers between 30 and 50. They are 31,37,41,43 and 47. Therefore the required average = ( )/5 = 199/5 = 39.8 Ex.2. Find the average of first 40 natural numbers? Sol: Sum of first n natural numbers = n(n+1)/2 ; So, sum of 40 natural numbers = (40 41)/2 = 820. Therefore the required average = (820/40) = 20.5 Ex.3. Find the average of first 20 multiples of 7? Sol: Required average =7( )/20 or ( )/(20 2) = (147/2) = 73.5

88 Ex.4. The average of four consecutive even numbers is 27. find the largest of these numbers? Sol: Let the numbers be x,x+2,x+4 andx+6. then, [{x+(x+2)+(x+4)+(x+6)}/4] = 27 or (4x+12)/4 = 27 or x+3 =27 or x = 24. Therefore the largest number = (x+6) = 24+6 = 30. Ex.5. There are two sections A and B of a class consisting of 36 and 44 students respectively. If the average weight of section A is 40kg and that of section B is 35kg, find the average weight of the whole class? Sol: Total weight of ( ) students = ( )kg = 2980kg. Therefore weight of the total class = (2980/80)kg = 37.25kg. Ex:6. Nine persons went to a hotel for taking their meals 8 of them spent Rs.12 each on their meals and the ninth spent Rs.8 more than the average expenditure of all the nine.what was the total money spent by them? Sol: Let the average expenditure of all nine be Rs.x Then (x+8) = 9x or 8x = 104 or x = 13. Total money spent = 9x = Rs.(9 13) = Rs.117. Ex.7: Of the three numbers, second is twice the first and is also thrice the third. If the average of the three numbers is 44.Find the largest number. Sol: Let the third number be x. Then second number = 3x. First number = 3x/2. Therefore x+3x+(3x/2) = (44 3) or x = 24 So largest number = 2nd number = 3x = 72. Ex.8: The average of 25 results is 18.The average of first 12 of them is 14 & that of last 12 is 17.Find the 13th result. Sol: Clearly 13th result = (sum of 25 results) (sum of 24 results) =(18 25) (14 12)+(17 12) = 450 ( ) = = 78. Ex.9: The Average of 11 results is 16, if the average of the 1st 6 results is 58 & that of the last 63. Find the 6th result? Sol: 6th result = ( ) = 66 Ex.10:The average weight of A,B,C is 45 Kg. The average weight of A & B be 40Kg & that of B,C be 43Kg. Find the weight of B. Sol. Let A, B, C represent their individual weights. Then, A + B+ C = (45 3) Kg = 135 Kg A+ B = (40 2) Kg = 80 Kg & B + C = (43 2)Kg = 86Kg B = (A+B)+(B+C) (A+B+C) = ( )Kg = 31Kg.

89 Ex. 11. The average age of a class of 39 students is 15 years. If the age of the teacher be included, then the average increases by3 months. Find the age of the teacher. Sol. Total age of 39 persons = (39 15) years = 585 years. Average age of 40 persons = 15 yrs 3 months = 61/4 years. Total age of 40 persons = ((61/4 ) 40) years = 610 years. :. Age of the teacher = ( ) years=25 years. Ex. 12. The average weight of 10 oarsmen in a boat is increased by 1.8 kg when one of the crew, who weighs 53 kg is replaced by a new man. Find the weight of the new man. Sol. Total weight increased = (1.8 10) kg = 18 kg. :. Weight of the new man = ( ) kg = 71 kg. Ex. 13. There were 35 students in a hostel. Due to the admission of 7 new students, ; the expenses of the mess were increased by Rs. 42 per day while the average expenditure per head diminished by Rs 1. What was the original expenditure of the mess? Sol. Let the original average expenditure be Rs. x. Then, 42 (x 1) 35x = 42 or 7x = 84 or x =12. Original expenditure = Rs. (35 12) = Rs Ex.14. A batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. Find his average after 17th inning. Sol. Let the average after 17th inning = x. Then, average after 16th inning = (x 3). :. 16 (x 3) + 87 = 17x or x = (87 48) = 39. Ex.15. Distance between two stations A and B is 778 km. A train covers the journey from A to B at 84 km per hour and returns back to A with a uniform speed of 56 km perhour. Find the average speed of the train during the whole journey. Sol. Required average speed = ((2xy)/(x+y)) km / hr = ( )/(84+56)km/hr = ( )/140 km/hr =67.2 km/hr. Ex. 16. In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs? Sol. Required run rate = (3.2 x 10) = 250 = Ex. 17. A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs and Rs for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500? Sol. Total sale for 5 months = Rs. ( ) = Rs Required sale = Rs. [ (6500 x 6) ] = Rs. ( ) = Rs

90 Ex. 18. A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half (1/2). The number of pupils in the class is: Sol. Let there be x pupils in the class. Total increase in marks = x x 1 = x 2 2 x = (83-63) x = 20 x = Ex. 19. A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is: Sol. Since the month begins with a Sunday, to there will be five Sundays in the month. 510 x x 25 Required average = 30 = = 285 Ex. 20. The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team? Sol. Let the average age of the whole team by x years. 11x - ( ) = 9(x -1) 11x - 9x = 46 2x = 46 x = 23. So, average age of the team is 23 years. Ex. 21. The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is: Sol. Sum of the present ages of husband, wife and child = (27 x x 3) years = 90 years. Sum of the present ages of wife and child = (20 x x 2) years = 50 years. Husband's present age = (90-50) years = 40 years. Ex. 22. In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother doest not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun? Sol. Let Arun's weight by X kg. According to Arun, 65 < X < 72 According to Arun's brother, 60 < X < 70. According to Arun's mohter, X <= 68

91 The values satisfying all the above conditions are 66, 67 and Required average = = = 67 kg. 3 3 Ex. 23. The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero? Sol. Average of 20 numbers = 0. Sum of 20 numbers (0 x 20) = 0. It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a). Ex. 24. The average of k numbers is 20, and the average of 10 of these numbers is 10. In terms of k, find the average of the remaining numbers. Sol. The total of the k numbers is 20k, and the total of 10 out of the 20 numbers is = 100. So, the total of the remaining (k 10) numbers is 20k 100, and their average is 20(k 5)/ (k 10) Ex. 25. The average score of 11 students is 50. If the average of the first 6 students is 49 and that of the last 6 is 52, find the score of the 6 th student. Sol. Total score of 11 students = = 550 Total score of first 6 students = 6 49 = 294 Total score of last 6 students = 6 52 = 312 Total score of first 6 and last 6 students = = 606 Score of the 6 th student = = 56 Ex. 26. Out of 36 students in a hostel, 19 left. Due to this, the expenditure of the mess decreased by $48 per day, while the average expenditure per head increased by $5. What is the sum of original expenditure and the new expenditure per day of the mess? Sol. Let the average mess expenditure per student be x. Then, total expenditure = 36 x. After 19 students left, new expenditure = 36x 48 According to the question &honie Kg = (x + 5) 36x 48 = 17x x = 133 x = 7 Sum of original expenditure and new expenditure of mess = 36x + 36x 48 = 72x 48 = 456 Ex. 27. The average temperature for Monday, Tuesday and Wednesday was 55 o ; the average for Tuesday, Wednesday and Thursday was 60 o ; and the temperature for Thursday was 56 o. Find the temperature on Monday. Sol. Total temperature for (M + T + W) = 3 55 o = 165 o (1) Total temperature for (T + W + Th) = 3 60 o = 180 o (2)

92 On solving (1) and (2) Th M = M = 15 Mon = 41 o. Ex. 28. If a, b, c, d, e are five consecutive odd numbers, find their average in terms of a. Sol. a, b, c, d, e are five consecutive odd numbers. Therefore, b = a + 2, c = b + 2 = a + 4 d = c + 2 = a + 6 and e = d + 2 = a + 8 Average of the five numbers = (5a + 20)/5 = a + 4. Ex. 29. Four years ago, the average age of four friends was 21 years. Now, with a fifth friend joining the group, their average is 24 years. What is the present age of the new friend? Sol. Total age of all friends including new friend = 24 5 = 120 years Present age of the new friend = new total of ages old total of ages = 20 years Ex. 30. A tree is 8.4 m tall, another is half its height and the third one is its quarter. What is the average height of the trees? Sol. Height of the first tree = 8.4 m tall Height of the second tree = ½ 8.4 = 4.2 m Height of the third tree = ¼ 8.4 = 2.1 m Total height = = 14.7 Average height = Ki.g = 4.9 m. & Ex. 31. The average monthly rainfall for a year in Tokyo is 2.7 inches. The average for the first 7 months is 1.1 inches less than the annual average. If the total rainfall for the next 4 months is 20.8 inches, calculate the rainfall in the last month. Sol. The total rainfall for a year = = 32.4 inches. For the first 7 months average rainfall is 1.1 inches less than the average annual rainfall Hence, average rainfall for the first 7 months = = 1.6 inches Total rainfall for the first 7 months = = 11.2 Total rainfall for the last 4 months = 20.8 inches Hence, rainfall in last month = Annual rainfall [total rainfall of first 7 and next 4 months] = 32.4 [ ] = = 0.4 inches Ex. 32. Average number of marks scored by 60 students in a class was The marks of one student were altered from 88 to 78 and marks of another were altered from 13 to 43. What was the new average? Sol. Total marks of 60 students = = 1950 Decrease in marks of the first student = = 10 And increase in marks of the other student = = 30 Total variation in marks = = Now new average = KjgM hm =

93 Ex. 33. A shopkeeper earned $504 in 12 days. His average income for the first 4 days was $40 a day. His average income for the remaining days was : Sol. Money earned in 12 days = $504 Money earned in first 4 days = 4 40 = 160 Money earned in last 8 days = = 344 Average for remaining days = &ii e = $43. Ex. 34. A team of eight marksmen entered a shooting competition. The best marksman scored 85 points. If he had scored 92 points, the average for the team would have been 84. How many points altogether did the team score? Sol. Let the average points scored by the team of 8 marksmen be x Total points = 8x Now if best marksman scored 92 points instead of 85, then change in points = = 7 New average = eopg e = 84 8x + 7 = 672 8x = 665 Total points scored by the team = 8x = 665 points. Ex. 35. It takes Bob 2t minutes to complete form A, which has k questions, and 3t minutes to complete form B which has k/2 questions. If Bob answers all the questions, his average time to answer a form B question is how many minutes longer than his average time to answer a form A question? Sol. k questions in form A take 2t minutes. So, one question in form A will take 2t/k minutes. Similarly k/2 questions in form B take 3t minutes. So, one question in form B takes 3t/(k/2) or 6t/k minutes. So, the difference in time = 6t/k 2t/k = 4t/k minutes. Ex. 36. A publisher incurs a fixed expense of $2000 towards composing and other activities in printing a book. A variable cost of $2.25 per book is incurred towards paper, binding charges etc. What is the average cost per book if copies are printed? Sol. Fixed expenses = 2000 Now total variable cost = $ Total average cost = = $MMMP$.% immm immmm XdO\c qw`_pz^]d^p{\ qw`_ YW.WX pwws` = $MMM immmm = K $M = $2.30. Ex. 37. The average monthly salary of an office consisting of 120 clerks, 1 superintendent and one manager is $1500. If the manager and the superintendent have monthly salaries of $3000 and $2000 respectively, what is the average monthly salary of the clerks? Sol. Total monthly salary = $1500 (120 clerks + 1 suptt. + 1 manager) = = $ Total salary after Suptt. & Manager leave = ( ) = = $ Now new monthly average salary = KgeMMM K$M = $

94 Ex. 33. Three dogs weigh 6, 12 and 9 kg respectively. The weight of the heaviest dog is how much more than the average weight of the three dogs? Sol. Weight of three dogs 6, 12 and 9 kg. Average of weight of three dogs = hpk$pj = 9 kg Heaviest dog is 12 kg Difference = 12 9 = 3 kg & = $g & Ex. 34. The average of 3 integers s, t and 8 is 12. Which of the following cannot be the value of the product st? (a) 27 (b) 52 (c) 56 (d) 160 Sol. It is given that (s + t + 8)/3 = 12. So s + t = 36 8 = 28. Since all the answer choices are positive integers, we can assume that s and t are positive integers. Now we will have to try and the eliminate the answers choices whose values can be equal to s t such that s + t = 28. (1) is possible because 27 1 = 27, (2) is also possible because, 26 2 = 52, (5) is possible as = 196. Also (4) is possible as 20 8 = 160. So we are left with choice (3) only and it is the right answer Ex. 35. Ann, Jack, Carl and Linda own a total of 43 T-shirts. If 10 of the T-shirts belong to Ann, what is the average number of T-shirts owned by each of the others? Sol. The number of T shirts owned by Jack, Carl and Linda is = 33. So, the average number owned by them is && & = 11. Ex. 36. The average score of a cricketer in a certain number of innings is 15. When he scored zero in one innings, his average dropped to How many innings have been played including the last one? Sol. Let the number of innings = x Total runs scored by the player = 15x New number of innings = x + 1 Also new average = 13.5 So, (15x + 0)/(x + 1) = 13.5 Number of innings played including the latest one = = 10. Ex. 37. The average age of a class of 30 students and a teacher reduces by 0.5 years, if we exclude the teacher. If the initial average is 14 years, find the age of the class teacher. Sol. Average age of 30 students and 1 teacher = 14 years. Total age of 31 persons = = 434 years. Average age of 30 students = = 13.5 years Total age of 30 students = = 405 years Age of teacher = = 29 years.

95 Exercise 1. The average runs scored by the eleven players of a cricket team is 60. If the runs scored by the captain are neglected, the average of runs scored by the remaining players increases by 5. How many runs were scored by the captain? a. 8 b. 9 c. 8.5 d. 10 e. None of these 2. In a certain company, the average production by 18 employees is 400 units per month. If the total production per month is increased by 180 units, then will be the average production of each employee? a. 400 b. 490 c. 450 d. 420 e. None of these 3. The number of students in each section of a school is 24. After admitting the new students, three new sections were started. Now the total number of section is 16 and there are 21 students in each section. The number of new students admitted is a. 18 b. 20 c. 16 d. 24 e. None of these 4. The total of three consecutive even numbers is 36 more than the average of these three numbers. Which of the following is the second of these three numbers? a. 18 b. 20 c. 16 d. 24 e. None of these 5. A man walks for 1 hours at the speed in 6km per hour. Then he walks 10km in 1 hours. What is his average speed in km per hour for the whole journey? a. 8 b. 9 c. 6 d. 10 e. None of these 6. The average salary of Prakash, Jayesh and Kailash is Rs The average salary of Prakash and Kailash is Rs What is Jayesh s salary? a b c d e. None of these 7. The average of the runs made by 10 players in a cricket team of 11 players is 43. If the eleventh player s runs are also considered, the average of 11 players runs decreases by 1. How many runs did the eleventh player make? a. 32 b. 20 c. 16 d. 24 e. None of these 8. The average of 7 numbers is 25. If each of the numbers is multiplied b 5, find the average of new set of numbers a. 132 b. 120 c. 116 d. 125 e. None of these 9. The number of students in each section of a school is 24. After admitting the new students, three new sections were started. Now the total number of sections is 16 and there are 21 students in each section. The number of new students admitted is a. 32 b. 20 c. 16 d. 24 e. None of these

96 10. The average of runs scored by the 11 players of a cricket team is 60. If the runs scored by the captain are neglected, the average of runs scored by the remaining players increases by 5. How many runs were scored by the captain? a. 8 b. 9 c. 6 d. 10 e. None of these 11. Average scores of 50 students in a class is 44. Later on, it was found that a score 23 was incorrectly recorded as 73. The correct average score is a. 43 b. 20 c. 16 d. 24 e. None of these 12. The average weight of 10 oarsmen in a boat is increased by 2.5 kg when one of the crew, who weights 64 kg is replaced by a new man. Find the weight of the new man a. 43 b. 89 c. 80 d. 24 e. None of these 13. If the sum of three consecutive odd numbers is 213. Find the first term. a. 69 b. 89 c. 80 d. 50 e. None of these 14. Of the three numbers, the average of the first and the second is greater than the average of the second and the third by 15. What is the difference between the first and the third of the three number a. 30 b. 20 c. 16 d. 24 e. None of these 15. If the average of n terms is X then the sum of n terms= a. n/x b. n 2 X c. nx d. X/n e. None of these 16. Find the average of first 6 odd multiples of 4 a. 143 b. 189 c. 180 d. 124 e. None of these 17. A student walks to his school at 5 km/hr and then goes back on a cycle at 15 km/hr. Find the average speed for the whole journey in km/hr a. 4.3 b. 8.9 c. 8 d. 7.5 e. None of these 18. Find the average of first 200 natural numbers. a. 100 b. 90 c d e. None of these 19. I went to a hotel along with 12 friends. I paid Rs. 145 and all the others paid an equal amount. In the end when we did some calculations, we found that the average sum paid by all of us was Rs. 5 more than what was originally paid by each of my friends. How much money did each of my friends pay? a. 100 b. 90 c. 80 d. 110 e. None of these 20. There are 20 students in Mr. Talwar s class. Once he conducted an examination out of 100. He then arranged the marks in the ascending order. He found that Rohit, the topper of the class slipped to the tenth position. When he was adding the scores of the last 11 students the average was 64 and that of the top 10 was 67. If the average mark obtained by all the students of his class was 65, how many marks did Rohit score?

97 a. 75 b. 90 c. 74 d. 73 e. None of these 21. Mr. Rawat was sitting idle in his home. He was making some calculations and he found that his age was 58 years less than the square of the age of his son. He further made some calculations and made a statement that the sum of the squares, of the sum and the product of the digits in his age, was 100. Find the age of his son a. 8 b. 9 c. 8.5 d. 10 e. None of these 22. In a recent survey of 800 socialities, it was found that 3% of them attended one party a week on an average. Of the other 97% it was found that half attended an average of two parties a week and the other half did not attend any party. How many party attenders were there in an average week? a. 800 b. 900 c. 850 d e. None of these 23. If average of 8 consecutive even numbers is 64, what is the difference between the smallest and the largest numbers? a. 18 b. 19 c. 14 d. 10 e. None of these 24. The average of 3 numbers is 27 and that of first two is 18. The third number is a. 42 b. 45 c. 48 d. 36 e. None of these 25. Nine men went to a hotel. 8 of them spent Rs. 3 each over their meals and the ninth spent Rs. 2 more than the average expenditure of all the nine. The total money spent by all of them was a. 28 b. 29 c. 24 d e. None of these 26. The salary bill for the month of May in respect of 4 stenos and 2 clerk in an office was Rs. 16,000. On June 1st, one steno left and one clerk was appointed by which, the monthly salary bill was reduced by Rs. 1,000. What is the monthly salary of each of the clerk? a b c d e. None of these Directions for Q 27 & 28: Read the given data and answer the questions given. A batsmen s average score for a certain number of innings was per innings. He played 3 innings more and scored 28, 34 and 37 runs respectively. Thus increasing his average by How many innings did he play in all? a. 28 b. 29 c. 24 d. 27 e. None of these 28. What was his total score? a. 800 b. 600 c. 620 d. 621 e. None of these 29. In the month of July of a certain year, the average daily expenditure of an organization was Rs. 68. For the first 15 days of the month, the average daily expenditure was Rs. 85 and for the last 17 days, Rs. 51. Find the amount spent by the organization on the 15th of the month a. 28 b. 29 c. 34 d e. None of these 30. In 1919, W. Rhodes, the Yorkshire cricketer, scored 891 runs for his country at an average of

98 34.27, in 1920, he scored 949 runs at an average of 28.75, in 1921, 1329 runs at an average of and in 1922, 1101 runs at an average of What was his batting average for the four years? a. 35 b c d e. None of these 31. In hotel Jaysarmin, the rooms are numbered from 101 to 130 on the first floor, 221 to 260 on the second floor and 306 to 345 on the third floor. In the month of June 2002, the room occupancy was 60% on the first floor, 40% on the second floor and 75% on the third floor. If it also known that the room charges are Rs. 200, Rs. 100 and Rs. 150 on each of the floors, then find the average income per room for the month of June 2002 a. 88 b c d e. None of these 32. A salesman gets a bonus according to the following structure: If he sells articles worth Rs. X then he gets a bonus of Rs. (x/100-1). In the month of January, his sales value was Rs. 100, in February it was Rs. 200, from March to November it was Rs. 300 for every month and in December it was Rs Apart from this, he also receives a basic salary for Rs. 30 per month from his employer. Find his average income per month during the year a. 35 b c d e. None of these 33. The weight of a body as calculated by the average of 7 different experiments is g. The average of the first three experiments is g of the fourth is g greater than the fifth, while the average of the sixth and seventh experiment was g less than the average of the first three. Find the weight of the body obtained by the fourth experiment a. 35 b c d e. None of these 34. There are five boxes in a cargo hold. The weight of the first box is 200 kg and the weight of the second box is 20% higher than the weight of the third box, whose weight is 25% higher than the first box s weight. The fourth box at 350 kg is 30% lighter than the fifth box. Find the difference in the average weight of the four heaviest boxes and the four lightest boxes a. 35 kg b. 70 kg c. 75 kg d. 80 kg e. None of these 35. For question 34 find the difference in the average weight of the heaviest three and the lightest three a kg b kg c kg d kg e. None of these persons went to a hotel for a combined dinner party. 13 of them spent Rs. 79 each on their dinner and the rest spent Rs. 4 more than the average expenditure of all the 19. What was the total money spent by them? a b c d e. None of these 37. The average weight of 5 men is decreased by 3 kg when one of them weighting 150 kg is replaced by another person. This new person is again replaced by another person whose weight is 30 kg lower than the person he replaced. What is the overall change in the average due to this dual chance? a. 10 b. 9.5 c d. 9 e. None of these

99 38. The weight of person is partly proportional to the amount of daily intake of calories and partly proportional to his age. A person aged 36 yrs weighs 61 kg when his intake is 2500 calories everyday while a person aged 60 yrs weighs 70 g when he takes in 1000 calories per day. How many would a person of age 24 yrs weight when his calorie intake is 3000 per day? a. 28 b. 54 c. 50 d. 27 e. None of these 39. The average score of the first 3 students in an examination is greater than that of the first 5 students by 20, but the total score of the first 3 is less than that of the first 5 by 70. Find the total score of the first 3 students a. 288 b. 254 c. 250 d. 227 e. None of these 40. A man s average expenditure for the first 4 months of the year was Rs For the next 5 months the average monthly expenditure was Rs more than what it was during the first 4 months. If the person spent Rs. 760 in all during the remaining 3 months of the year, find what percentage(approximate) of his annual income of Rs he saved in the year a. 8 % b. 5 % c. -5 % d. -7 % e. None of these 41. There were 42 students in a hostel. Due to the admission of 13 new students, the expenses of mess increase by Rs. 31 per day while the average expenditure per head diminished by Rs. 3. What was the original expenditure of the mess? a. 688 b. 654 c. 650 d. 637 e. None of these 42. The average weight of boys of Anil s age and height is 48 kg and if Anil weights 110% of average, then Anil s weights a. 28 kg b kg c kg d. 27 kg e. None of these 43. The average of n consecutive whole numbers will always be a whole number if a. n is even b. n is odd c. both a. and b. is possible d. data inadequate e. None of these 44. The average of 13 results is 60. If the average of seven results is 57 and that of last seven is 58, find the 7th result a. 28 b. 25 c. 50 d. 27 e. None of these 45. The average score of a cricketer for 10 matches is 49.7 runs. If the average for the first seven matches are 45. What is the average for the last 3 matches? a b. 60 c d. 65 e. None of these 46. Number B is 70 more than twice the number A. Number C is 10 more than twice the number B. If the average of A, B and C is 120. What is the value of C? a. 228 b. 254 c. 250 d. 227 e. None of these 47. The average of first five multiplies of 9 is a. 28 b. 54 c. 50 d. 27 e. None of these

100 48. The average monthly salary of 30 workers in a firm is Rs If the salary of the officer is added the average increases by Rs. 30. What is the monthly salary of the officer? a b c d e. None of these 49. The average weight of P, Q and R is 75 kg. If the average weight of P and Q is 60 kg and that of Q and R is 67 kg then the weight of Q is a. 28 b. 29 c. 50 d. 27 e. None of these 50. The average marks obtained by 35 students was 120. If the average of passed candidates was 130 and that of failed candidates was 60, the number of candidates who passed the examination a. 28 b. 29 c. 30 d. 27 e. None of these 51. The average of 6 numbers is 32. If one number is excluded the average becomes 30. The excluded number is a. 42 b. 54 c. 50 d. 27 e. None of these 52. The average of 21 numbers is 32. The average of first 11 of these numbers is 23 and the average of the last 11 of these numbers is 32. What is the twelth number? a. 28 b. 54 c. 50 d. data inadequate e. None of these 53. The average marks of 50 students in a class is 60. The average of those who pass is 75 and those who fail is 50. How many students did fail? a. 28 b. 34 c. 30 d. 27 e. None of these 54. The average marks of 40 students in a subject is 65. The average of those who passed is 75 and those who failed is 35. How many students did pass? a. 28 b. 54 c. 50 d. 27 e. None of these 55. The average of runs made by 11 players in a cricket team is 75, If the captain s runs is not considered, the average of the runs made by the remaining players increases by 4. How many runs did captain make? a. 28 b. 35 c. 36 d. 27 e. None of these 56. The average of three whole numbers is 36. The smallest number is one-third of the largest number. The middle number is twice the smallest number. Find out the middle number a. 28 b. 35 c. 36 d. 27 e. None of these 57. The average marks in Tamil, History and Maths is 45. The average marks in Science, Maths and History is 50. If marks in Science are 40, what should be the marks in Tamil? a. 28 b. 25 c. 26 d. 27 e. None of these

101 58. The average score of a cricketer for 10 matches is 39.9 runs. If the average for the first five matches is 39 then what is the average score for the last five matches? a. 40 b c d e. None of these 59. Of the three number, the first is twice the second and the second is thrice the third. If the average of the three numbers be 20, find the first number a. 28 b. 35 c. 36 d. 27 e. None of these 60. The average salary per head of the entire staff of an office including the clerks is Rs. 60. The average salary per head of the officer is Rs. 400 and that of the clerks is Rs. 56. Given that the number of officers is 13, find the number of clerks in the office a b c d e. None of these 61. The average collection in a week of Cinema hall is Rs.10, 504. The average collection of 6 days excluding Friday is Rs.9, 544. What is the collection on Thursday? a b c d e. None of these 62. The average marks of 96 students in a class is 90. The average marks of the girls in the class is 80 and the average marks of the boys in the class is 100. What is the ration between the number of boys and girls in the class? a. 1:2 b. 1:1 c. 2:1 d. 3:2 e. None of these 63. The average of two numbers is x. If one number is y then the other number is a. 2x - y b. 2y - x c. x - y d. y - x e. None of these 64. The average weight of three men X, Y and Z is 80 kg. Another man W joins the group and the average of the group becomes 78 kg. If another man V whose weight is 5 kg more than of W replaces X, then the average weight of Y, Z, W and V becomes 76 kg. The weight of X is a. 88 b. 85 c. 86 d. 87 e. None of these 65. The daily earnings of a Auto rickshaw driver a week are Rs , Rs , Rs , Rs , Rs , Rs and Rs respectively. Find his average daily earning for the week? a b c d e. None of these Tables and 8 Chairs were bought for Rs If the average price of a table be Rs 206, what is the average price of a chair? a. 388 b. 385 c. 375 d. 376 e. None of these 67. The average monthly expenditure of a family was Rs 2100 during first 3 months, Rs 2520 during next 4 months and Rs 2652 during last 5 months of the year. If the total saving during the year be Rs. 1440, find average monthly income? a b c d e. None of these

102 68. The average weight of 40 students of a class is 55 kgs. One of the students whose weight is 47 kgs is replaced by another student and the average weight was increased by 300 grams. Find the weight of the new student. a. 59 b. 60 c. 61 d. 62 e. None of these 69. The average weight of 8 children is 21.5 kgs. Two of them weight 20.4 kg and 22 kgs respectively. What is the average weight of remaining six? a b c d e. None of these 70. There are three numbers, the second is 1/4th of the first and the third is half of the second. If the average of the three numbers is 99, find the second number a. 59 b. 60 c. 55 d. 54 e. None of these 71. The average of the first 150 natural numbers is a b c d e. None of these 72. What is the average of the cubes of the first six counting numbers (i.e.) find the average of 13, 23, 33, 43, 53, 63 a b c d e. None of these 73. The average of 8 numbers is 6. What is the 9th number so that average becomes 8. a. 24 b. 25 c. 26 d. 27 e. None of these 74. A batsman in his 14th innings makes a score of 73 runs and there by increases his average score by 3. What is his average after the 14th innings? He had never been not out? a. 34 b. 35 c. 36 d. 37 e. None of these 75. The average of first six prime numbers is a. 4 b. 5 c. 6 d. 7 e. None of these 76. The average score of a cricketer for 10 matches is 49.8 runs. If the average for the first 6 matches is 54, find the average for the last 4 matches a b c d e. None of these 77. The average weight of 9 persons is increased by 4 kg when one of them whose weight is 65 kg is replaced by a new man. The weight of the new man is a. 101 b. 102 c. 103 d. 104 e. None of these 78. The sum of three numbers is 175. If the ration between first and second be 5:6 and that between second and third be 9:11, then the second number is a. 56 b. 57 c. 55 d. 58 e. None of these

103 79. The average temperature of first 3 days in 32 C and of the next 3 days is 29 C. If the average of the whole week is 31.5 C, the temperature of the last day is a C b C c C d C e. None of these 80. On a journey across Jaipur, a car s average speed is 40km/hr for 50% of the distance, 30 km/hr for 20% of it and 20 km/hr for the remaining distance. The average speed for the whole journey is a. 25 km/hr b. 29 km/hr c. 30 km/hr d. 35 km/hr e. None of these Answers 1. D 2. E 3. D 4. A 5. C 6. C 7. A 8. D 9. D 10. D 11. A 12. B 13. A 14. A 15. C 16. E 17. D 18. C 19. C 20. C 21. D 22. A 23. C 24. B 25. D 26. D 27. D 28. D 29. C 30. C 31. C 32. B 33. B 34. C 35. C 36. E 37. D 38. B 39. E 40. C 41. E 42. C 43. B 44. B 45. A 46. E 47. D 48. D 49. B 50. C 51. A 52. D 53. C 54. E 55. B 56. C 57. B 58. D 59. C 60. E 61. D 62. B 63. A 64. B 65. C 66. C 67. C 68. A 69. B 70. D 71. A 72. B 73. A 74. A 75. C 76. B 77. A 78. B 79. B 80. B

104 Percentage The word "percentage" literally means "per hundred" or "for every hundred." Therefore, whenever you calculate something as a part of 100, that part is numerically termed as percentage. In other words, percentage is a ratio whose second term is equal to 100. For example, 1:4 can be written as 25: 100 or 25%, 3: 8 can be written as 37.5: 100 or 37.5%, 3: 2 can be written as 150: 100 or 150%, and so on. IMPORTANT CONCEPTS ASSOCIATED WITH PERCENTAGE Basic formula of percentage: Percentage of: Commodity Price Increase/Decrease: If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is: R (100 + R) 100 % If the price of the commodity decreases by R%, then to maintain the same expenditure by increasing the consumption is: R (100 R) 100 %

105 Percentage increase/decrease: Percentage increase/decrease when a quantity a increase/decreases to become another quantity b is Percentage less than/greater than: Have a look at the picture given below: You can see that Johnny is taller than Vicky. What will your answer be if I ask (a) By what percentage is Johnny taller than Vicky? (b) By what percentage is Vicky shorter than Johnny? Answer:

106

107

108 Therefore, to find the final quantity after a 20% increase, we can directly multiply the old quantity by a factor of 1.2 and get the new quantity. Similarly, for a 20% decrease, we can multiply the old quantity by 0.8 and get the new quantity. The factors to be multiplied for various percentage increase/decrease are given below: The biggest advantage of using the factors is that for subsequent percentage increase/decrease, we just keep on multiplying the corresponding factors and get the final quantity. Solved examples Ex. 1. There has been a 30% increase in the production of milk chocolates in Amul Dairy this month. If the production this month is 9,100 boxes of milk chocolates, what was it one month ago? Sol. Increased amount = Original amount KMMP", where p = 30%, Original amount = Increased amount = 9,100 9,100 = N x KMMP&M KMM N = jkmm KMM K&M N = 7,000 boxes/month. Ex. 2. A and B are two numbers such that B A = P and B = P% more than A. Find B in terms of P Sol. If A = 100, then B = P and hence B A = P, satisfies. KMM Ex. 3. a, b, c, d are 4 numbers, such that b is 50% of a, c is 25% less than b and d is 10% more than c. If a is k% higher than d, what is the value of k? Sol. b = ^ $ c = & i b d = KK c = KK & b = && a KM KM i em So a is ^Nc 100% more than d c i.e. ^NSS }^ SS } 100

109 ig && 100 = 142% Ex. 4. The population of a town increased by 3% in a given interval but it would have been 1,500 less, if there had been a decrease of 2%. Find the original population Sol. Earlier there was an increase of 3%, instead if then there is a decrease of 2%, it means there is net change of 5% in population. Thus 5%of total population = 1,500. Total population = 30,000. Ex. 5. India scored a total of x runs in 50 overs. Sri Lanka tied the scores in 10% less overs. If India had scored 30 runs more, then Sri Lanka would have tied it in 50 overs at the same run rate. Find how many runs were scored by India. Sol. Average run rate of India = O %M Avg. run rate of Sri Lanka = O %M If India scored (x + 30) runs, then, Sri Lanka tied the match in 50 overs. That means, O 50 = x + 30 i% 50x = 45x x = i% &M = 270. % Ex. 6. The number of boys and girls who appeared in SSC CGL 2012 examination were in the ratio 16 : 9 and the number of boys and girls passing the examination were in the ratio 4 : 3. If 75% of the girls passed the examination, find the % of boys who passed the examination Sol. Let the number of boys & girls be 16x & 9x respectively. and let boys and girls passing are 4y and 3y respectively. %age of girls passing = (3y/9x) 100 = 75 y/x = 9/4 % of boys passing = (4y/16x) 100 = 25 (y/x)%. Substitute the value of y/x % of boys passing = 25 9/4 = 56 1 / 4 %. Ex. 7. A college has raised 75% of the amount it needs for a new building by receiving an average donation of Rs. 600 from the people already solicited. The people already solicited represent 60% of the people, the college asked for donations. If the college is to raise exactly the amount needed for the new building, what should be the average donation from the remaining people to be solicited? Sol. Let x be the total number of people the college will ask for donations. People already solicited = 0.6x Amount raised from the people solicited = x = 360x Now 360x constitutes 75% of the amount. Hence, remaining 25% = 120x Average donation from remaining people = K$MO M.iO = 300.

110 Ex. 8. If Manu writes atleast hundred lines, Jaspreet does not and if Jaspreet writes, Manu does not. Either of them writes at least 150% and at the most 250% of what the other writes. What is the maximum number of lines that the two can write? Sol. Maximum that one can write without writing 100 lines = 99. Then, the other can write, a maximum of lines. Maximum total = 347 lines Ex.9. Four friends A, B, C, D have some candies with them. B has 10% more candies than A, whereas C has 10% more than B and D has 10% more than C. The number of candies with A and D together is 21 more than B and C together. How many candies does B have? Sol. If A has x candies, A x B 1.1x C 1.21x D 1.331x A + D = 2.331x B + C 2.31x 0.021x = 21 x = 1,000 B has 1,100 candies. Ex. 10. A man earns x% on the Rs. 2,000 and y% on the rest of his income. If he earns Rs. 700 from Rs. 4,000 and Rs. 900 from Rs. 5,000 of income, find x. Sol. The two equations can be written O Q KMM KMM = 700, and O Q = 900. KMM KMM The equations can be simplified to x + y = 35 and 2x + 3y = 90. Solving these two equations simultaneously, we get x = 15 Ex. 11. A gambler has 125 coins with him. Whenever he has coins 100 in hand, he lent 20% of the coins. When he has 80 coins, he borrows 25% of the coins in hand. What will be the number of coins in his hand at the end of 29 th transaction? Sol. He has 125 coins (More than 100). So he will lent 20% of the coins. After the Ist transaction remaining coins = $M KMM = 100 Again he will lend out 20% of the coins. Remaining coin after IInd transaction = 80 Now, he will borrow 25% coins. No. of coins after IIIrd transaction = 100. So, if you see that after every even transaction he will have 80 coins, and after every odd transaction he will have 100 coins. So, after 29 th transaction he will have 100 coins.

111 Ex. 12. Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paise went on sales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax free items? Sol. Let the amount taxable purchases be Rs. x. Then, 6% of x = x = x = 5. Cost of tax free items = Rs. [25 - ( )] = Rs Ex. 13. A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had: Sol. Suppose originally he had x apples. Then, (100-40)% of x = x x = x 100 x = 60 = 700. Ex. 14. In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2 of the number of students of 8 years of age which is 48. What is the 3 total number of students in the school? Sol. Let the number of students be x. Then, Number of students above 8 years of age = (100-20)% of x = 80% of x. 80% of x = of x = x = 100. Ex. 15. Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third of the sum of 6% of A and 8% of B. Find the ratio of A : B. Sol. 5% of A + 4% of B = 2 (6% of A + 8% of B) A + B = A + B A + B = A + B

112 A = A = 1 B A = =. B 75 3 B Required ratio = 4 : 3 Ex. 16. In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was: Sol. Number of valid votes = 80% of 7500 = Valid votes polled by other candidate = 45% of = x 6000 = Ex. 17. Three candidates contested an election and received 1136, 7636 and votes respectively. What percentage of the total votes did the winning candidate get? Sol. Total number of votes polled = ( ) = Required percentage = x 100 = 57% Ex. 18. Two tailors X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120 percent of the sum paid to Y, how much is Y paid per week? Sol. Let the sum paid to Y per week be Rs. z. Then, z + 120% of z = 550. z z = z = x 5 z = 11 = 250. Ex. 19. Rajeev buys good worth Rs He gets a rebate of 6% on it. After getting the rebate, he pays sales 10%. Find the amount he will have to pay for the goods. Sol. 6 Rebate = 6% of Rs = Rs. x 6650 = Rs Sales tax = 10% of Rs. ( ) = Rs. 10 x 6251 = Rs Final amount = Rs. ( ) = Rs

113 Ex. 20. The population of a town increased from 1,75,000 to 2,62,500 in a decade. The average percent increase of population per year is Sol. Increase in 10 years = ( ) = Increase% = x 100 = 50% Required average = 50 = 5%. 10 Ex. 21. It costs Rs. 5,000 to paint the total surface of a cube. But in summer, each side expands by 20%. What will be the cost of painting the cube in the summer? Sol. The total surface area of cube = 6a 2 Since, each side is increased by 20% New side = 1 + $M KMM a = h % a Total surface area = 6 &h $% a2 So the cost also increases by &h So, answer = &h $% times. $% 5,000 = Rs. 7,200 Ex.22. A is 10% more than B, and C is 10% lesser than B. If the difference between A and C is 18, what is the value of A? Sol. A = 10% more than B = 1.1B C = 10% less than B = 0.9B A C = B 0.9B = 18 B = 90 A = 99 Ex. 23. Instead of finding 28% of a number Saloni incorrectly found 82% of the number. She got an answer that is 540 more than the answer that she is supposed to get. What is the answer she is supposed to get? Sol. Let the number be k Original answer should be 0.28k Her answer is 0.82k Difference = 0.54k It is given as 540 k = 1,000 Answer = 0.28k = 280 Ex. 24. Mr. Miranda saves 40% of his salary every month. But in April his expenses increase by 10% and he saves Rs. 1,200 less than the amount he saves normally. What is his salary (in Rs.)? Sol. Let x be the salary, 0.6x be the expenses. This month, the expenses increase by 10% and so become 0.66x. So savings decrease by 0.06x. 0.06x = 1,200 x = Rs. 20,000

114 Ex.25. In every box of 120 bulbs 5% are defective. If a company needs 2,000 non-defective bulbs, what is the minimum number of boxes it has to purchase? Sol. Each box contains 114 non-defective bulbs. Answer = $MMM KKi = 17.4 So the company has to purchase a minimum of 18 boxes to meet its requirement of bulbs. Ex.26. Padma purchased a box of chocolates. She ate 20% of them and gave 25% to her friends from the remaining. Again, she ate 20% of it and gave 25% to her friends from the remaining. Then she gave 8 chocolates to a guest. Finally, she is left with 35% of the chocolates that she initially had. How many chocolates did Padma eat? Sol. Assume the box contains x number of chocolates. 1 st time 0.2x eaten 2 nd time 0.12x eaten Given to friend 0.2x To friend = 25% of 0.48x = 0.12x Left with him = 0.6x Left = 0.36x Remaining = 0.36x It is given, 0.36x - 8 = 0. 35x x = 800 She ate 800 chocolates. 0.2x x = 0.32x = 256 Ex. 27. The cost of an air ticket is Rs. 3,000 + tax. If a discount of 50% is allowed on the cost of the air ticket (not tax), the cost of the ticket decreases by 30%. What is amount of tax charged on the purchase of the ticket? Sol. 70% of (3,000 + tax) = 1,500 + tax 2, tax = 1,500 + tax tax = 2,000 Ex. 28. If a student scored 30% marks in a paper, he fails by 6 marks but if he scores 40% marks he passes by 2 marks. What are the maximum marks? Sol. Same things on two different scales given (Difference of percentage) of total marks = (6 + 2) 10% of maximum marks = 8 Maximum marks = 80 Ex. 29. If a quantity is increased by 30% and then decreased by 50%, by how much percentage does it have to increase to bring it back to the original quantity? Sol. Assume the quantity as 100.

115 If it is increased by 30%, it will be 130 and if it decreases by 50%, it will become 65. So it has to be increased by 35 to come to 100 again. So answer = &% %. h% Ex. 30. The salary of a person is Rs. 15,000. But in the next month he gets a bonus of Rs. 1,000. He decided to save the bonus and in the process saved 10% more than the amount he saved the previous month. If there is no change in his expenses from the last month, what are his monthly expenses? Sol. Rs. 1000, increases his savings by 10% His previous month savings were Rs So expenses = = Rs Ex. 31. x is a positive integer. How many values x will take such that x% of 250 is less than 60? Sol. 24% of 250 is 60. x should be less than 24. It can be any integer from 1 to 23. So, it takes 23 values. Ex.32. In a certain cinema hall, there are 4 kinds of tickets, AC Boxes, Boxes, Balcony and DC. The relationship between the rates of tickets is: price of boxes is 20% higher than DC, DC is 40% less than AC boxes and price of balcony is 30% of the sum of the price of DC and AC Boxes. If the sum of the price of boxes and AC boxes is Rs. 86 then find the price of the balcony. Sol. Let say the price of the AC boxes = Rs. x Then the price of the DC = Rs. 0.6 x The price of the boxes = 0.72 x And, the price of the balcony = &M (0.6x + x) KMM = 0.48 x And, 0.72x + x = 86 x = eh K.g$ = 50 So the price of balcony = = Rs. 24. Ex. 33. In an election there are two candidates. A candidate gets 40% of the total votes and loses the election by 9,800 votes. Find the total number of votes. Sol. Winner must got 60% of total votes %age difference = 20% And votes difference = 9800 So, total votes 20% = 9800 Total votes = Ex. 34. The ratio of the number of males and females in a village is 3 : 2. 20% of the males and 25% of the females are graduates. Then find the % of the total population of the village who are not graduates?

116 Sol. Lets say that the male population = 300 Lets say that the female population = 200 Graduate Male population = 300 $M = 60 KMM Graduate Female population = 200 $% = 50 KMM Note educated = = 390 % of not educated = &jm 100 = 78%. %MM Ex. 35. Mr. Raj was throwing a party for his friends, and for that he required 37 kg of sweets. Assuming 19% wastage and knowing that the sweets are available in 2 kg packets only, what would be the cost of the sweets purchased, if one packet costs Rs.15? Sol. He has to buy X kg such that 81% of X = 37 kg X = 37 KMM = kg ek He has to buy 46 kg (multiple of 2 kg), i.e. 23 packets of sweets. Total cost = = Rs. 345 Ex. 36. If the salary of A is 20% more than that of B, by what percent salary of B is less than that of A? Sol. Let B's salary = 100 Thus, A's salary = 100 x 1.2 = 120 Therefore, $M KMM x 100 = = 16.66% K$M h Hence, B's salary is 16.66% less than A Ex. 37. A salesman s commission is 5% on all sales upto Rs. 10,000 and 4% on all sales exceeding this amount. He remits Rs. 31,100 to the parent company after deducting his commission. His sales is worth Sol. Let the total sales amount = Rs x Then, the commission, which salesman got in given as 5% of % of (x 10000) = x 31, 100 By solving, we get x = Ex. 38. In the month of January, the railway police caught 4000 ticketless travelers, in February, the number rose by 5%. However, due to constant vigil by the police and the railway staff, the number reduced by 5% in March and in April it further reduced by 10%. The total number of ticketless travelers caught in the month of April was Sol. Total number of ticketless travelers In February = % = 4200 In March = % = 3990 In April = % = 3591

117 Exercise 1. When the price of T.V. sets was increased by 30%, the number of T.V. sets sold decreased by 20%. What was the effect on the sales? a. 5 % increase b. 7 % increase c. 4 % increase d. 8 % increase e. None of these 2. Kiran earns 50% of Raghu s salary. Raghu earns 80% of Dinesh s salary. If the total salary of the three for a month is Rs. 22,000, how much did Kiran earn that month? a b c d e. None of these 3. Janaki spends 20% of her monthly salary for her son s education. When she will get her increment of Rs. 170 next month she plans to spend half of that also for her son s education. The total she would spend then would be Rs What is her present salary? a b c d e. None of these 4. The cloth shop has announced a reduction in price by 25%. How many kurtas priced at Rs. 28 should one buy to avail a total reduction of at least Rs. 40? a. 5 b. 6 c. 4 d. 7 e. None of these 5. One-fourth of a number is 10 less than 30 percent of the same number. What is that number? a. 600 b. 200 c. 800 d. 500 e. None of these 6. The length of a rectangle is increased by 20%. By what percent should the width be reduced to maintain the same area? a b c d. 18 e. None of these 7. What will be 40% of a number whose 200% is 90? a b c d. 18 e. None of these 8. Gopal spends 60% of his salary on food items, 50% of the remaining on clothes and transport and saves the remaining amount every month. If his annual savings are Rs. 9,600 how much money does he spend on the transport per month? a b c d. data inadequate e. None of these 9. If one-third of one-fourth of a number is 15, what is 30% of that number? a. 56 b. 54 c. 58 d. 50 e. None of these 10. If three-fourth of a number is 244, what will be the approximate value of 136 percent of that number? a. 650 b. 250 c. 850 d. 450 e. None of these 11. What will be the approximate value of 161% of 841? a b c d e. None of these

118 12. When the price of a toy was increased by 30%, the number of toys sold was decreased by 20%. What was the effect on the sales? a. 6% increase b. 4% deacrease c. 4% increase d. 5% deacrease e. None of these % of A s salary is equal to the 20% of & th of B s salary. If B s salary is Rs. 2400, what is A s % salary? a. 960 b. 200 c. 800 d. 500 e. None of these 14. In a library, 40% of the total numbers of books are in English, 80% of the rest are in Hindi and the balance 300 books are in other languages. What is the total number of books in the library? a b c d e. None of these 15. A businessman invests 70% in machinery, 20% in raw material and has Rs cash with him. What is the total investment? a. 1,60,000 b. 1,20,000 c. 1,80,000 d. 1,50,000 e. None of these 16. A man saves 25% of his monthly salary. If on account of rise in prices he is to increase his monthly expenses by 25% he is only able to save Rs. 250 per month. What is his monthly salary? a b c d e. None of these 17. Atul s marks in science are 20 less than 60% of the total marks obtained by him in science and maths together. If the total marks are 250, what are his marks in maths? a. 360 b. 240 c. 480 d. 120 e. None of these 18. In a selection process comprising of written test, group tasks and interview, only 40% qualified in written test, 80% of those passed in written test qualified in group tasks and 65% of these passed in group tasks qualified for interview. If the number of those qualified finally was only 78, what was the total number of candidates who appeared? a. 450 b. 225 c. 375 d. 325 e. None of these 19. A candidate should score 45% marks of the total marks to pass the examination. He gets 520 marks and fails by 20 marks. The total marks in the examination are a b c d e. None of these 20. If the population of the town is and its annual increase is 10%, then its correct population at the end of 3 years will be a b c d e. None of these 21. If x% of 1200 is 20 then x is a. 2.5 b. 2 c d. 1.5 e. None of these 22. If y exceeds x by 30% then x is less than y by

119 a. 23 K $ % b. 23 % K& K& K c. 23 % KM K d. 24 % K& e. None of these 23. When 80 is subtracted from 80% of a number, the result is 80, and the number is a. 400 b. 200 c. 800 d. 500 e. None of these 24. The price of an article is cut by 25%. To restore it to its original price, the new price must be increased by a b c d e. None of these 25. A and B s salaries together amount to Rs A spends 95% of his salary and B, 85% of his. If now their savings are the same, what is A s salary? a b c d e. None of these 26. In an examination, 15,000 candidates appeared. Each offered either maths or computer science or both. If 75% offered maths and 45% computer science how many offered both? a b c d e. None of these 27. A reduction in the price of orange enables a person to purchase 5 oranges for Re. 1 instead of Rs What is the percentage reduction in price? a % b. 20 % c. 25 % d. 50 % e. None of these 28. A s income is 25% less than B s income and B s income is 20% less than C s income. If C s income is Rs. 200, then the income of A is a. 150 b. 180 c. 120 d. 80 e. None of these 29. Ramesh s salary was reduced by 20%. In order to have his salary back to original amount it must be raised by a % b. 20 % c. 25 % d. 50 % e. None of these 30. A shop s receipts on the 1st day of the week were Rs. 20,000 on the second day 25% more than this and on the 3rd day 75% of the sum of the receipts of the first 2 days. What was the average of the receipts during this three day period (in rupees)? a b c d e. None of these 31. A man pays 10% of his salary as tax. If after spending 90% of the remainder, he has Rs. 180 left with him, what is his salary before taxation? a b c d e. None of these 32. The amount of water that should be added to reduce 8 ml mixer containing 50% alcohol, to a mixer containing 40% alcohol is a. 4 ml b. 2 ml c. 8 ml d. 3 ml e. None of these 33. A man by means of his false balance defrauds his costumers to the extent of 30% by purchasing the goods as well as by selling the goods find his net profit a. 60 b. 65 c. 70 d. 69 e. None of these

120 34. The salary of a typist was first raised by 20% and then the same was reduced by 10%. If he presently draws Rs. 1100, what was his original salary? a b c d e. None of these 35. In an examination 65% of the candidates passed in Maths and 75% passed in Science. If 22% failed in both, find the pass percentage who passed in both subjects. a. 62% b. 20 % c. 25 % d. 50 % e. None of these 36. In an election, a candidate secured 42% of the votes. The other candidates secured 58% and defeated him by 480 votes. Find the number of votes polled a b c d e. None of these 37. Two numbers are 25% and 50% less than a third number. Express first number as a percentage of the second a. 162% b. 120 % c. 125 % d. 150 % e. None of these 38. A man earns Rs in a month and spends 70% of his income. How much does he save in a year? a. 25,200 b c. 22,500 d. 26,500 e. None of these 39. The salary of a worker is first increased by 50% and afterwards reduced by 50%. What is net change in his salary? a % b. 20 % c. 25 % d. 50 % e. None of these 40. Two numbers are respectively 10% and 20% more than a third number, the percentage that is first of the second is a. 91.6% b. 80 % c. 75 % d. 95 % e. None of these 41. A person saves every year 10% of his income, if his income increased every year by 5% then his savings increase every year by a. 5% b. 2 % c. 4 % d. 6 % e. None of these 42. If 40% of a number is equal to 240. Find the number a. 500 b. 600 c. 800 d. 750 e. None of these 43. If A% of a number is equal to B% of second number, then the ration between first and second number is a. A:B b. B:A c. (A+B):A d. (A+B):B e. None of these 44. What is 60% of 30% of 900 grams? a. 163 gms b. 164 gms c. 170 gms d. 162 gms e. None of these 45. If 40% of 7 K = x% of 80. What is the value of x? $ a. 3.5 b c d. 4.5 e. None of these

121 46. What rate percent is 1 minute 48 seconds to an hour? a. 5% b. 2 % c. 4 % d. 6 % e. None of these 47. If two numbers are respectively x% and y% more than a third number, the first as a percent of second is a. KMMPO 100% KMMPQ e. None of these KMMPO b. 100% KMMNQ KMMNO c. 100% KMMPQ KMMNO d. 100% KMMNQ 48. Two numbers are 20% and 40% less than a third number. Then first number as percentage of the second is.. a % b. 120 % c. 125 % d. 150 % e. None of these 49. Adding 12% of x to x is equivalent to multiplying x by what? a % b. 120 % c. 112 % d. 150 % e. None of these 50. Subtracting 21% of x from x is equivalent to multiplying x by what a b. 0.5 c d e. None of these 51. If A s salary is 50% less than that of B, then how much percent is B s salary more than that of A? a. 100 % b. 120 % c. 125 % d. 150 % e. None of these 52. If the price of a commodity increases by x%, then reduction in consumption, so as not to increase the expenditure is O a. 100 % (KMMNO) b. 100 % (KMMNO) c. 100 % d. O 100 % KMMNO KMMPO KMMNO KMMPO e. None of these 53. If the price of coffee seeds is increased by 50%, find by how much percent a householder must reduce the consumption of coffee so as not to increase the expenditure? a % b. 20 % c. 25 % d. 50 % e. None of these 54. If the price of rice falls down by 20% by how much percent must a householder increase his consumption, so as not to decrease expenditure on this item? a % b. 20 % c. 25 % d. 50 % e. None of these 55. The price of commodity is diminished by 40% and its consumption increases by 16% find the effect on the revenue. a. 3% increase b. 2% increase c. 5% decrease d. 4% decrease e. None of these 56. A% of B is B% of? a. A b. B c. (A+B) d. (A-B)% e. None of these

122 57. The price of a radio is cut by 20%. To restore it to the former value the new price must be increased by a % b. 20 % c. 25 % d. 50 % e. None of these 58. In an examination 65% of the students passed in Science, 68% passed in Maths and 25% failed in both subject. Find the pass percentage? a. 50% b. 58 % c. 60 % d. 75 % e. None of these 59. In an examination 75% of the students passed in English and 55% passed in Science and 33% failed in both the subjects. Find the pass percentage? a. 50% b. 58 % c. 63 % d. 75 % e. None of these 60. If A% failed in subject 1, and B% is failed in subject 2, and C% failed in both the subject. Find the pass percentage a. [100-(A-B-C)] % b. [100+(A+B-C)] % c. [100-(A-B+C)] % d. [100-(A+B-C)] % e. None of these 61. A reduction of 40% in the price of oranges would enable a person to get 2 dozen more for Rs. 5. Find the reduced price per dozen a. Re. 4 b. Re. 1 c. Re. 2 d. Re. 3 e. None of these 62. Two numbers are 40% and 44% less than a third number. How much percent is the second number less than the first? a. 5% b. 6.5% c % d % e. None of these 63. The salary of a person is first increased by 20% and then reduced by 20% what is net change in his salary? a. 3% increase b. 4% increase c. 5% decrease d. 4% decrease e. None of these 64. If the price has fallen by 25%, what percent of its consumption be increased so that the expenditure may be the same as before? a % b. 20 % c. 25 % d. 50 % e. None of these 65. The salary of Arun is first decreased by 20% and then is increased by 30%. Find the net effect? a. 3% increase b. 4% increase c. 5% decrease d. 4% decrease e. None of these 66. Selling price of a T.V. is increased by 10%. As a result of it, the sale is decreased by 20%. Find the effect on sales a. 13% increase b. 14% increase c. 12% decrease d. 14% decrease e. None of these 67. If Y s salary is 20% more than X s and Z s salary is 10% less than Y s. How many percent Z s salary is more than X s? a. 5% b. 8% c. 6% d. 7% e. None of these

123 68. Y s income is 30% more than X s and Z s income is 40% more than Y s. How much percent is Z s salary is more than X s? a. 50% b. 85% c. 82% d. 90% e. None of these 69. Price of Tea is increased by 50%. By how much, new price must be reduced to restore it to the original price? a % b. 20 % c. 25 % d. 50 % e. None of these 70. Price of Coffee is reduced by 25% by how much new price must be increased by to restore it to original price? a % b. 20 % c. 25 % d. 50 % e. None of these 71. When 70 is subtracted from 70% of a number, the result is 70. Find the number a. 400 b. 800 c. 250 d. 200 e. None of these 72. The price of cooking gas has increased by 50% the percentage of reduction that a family should effect in the use of gas so as not to increase the expenditure on this account is a % b. 20 % c. 25 % d. 50 % e. None of these 73. The price of oil is increased by 25%. If the expenditure is not allowed to increase, the ratio between the reduction in consumption and original consumption is a. 5:1 b. 1:5 c. 3:2 d. 2:3 e. None of these 74. The population of a village is 1600 in It increases 20% seats annually. How many students will be there in 1999? a b c d e. None of these 75. x is six times as large as y. The percentage by which y is less than x is a % b. 80 % c. 82 % d. 85 % e. None of these 76. One litre of water is evaporated from 20 litres of solution containing 5% salt. Percentage of salt in the remaining solution is a % b. 20 % c. 25 % d. 50 % e. None of these 77. Which number is 70% less than 90? a. 30 b. 33 c. 27 d. 36 e. None of these 78. A number exceeds 30% of itself by 60. The number is a. 85 % b. 85 % c. 85 i d. 85 & e. None of these e g g g 79. What percent of 5% of 10% a % b. 20 % c. 25 % d. 50 % e. None of these 80. Subtracting 8% of A from A is equivalent to multiplying A by how much? a b. 0.5 c d. 0.8 e. None of these

124 Answers 1. C 2. C 3. C 4. B 5. B 6. C 7. D 8. D 9. B 10. D 11. B 12. C 13. A 14. B 15. A 16. B 17. D 18. C 19. B 20. C 21. C 22. A 23. B 24. B 25. D 26. D 27. A 28. C 29. C 30. C 31. B 32. B 33. D 34. A 35. A 36. D 37. D 38. A 39. C 40. A 41. A 42. B 43. B 44. D 45. C 46. C 47. A 48. A 49. C 50. A 51. A 52. D 53. A 54. C 55. D 56. A 57. C 58. B 59. C 60. D 61. B 62. C 63. B 64. A 65. B 66. C 67. B 68. C 69. A 70. A 71. D 72. A 73. B 74. E 75. A 76. B 77. C 78. B 79. D 80. A

125 Profit & Loss Cost Price: The price, at which an article is purchased, is called its cost price, abbreviated as C.P. Selling Price: The price, at which an article is sold, is called its selling prices, abbreviated as S.P. Profit or Gain: If S.P. is greater than C.P., the seller is said to have a profit or gain. Loss: If S.P. is less than C.P., the seller is said to have incurred a loss. IMPORTANT FORMULAE 1. Gain = (S.P.) - (C.P.) 2. Loss = (C.P.) - (S.P.) 3. Loss or gain is always reckoned on C.P. 4. Gain Percentage: (Gain %) Gain x 100 Gain % = C.P. 5. Loss Percentage: (Loss %) Loss x 100 Loss % = C.P. 6. Selling Price: (S.P.) SP = 7. Selling Price: (S.P.) (100 + Gain %) x C.P 100 (100 - Loss %) SP = x C.P Cost Price: (C.P.) 100 C.P. = x S.P. (100 + Gain %) 9. Cost Price: (C.P.) 100 C.P. = x S.P. (100 - Loss %) 10. If an article is sold at a gain of say 35%, then S.P. = 135% of C.P. 11. If an article is sold at a loss of say, 35% then S.P. = 65% of C.P. 12. When a person sells two similar items, one at a gain of say x%, and the other at a loss of x%, then the seller always incurs a loss given by: Loss % = Common Loss and Gain % 2 = x 2.

126 If a trader professes to sell his goods at cost price, but uses false weights, then Error Gain % = x 100 (True Value) - (Error) %. If shopkeeper marks the item by m% (of CP) and gives d% discount then, Profit % = m d m d 100 Solved examples Ex. 1. On selling an article for Rs. 240, a trader incurs a loss of 4%. At what amount must he sell the article for so that he makes a profit of 10%? Sol. S.P. = Rs. 240 Loss = 4% 0.96C.P = 240 C.P = 250 Now trader wants to earn 10%. Thus new S.P = = Rs K Short Cut method: = $ where S 1 = Rs. 240, x 1 = 4%, x 2 = 10% KMMPOK KMMPO$ Solving we get S 2 = Rs Ex. 2. Mohan sold two books at Rs. 140 each. He made a profit of 20% on selling one of the books, and incurred a loss of 20% on selling the other. What is the outcome of selling both the books? Sol. Let the combined cost price of both articles be Rs. Y. Selling price of both articles is Rs Overall loss % = O 5 KMM % = (400/100) = 4 % (where x = 20) 0.96 Y = 280 Y = Rs Overall loss = = Rs = Rs. 12 (appx.) Ex. 3. If the SP of 12 articles = CP of 10 articles, what is the loss %? Sol. There is a loss of 2 articles for every 12 articles Therefore % loss = 2/ = 16.66%. Ex. 4. If a toy is sold for Rs , there would be a loss of 10%. At what price should it be sold to earn a profit of 20%? Sol. According to the given condition: 0.9x = 10.8 (where x is the cost price of the toy) x = 12 Selling price to gain 20% = = 14.4 Ex. 5. A cycle is sold for Rs. 880 at a loss of 20%. At what price should it be sold so that there

127 is a gain of 10%? Sol. S.P. = Rs. 880 Loss = 20% Let C.P. be x. Then 0.8x = 880 x = Rs Now gain = 10% New S.P. = = Rs. 121 Ex. 6. By selling a transistor for Rs. 572, a shopkeeper earns a profit equivalent to 30% of its cost price. Find its cost price. Sol. Let C.P. = x Given S.P. = 572 Profit = &M KMM x = 0.3x C.P. + Profit = S.P. x + 0.3x = x = 572 x = Rs. 440 Ex. 7. A milkman buys 10 equal sized milk drums. If he sells milk at Rs. 5 per litre, he loses Rs. 200 but if he sells it at Rs. 6 a litre, he will gain Rs What is the capacity of each drum in litres? Sol. Let total milk in one vessel be x litre. Milk in 10 vessels will become 10 x. For Rs. 5 per litre, total milk cost = 10x 5 = 50x For Rs. 6 per litre, total milk cost = 10x 6 = 60x From the given information, we can say that 60x 50x = x = 350 x = 35 litres Ex.8. A businessman marks his goods in such a way that even after allowing 12.5% discount on a cash purchase he gains 20%. If the cost price is Rs. 140, the marked price is Sol. 12.5% (1/8) discount is on the marked price (M.P) S.P = 7/8 M.P (1) Also C.P = Rs. 140 Profit = 20% S.P. = = 168 (2) From (1) and (2) 7/8 M.P = 168 M.P = Rs Ex. 9. A dealer marks articles at a price that gives him a profit of 30%. 6% of the goods were lost in a fire, 24% were soiled and had to be sold at half the cost price. If the remainder were sold at the marked price, what percentage profit or loss did the dealer earn? Sol. Let say total article = x

128 Let say the cost of each article = Rs. 100 Marked price = 130 Selling price of soiled article (24%) = x $i 50 = 12 x KMM Selling price of rest of the article = x gm 130 = 91 x KMM Total selling price = 91x + 12x = 103 x Profit = &O 100 = 3%. KMMO Ex. 10. From a batch of iron ore, 80% iron can be extracted. The cost of the ore is Rs.0.30 per kg and the cost to produce iron from 100 kg of ore is Rs.40. If each kg of iron is sold at Re.1, what is the profit percentage? Sol. If we take 100 kg ore, it costs Rs.30 and production cost is Rs.30. From this we will get 80 kg of iron So, to get 80 kg of iron, Rs.70 is required. By selling this 80 kg iron at Re. 1 per kg, it comes Rs.80 So percentage profit = emngm 100 = 14.28% gm Ex. 11. A shopkeeper is able to make a profit of 25% even after allowing a discount of 25%. How much profit will he be able to make by not allowing any discount? Sol. Let x be the M.P. and y be the C.P. So, if he gives 25% discount, his S.P. will be 0.75x Since he got 25% profit, his S.P. should be 1.25y 0.75x = 1.25y x = % & y 66.66% more than y i.e. the M.P. is 66.66% more than the C.P. Ex. 12. A businessman purchased an article and marked its price as 30% more. After allowing 10% discount, he earned Rs. 1,020 as profit. What is the cost price of the article? Sol. Let C.P. of the article be Rs. x. So, M.P. will become 1.3x. After giving 10% discount, his S.P. = x. = 1.17x Profit = 0.17x But the profit is given as 1,020, 0.17x = 1,020 x = Rs. 6,000 Ex. 13. A man purchased a packet of 10 pens and got 2 pens free with it. He marked the price 20% more than the C.P. If he did not allow any discount and sold all the 12 pens at this price, what was his profit percentage? Sol. Assume C.P. of each pen is Re. 1. So, C.P. is Rs. 10 ( he purchased 10 pens).

129 M.P. of each pen is Rs and he has 12 pens now. So, he will get 12 (1.20) = Rs The percentage profit = Ki.iNKM 100 = 44% KM Ex. 14. The cost price of 20 articles is the same as the selling price of x articles. If the profit is 25%, then the value of x is: Sol. Let C.P. of each article be Re. 1 C.P. of x articles = Rs. x. S.P. of x articles = Rs. 20. Profit = Rs. (20 - x) x x 100 = 25 x x = 25x 125x = 2000 x = 16. Ex. 15. In a certain store, the profit is 320% of the cost. If the cost increases by 25% but the selling price remains constant, approximately what percentage of the selling price is the profit? Sol. Let C.P.= Rs Then, Profit = Rs. 320, S.P. = Rs New C.P. = 125% of Rs. 100 = Rs. 125 New S.P. = Rs Profit = Rs. ( ) = Rs Required percentage = 295 x 100 = 1475 % = 70% (approximately) Ex. 16. The percentage profit earned by selling an article for Rs is equal to the percentage loss incurred by selling the same article for Rs At what price should the article be sold to make 25% profit? Sol. Let C.P. be Rs. x. Then, x x 100 = x x 100 x x x = x x = 3200 x = 1600 Required S.P. = 125% of Rs = Rs. 125 x 1600 = Rs Ex. 17. Sam purchased 20 dozens of toys at the rate of Rs. 375 per dozen. He sold each one of them at the rate of Rs. 33. What was his percentage profit? Sol. 375 Cost Price of 1 toy = Rs. = Rs Selling Price of 1 toy = Rs. 33

130 So, Gain = Rs. ( ) = Rs Profit % = 1.75 x 100 = 28 % = 5.6% Ex. 18. When a plot is sold for Rs. 18,700, the owner loses 15%. At what price must that plot be sold in order to gain 15%? Sol. 85 : = 115 : x x 115 x = = Hence, S.P. = Rs. 25,300. Ex. 19. A trader mixes 26 kg of rice at Rs. 20 per kg with 30 kg of rice of other variety at Rs. 36 per kg and sells the mixture at Rs. 30 per kg. His profit percent is: Sol. C.P. of 56 kg rice = Rs. (26 x x 36) = Rs. ( ) = Rs S.P. of 56 kg rice = Rs. (56 x 30) = Rs Gain = x 100 = 5% Ex. 20. A sold a watch to B at a gain of 5% and B sold it to C at a gain of 4%. If C paid Rs. 1,092 for it, the price paid by A is Sol. Given C paid Rs. 1,092 Let A paid Rs. X Then amount paid by B to A = Rs. 1.05X Amount paid by C to B = X = 1,092 X = Rs. 1,000 Short Cut method: From options: Let A paid Rs. 1,000 Then B paid to A = 1,000 % + 1,000 = Rs. 1,050 KMM Moreover, C paid to B = 1, i It satisfies the given condition. KMM = Rs Ex. 21. A sells an article to B at a profit of 20%. B sells it to C at a profit of 15%. C paid Rs.190 more than it cost A. What profit did A make? Sol. Let the article cost Rs.X to A A s selling price = B s cost price = X 1.2 B s selling price = C s cost price = X Now, X X = X X = X = 190 X = KjM KMM = 500 &e A s profit = 20% of 500 = Rs.100

131 Ex. 22. A tradesman, by means of a false balance, defrauds 10% in buying goods and 10% in selling. What % does he gain on his outlay by his dishonesty? Sol. Assume the tradesman buy 100 gm for Rs Since he defrauds 10%, he will get 110 gm for Rs While selling, he will give only 90 gm for Rs So, on 110 gm, he can earn KMM KKM = , so, the profit is 22.22% jm Ex. 23. A cloth merchant says that due to slump in the market, he sells the cloth at 10% loss, but he uses a false metre scale and actually gains 15%. Find the actual length of the scale. Sol. Please note that in this question he is gaining and losing at the same time. It means his selling price and cost price will be different. Now he is gaining 15% on the actual scale he is using (false scale) and he is showing his loss on the true scale of one metre. Let false scale length = L cm. According to the given conditions 1.15 L = L = jm 100 = cm. KK% Ex. 24. A pharmaceutical company produces 6,000 bottles of cough syrup for Rs. 132,000. It distributed 20% of the bottles to the doctors as a sample and sold 2/3 of the remaining bottles at 25% discount. The remaining bottles were sold to shopkeepers at 10% discount on the printed price of the bottle that was 50% more than the cost price. Find the profit/loss. Sol. Cost price = K&$MM hmmm = Rs. 22 Selling Price = 22 K%M KMM = 33 1,200 bottles are given for free. Selling price of rest of the bottles = 4,800 $ & g% SP of the remaining = 1, jm = 47,520 KMM Total S.P. = 126,720 Loss (%) = %$em 100 = 4% K&$MMM KMM 33 = 79,200 Ex. 25. A fruit vendor purchased 100 apples from wholesale market at Rs. 50 each. If 10 of them were damaged, what price should be charged on the remaining apples to earn 20% profit? Sol. His investment = = 5,000 If he wants to get 20% profit, he has to earn Rs. 6,000 on 90 apples. So, each apple costs hmmm = Rs jm Ex. 26. There are two types of iron ore A and B. Ore A has 80% iron and B has 60% iron. The cost of 1 kg of ore A is Rs and of ore B is Rs (Assume no other product except iron will come from the ore).if the processing cost is negligible and there is no other cost, which of the two ores is more profitable to process? Sol. 100 kg of ore A gives 80 kg of iron and costs Rs kg of ore B gives 60 kg of iron and costs Rs. 120.

132 So, A is better. Ex. 27. By selling 12 mangoes, a fruit seller gains a profit equal to selling price of 2 mangoes. What is his profit percent? Sol. Consider S.P. of a mango as Re.1 P = 2, S.P. = 12, C.P. = 10 Profit = $ 100 = 20% KM Ex. 28. If milk man mixes 1 litre pure water in every 4 litre of pure milk and sold it at the cost price. What is the profit percent earned by him? Sol. Find = % making 25% profit U i Ex. 30. By selling 12 articles, a shopkeeper gains a profit equal to cost price of 2 articles. What is his profit percent? Sol. Assume, CP of one article is Re.1. So, by selling 12 articles (worth Rs.12) he can make a profit of Rs.2 So, profit % = $ 100 = 16.66% K$ Ex. 31. A manufacturer estimates that on inspection, 12% of the articles produced will be rejected. He accepts an order to supply 22,000 articles at Rs each. He estimates the profit on his outlay including the manufacturing of rejected articles, to be 20%. Find the cost of manufacturing each article. Sol. When the manufacturer makes 100 articles, he sells only 88. Revenues = = Rs. 660 But profit being 20% of outlay, total outlay cost of 100 articles is Rs So the cost of per article is Rs.5.50 Ex. 32. There would be 10% loss, if a toy is sold at Rs per piece. At what price should it be sold to earn a profit of 20%? Sol. According to the given condition: 0.9x = 10.8 (where x is the cost price of the toy) x = 12 Selling price to gain 20% = = 14.4 Ex. 33. A sweet maker made 620 sweets at a cost of 80 paise per sweet. Out of these, 120 sweets got spoiled and he sold rest of the sweets in such a way, that when he sold 400 sweets, he would have made a profit of 50% on the total cost of 620 sweets. Find his actual profit % on the total outlay. Sol. Cost of the sweets = = Rs. 496 Let the selling price per sweet = Rs. x Selling price of 400 sweets = 400 x 400x = 496 K%M x = K%M ijh KMM imm KMM = Rs Selling price of 500 sweets = = 930.

133 Profit (%) = i&i 100 = 87.5 ijh Ex. 34. Rajgopalan purchased a second hand TV for Rs. 4,800 and spent 20% of the cost on its repairs. If he wants to earn Rs.1,440 as net profit on it, how much percentage should he add to the purchase price of the TV? Sol. Repair cost = 0.2 4,800 = 960 To earn net profit of Rs. 1,440 he should also recover his repair cost. It means he has to add Rs. 2,400 (1, ) to his purchase price to earn 1,440 as net profit. % markup = (2400/4800) 100 = 50%. Ex. 35. A wholesaler sells 30 pens for the price of 27 pens to a retailer. The retailer sells the pens at their marked price. Find the % gain for the retailer. Sol. Assume the marked price of one pen is Re. 1. So the retailer gets 30 pens for Rs. 27 and he sells those for Rs. 30. So profit = & $g 100 = 11K j % Ex. 36. A man sells an article at a gain of 10%. If he had bought it at 10% less and sold it for Rs.132 less, he would have still gained 10%. The cost price of the article is Sol. First profit = 10% SP = CP + KM of CP = 1.1 CP..(1) KMM New cost price = 0.9 CP New selling price = S.P 132 Now, S.P = CP From (1) and (2) 1.1 CP 132 = 0.99 CP 0.11CP = 132 CP = Rs. 1,200 (2) Alternate method: Let CP 1 = Rs. 100 SP 1 = 110 New CP 2 = 90 SP 2 = 99 Difference in SP = SP 1 SP 2 = 11 (in second case) then original CP = 100. If he is selling far Rs. 132 less then CP = KMM 132 = Rs. 1,200 KK

134 Exercise 1. Kamalesh purchased 120 reams of paper at Rs. 100 per ream and the expenditure on transport was Rs He had to pay an octrol duty of 50 paise per ream and the coolie charges were Rs. 60. What should he charge per ream to gain 40%? a. 147 b. 127 c. 117 d. 137 e. None of these 2. On selling an article at Rs. 530, the gain is 20% more than the loss incurred on selling it at Rs In order to gain 20%, the selling price will be a. 400 b. 200 c. 800 d. 600 e. None of these 3. The marked price of a shirt and trousers are in the ratio of 1:2. The shopkeeper gives 40% discount on the shirt. If the total discount on both is 30%, the discount offered on the trousers is a % b. 20 % c. 25 % d. 50 % e. None of these 4. If a commission of 10% is given on the marked price of a book, the publisher gains 20%. If the commission is increased to 20%, the gain percent is a. 6.66% b. 5 % c. 8 % d. 6.5 % e. None of these 5. The marked price of an article is 20% more than the cost price and a discount of 20% is given on the marked price. In this kind of sale, the seller a % b. 20 % c. 25 % d. 50 % e. None of these 6. A shopkeeper sells a book whose marked price is Rs. 30 at a discount of 15% and gives a pen costing Rs free with each book. Even then he makes a profit of 20%. His cost price per book is a. 30 b. 20 c. 25 d. 50 e. None of these 7. A merchant purchases a calculator for Rs. 500 and fixes the list price in such a way that after allowing a discount of 10%, he earns a profit of 20%. Find the list price of the calculator a b c. 450 d. 500 e. None of these 8. Ramanlal a shopkeeper earns a profit of 33.33% after selling an article at 20% discount on the printed price. Find the ratio of the cost price and the printed of the article a. 5:3 b. 1:5 c. 3:5 d. 5:1 e. None of these 9. A trader has some watches in his stock. He marks his watches 20% above the cost price. He sold half the stock at marked price, one quarter at a discount of 20% on the marked price and the rest at a discount of 40% on the marked price. What is his gain percent? a. 3% b. 2% c. 5% d. 7% e. None of these 10. A sold a watch to B at 40% gain and B sold it to C at a loss of 20%. If C bought the watch for

135 Rs. 432, at what price did A purchase it? a b. 328 c d. 380 e. None of these 11. A shopkeeper allows a discount of 10% on the marked price of an item but charges a sales tax 8% on the discount price. If the customer pays Rs. 729 as the price including the sales tax, then what is the marked price of the item? a. 750 b. 600 c. 500 d. 450 e. None of these 12. The marked price of a watch is Rs The shopkeeper gives successive discounts of 10% and x% to the customer. If the customer pays Rs for the watch, find the value of x. a. 75 b. 10 c. 12 d. 15 e. None of these 13. A dealer sold three-fourth of his articles at a gain of 24% and the remaining at the cost price percentage of gain in the whole transaction is a. 18 b. 10 c. 12 d. 15 e. None of these 14. On a Rs payment order, a person has choice between three successive discounts of 10%, 10% and 30% and three successive discounts of 40%, 5% and 5%. By choosing the better one, he can save a. 275 b. 210 c. 125 d. 255 e. None of these 15. A trader marked the selling price of an article at 10% above the cost price. At the time of selling, he allows certain discount and suffers a loss of 2%. The discount allowed is a. 10 KM K& % 10 b. 10 % 11 j c. 10 % KK KM d. 9 % KK e. None of these 16. Due to 25% fall in the rate of eggs, one can buy 2 dozen eggs more than before by investing Rs What was original rate per dozen? a. Rs $MM & b. Rs $MM j c. Rs d. Rs KMM h e. None of these 17. A shop keeper purchased some books from a publication worth Rs Because of some reasons, he had to sell two-fifth part of the book at a loss of 15%. On which gain he should sell his rest of the books so that he gets neither gain nor loss? a. 10% b. 20 % c. 25 % d. 50 % e. None of these 18. The cost price of an item is two-third of its selling price. What is the gain or loss percent on that item? a b. 20 c. 25 d. 50 e. None of these 19. A bookseller sells a book at a gain of 20%. If he had bought it at 20% less and sell and sold it for Rs. 6 less, he would have gained 20%. The CP of the book is a. 15 b. 20 c. 25 d. 50 e. None of these 20. By selling 32 oranges for Rs. 30 a man loss 25%. How many oranges should be sold for Rs. 24 so as to gain 20% in the transaction? a. 15 b. 20 c. 25 d. 16 e. None of these

136 21. A toy is sold at 10% profit. Had it been sold for Rs.10 less, the profit may be 5% only. Find the cost price. a. 200 b. 600 c. 500 d. 400 e. None of these 22. An article costing Rs.2500 is sold at 20% loss. Find the selling price a b c d e. None of these 23. If selling an article for Rs. 880 causes 10% loss on the selling price. Find its cost price a. 896 b. 869 c. 968 d. 986 e. None of these 24. Find the marked price when selling price = 2880 and discount = 4% a b c d e. None of these 25. Find the rate of discount when marked price=rs. 250 and selling price Rs. 235 a. 3% b. 2% c. 4% d. 6% e. None of these 26. A person purchased a Toy at 25% discount on its marked price but sold it at the marked price. Find the gain percent? a % b. 20 % c. 25 % d. 50 % e. None of these 27. A person allows 4% discount on the marked price of his goods and still earns a profit of 20%. Find the cost price for his marked at Rs. 850? a. 860 b. 680 c. 780 d. 880 e. None of these 28. A man marks his goods at such a price that he can deduct 87.5%cash and yet make 20% profit. What is the marked price of an article which cost minimum him Rs. 140 a. 190 b. 189 c. 192 d. 195 e. None of these 29. A man sells T.V. to B at a gain of 10% and B cells it to C at a gain of 5%. If C pays Rs for it what did it cost to A? a b c d e. None of these 30. A man sells 2 toys at Rs.616 each. On one he gained 12% and on the other he lost 12%. Find his total gain or loss in the transaction. a. Rs. 18 loss b. Rs.16 loss c. Rs.18 profit d. Rs.16 profit e. None of these 31. A man bought a thing and sold it at a gain of 8%. If he had bought it for 10% less and sold it for Rs.240 less he would have made a profit of 16.66%. Find the cost price of the thing a. 860 b. 800 c. 780 d. 880 e. None of these 32. Gain or Loss is always counted on.. a. selling price b. marked price c. cost price d. data inadequate e. None of these

137 33. The retail list price of a radio is Rs.210. The seller allows a discount of 5% for cash and still makes a profit of 14% on his outlay. The manufacturer s price to the retailer is 26% higher than the cost price? a. 125 b. 139 c. 150 d. 110 e. None of these 34. A fan is sold for Rs Had it been sold for Rs. 940 there would have been an additional profit of 2% on the cost price. Find the cost price of the fan a. 860 b. 820 c. 800 d. 880 e. None of these 35. The difference in selling a toy at profit 8% and 12% is Rs.24. Find the cost price of toy a. 860 b. 680 c. 620 d. 600 e. None of these 36. A person buys 40 pencils from a wholesaler at the marked price of 36 pencils. If the person sells at the marked price, find his profit percent a. 11 $ % b. j 111 K % c. 11 % d. 9 KM 10K % e. None of these j 37. Single discount equivalent to a series of discounts of 20% and 30% is a. 44% b. 40 % c. 45 % d. 50 % e. None of these 38. A radio whose Tag price is Rs is available at 2 successive discounts of 20% and 25%. Find the single equivalent discount a. 44% b. 40 % c. 45 % d. 50 % e. None of these 39. The cost of a picture album is Rs. 500 it is available at two successive discounts of 15% and 20%. Find the amount paid by the customer a. 300 b. 200 c. 340 d. 360 e. None of these Ans: Rs Find single discount rate equivalent to a series of discounts 20%, 25% and 30% a. 58% b. 40 % c. 45 % d. 50 % e. None of these 41. If a shopkeeper allows x% discount on the marked price of an article then the selling price of the article = Ans: (100-x) % of marked price 42. A person allows 30% discount on all his articles, the number of the articles sold is increased by 30%. Find the effect on total sale? Ans: 9% decrease 43. A chair which costs Rs. 800 was sold at a loss of 5%. Find its selling price Ans: Rs. 760

138 44. I sold a pen at a loss of 25% on my outlay Express this loss as a percentage of the selling price a % b. 20 % c. 25 % d. 50 % e. None of these 45. Kabir buys an article with 25% discount on the original price. He makes a profit of 10% by selling it at Rs What is the original price of the article? Ans: Rs Vishnu bought 170 kg of sugar at the rate of Rs per kg and mixed it in 130 kg sugar purchased at Rs per kg. He sold the mixture with a profit of Rs The rate of the mixture per kg was Ans: Rs Subhash bought 20 kg of tea at the rate of Rs. 30 per kg and 30 kg at the rate of Rs. 25 per kg. He mixed the two and sold the mixture at the rate of Rs per kg. What was his loss in this transaction? Ans: Rs The amount of loss incurred by selling an article for Rs. 800 is 50% of the amount gained if it is sold for Rs. 1,400. At what price should it be sold to have 2% profit? Ans: Rs The loss incurred on selling an article for Rs. 270 is as much as the profit made when it is sold at 10% profit. What is the cost price of the article? Ans: Rs If the selling price of 40 articles is equal to the cost price of 50 similar articles, what is the percent gain or loss? a % profit b. 20 % loss c. 25 % loss d. 50 % profit e. None of these 52. Sukumar Chowdhury bought paper sheets for Rs and spent Rs. 100 on transport. Paying Rs. 300, he had 300 boxes made of them, which he sold at Rs. 16 each. What is his profit percent? Ans: Sohanlal purchased 120 reams of paper at the rate of Rs. 100 per ream. The expenditure on transport was Rs He had to pay an octroi duty of 50 paise per ream and the coolie charges were Rs. 60. What should be the selling price of each ream if he wants a profit of 20%? Ans: Rs Neeraj bought a car for a certain sum of money. He spent 10% of the cost of repairs and sold the car for a profit of Rs. 11,000. How much did he spend on repairs if he made a 20% profit? Ans: Rs By selling an article for Rs. 469, Mahesh gained 15%. What was the cost price of the article?

139 Ans: Rs A fruit vendor purchased 35 dozen apples at Rs. 10 per dozen. Out of this 2 dozen got rotten and he sold 28 dozen at Rs. 12 per dozen. Remaining apples were sold by sold by him at Rs. 15 per dozen. What profit did he make in this bargain? Ans: Rs There would be 10% loss if rice is sold at Rs per kg. What price per kg should it be sold to earn a profit of 20%? Ans: Rs Dilip prepared 50 show pieces of sandalwood. The cost of production of each piece was Rs. 10. Apart from this he has to pay Rs. 5 per piece as commission to the selling agent and Re. 1 per piece to the workshop assistant. What should be the selling price of each article if he wished to sell them at 30% profit? Ans: Rs Samir purchased 24kg of wheat at the rate of Rs per kg and 26kg of wheat at the rate of Rs per kg. He mixed the two and sold the mixture. Approximately at what price per kg should he sell the mixture to make 30 percent profit in this transaction? Ans: Rs If the chairs bought at price Rs. 300 to Rs. 450 are sold at prices ranging from Rs. 400 to Rs. 525, what is the maximum possible profit that might be made in selling 10 chairs? Ans: Rs The selling price of 50 articles is equal to the cost price of the same 60 articles. What is the percent profit or loss? a % profit b. 20 % profit c. 25 % profit d. 50 % profit e. None of these 62. A shopkeeper purchases 18kg of sugar at the rate of Rs per kg and 22 kg at the rate of Rs per kg. He mixes the two and sells the mixture at the rate of Rs per kg. What will be his gain or loss in the transaction? Ans: Rs loss 63. In a factory there are A grade and B grade employees. Each employee contributes as many rupees as the number of employees in his grade. The employer contributes Rs If the total contribution is Rs. 839 and total contribution made by grade B employees is Rs. 25, how many employees are there in the factory? Ans: A shop keeper earns a profit of Rs. 60 by selling an article. Had he sold the same article at Rs. 520, the profit percent would have been 30. At what price did he sell the article? Ans: Rs. 460

140 65. Prabhat bought 20 kg of rice at the rate of Rs per kg and 30 kg of rice at the rate of Rs per kg. He mixed the two for selling the mixture. Approximately at what price per kg should he sell the mixture to get 25% profit on the cost price? Ans: Rs By selling an article Sheetal earned a profit equal to the K i sold it at Rs. 375, what was the cost price? Ans: Rs. 300 of the price he bought it. If he 67. The percent profit made when an article is sold for Rs. 78 is twice as much as when it is sold for Rs. 69. What is the cost price of the article? Ans: Rs Pujari bought 20 quintals of rice at Rs. 250 per quintal and 30 quintals at Rs. 280 per quintal. He mixed the two and sold the mixture for Rs. 270 a quintal. What was his gain in the deal? Ans: Rs Prakash bought a suitcase with 25% discount on the price. At what price should he sell the suitcase if he wants to make a profit of 25% on his purchase price? Ans: Can t be determined 70. A grain merchant bought 50kg of wheat at the rate of Rs. 7 per kg and 20 kg at the rate of Rs. 8 per kg. He mixed both and sold at the rate of Rs. 10 per kg. What is his total profit? Ans: Rs The profit earned by selling a watch for Rs. 850 is as much as the loss incurred when it is sold for Rs What is the cost price of the watch? Ans: Rs A shopkeeper gives a discount of 25% on the marked price of an article and earns a profit of 20% on his outlay. If he had not given any discount, then the profit on his outlay would have been Ans: 60% 73. A fruit-seller sells mangoes at the rate of Rs. 50 for 10 mangoes. The number of mangoes purchased by the fruit-seller for Rs. 50 in order to get a profit of 60% is Ans: By selling goods for Rs. 380, A loses 5 percent on his outlay. The price at which A must sell them to gain 10 percent, would be Ans: Rs I sold my table for Rs. 144 earning profit percent equal to the cost price of the table numerically. The cost price of the table is Ans: Rs. 80

141 76. A person buys a book for Rs. 200 and sells it for Rs What will be his gain percent? Ans: 12.5% 77. A calculator is bought for Rs. 350 and sold at a gain of 15%. What will be the selling price of calculator in rupees? Ans: If the cost price is 95% of the selling price. What is the profit percent? Ans: The owner of a cell phone shop charges his customer 28% more than the cost price. If the customer paid Rs for the cell phone, what was the cost price of the cell phone? Ans: Rs Selling price of an article for Rs and the percent profit earned is 20%. What is the cost price of the article? Ans: Rs The profit earned after selling a pair of shoes for Rs is the same as the loss incurred after selling the same pair of shoes for Rs What is the cost price of the shoes? Ans: Rs A man sells an article at a profit of 40%. If he had bought it at 40% less and sold for Rs. 5 less, he would have gained 50%. Find the cost price of the article. Ans: Rs Charu purchased a dinner set at & Find the gain percent Ans: 10 KM of its selling price and sold it at 10% more than its SP. 84. Rajan sold an article for Rs at a loss of 25%. Find the cost price Ans: Rs The owner of a furniture shop charges his customer 18% more than the CP. If a customer paid Rs for a dining table, find its original price Ans: Rs Meena purchased two fans each at Rs She sold one fan at the loss of 5% and other at the gain of 10%. Find total gain of loss percent Ans: 2.5% profit 87. Kapil sold two bikes for Rs each. On one he gains 14% and on the other, he losses 14%, find his gain or loss percent in this whole transaction Ans: 1.96% loss

142 88. If the cost price of 20 articles is equal to the selling price of 15 articles. Find the profit percent Ans: 33.33% 89. A tradesman marks his goods at 20% above the cost price but allows purchasers a discount of 5%, What profit percent does he make? Ans: 14% 90. Profit earned by selling an article for Rs is same as the loss incurred by selling the article for Rs What is the cost price of the article? Ans: Rs If a man wants to sell his chair for Rs. 720 he gets 25% loss. To gain 25%, he should sell if for Ans: Rs The SP of an article after two successive discounts of 10% and 5% on marked price is Rs Find the marked price Ans: Rs The difference between the CP and SP of an article is Rs If the profit is Rs. 20%, the selling price is Ans: Rs Pankal purchased an item for Rs and sold it at the gain of 24%. From that amount he purchased another item and sold it at the loss of 20%. What is his overall gain/loss? Ans: Loss of Rs Sumit purchased an item for Rs and sold it at the gain of 35%. From that amount, he purchased another item and sold it at the loss of 20%. What is his overall gain/loss? Ans: Gain of Rs. 320

143 Time, Speed & distance General Information The terms Time and Distance are related to the speed of a moving object. General Formulae 1. Speed: It is obtained by dividing the distance covered by the object, by the time it takes to cover that distance. od`_^yq\ _]^z\{{\c Thus, Speed = d[\ _^s\y Distance = Speed Time Time = od`_^yq\ "\\c 2. Conversion of Units: One kilometer / hour = One meter / second = Ke km/hr. % Thus, x km/hr = x % m/sec. Ke And, x m/sec = x Ke % km/hr KMMM [_`. = % m/sec. hm hm `\q. Ke 3. (i) If the time taken is constant, the distance travelled is proportional to the speed, that is, more the speed; more the distance travelled in the same time. (ii) If the speed is constant, the distance travelled is proportional to the time taken that is more the distance travelled; more the time taken at the same speed. (iii) If the distance travelled is constant, the speed is inversely proportional to the time taken, that is, more the speed; less the time taken for the same distance travelled. 4. If two persons A and B start at the same time from two points P and Q towards each other and after crossing they take T1 and T2 hours in reaching Q and P respectively, then mš` `"\\c n Š` = $ `"\\c K 5. If a body travels d1, d2, d3,.. dn meters with different speeds s1, s2, s3,.., sn m/sec. in time T1, T2, T3,., Tn seconds respectively, then the average speed of the body throughout the journey is given by V^ = W_^{ cd`_^yq\ _]^z\{{\c W_^{ _d[\ _^s\y = ckpc$pc&p...pcy KP $P &P P Y 6. If the ratio of the speeds of A and B is a: b, then the ratio of the time taken by them to cover the same distance is K^ : K or b: a p

144 Some Important Shortcut Formulas Rule 1: If some distance is travelled at x km/hr and the same distance is travelled at y km/hr then the average speed during the whole journey is given by Rule 2: If a person travels a certain distance at x km/hr and returns at y km/hr, if the time taken to the whole journey is T hours then the one way distance is given by Rule 3: If two persons A and B start their journey at the same time from two points P and Q towards each other and after crossing each other they take a and b hours in reaching Q and P respectively, then Rule 4: If the same distance is covered at two different speeds S 1 and S 2 and the time taken to cover the distance are T 1 and T 2, then the distance is given by Ex. 1. A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour? Sol. 600 Speed = 5 x 60 m/sec. = 2 m/sec. Converting m/sec to km/hr (see formulas section) = 2 x 18 km/hr 5 = 7.2 km/hr. Ex. 2. An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 1 2 hours, it must travel at a speed of: 3 Sol. Distance = (240 x 5) = 1200 km. Speed = Distance/Time Speed = 1200/(5/3) km/hr. [We can write 1 $ hours as 5/3 hours] & Required speed = 1200 x 3 = 720 km/hr. 5 Ex. 3. Walking at three-fourth of his usual speed, a man covers a certain distance in two hours more than the time he takes to cover the distance at his usual speed. The time taken by him to cover the distance with his usual speed is: [SSC 2000]

145 Sol. New speed = & of usual speed i Old speed = o New time = old time (T) +2 od`_^yq\ New speed = Ž\ _d[\ 3 4 D T = D T + 2 3(T+2) = 4T 3T+6 = 4T T = 6 hours Alternate method: Let us take time taken by actual speed as T hours. New speed = & of usual speed i New time taken = i of normal time & Here new time = T+2 (T+2) = i T & 3T+6 = 4T T = 6 hours Ex. 4. The ratio between the walking speeds of A and B is 2 : 3 and therefore, A takes 10 minutes more than B to reach the destination. If A had walked at double the speed, he would have covered the distance in Sol. Let speed of A and B be 2x and 3x m/s respectively. Time taken by A and B to cover the same distance is 3x and 2x respectively. 3x 2x = 10 x = 10 A takes 30 minutes. When the speed is doubled, A takes 15 minutes Ex. 5. Two cyclists meet each other at 10 a.m. on Ferozepur road. After their meeting, one of them proceeded in the north direction and the other proceeded in the east direction. Exactly at noon, they were 60 km apart. If the difference between their speeds is 6 km/hr, find the speed of the faster cyclist. Sol. Let speed of the slower one be x km/hr. So, speed of the faster one will be (x + 6) km/hr. The distance travelled in 2 hr is 2x and 2(x + 6) km.

146 From the Pythagoras Theorem: (2x) 2 + [2(x + 6)] 2 = (60) 2 4x 2 + 4(x x) = x x = 3600 x 2 + 6x 432 = 0 By solving this, x = 18 km/hr So, the speed of the faster cyclist is 24 km/hr. Ex. 6. A bullock-cart overtakes two persons walking in the same direction, in a village, at the rate of 3 kmph and 5 kmph and passes them completely in 12 and 18 seconds respectively. Calculate the length of the bullock-cart (in metres). Sol. Length of bullock-cart = Difference of speed = 2 % K$ Ke = 20 m Ke h [Z{_d"{dq^_dWY WX _d[\ cdxx\]\yq\ WX _d[\ Ex. 7. A man can complete a journey in 10 hrs. He travels the first half of the journey at the speed of 21 km/hr and the second half at the speed of 24 km/hr. Find the total distance of the journey in kilometres. Sol. First half speed = 21 km/hr Second half speed = 24 km/hr Total time = 10 hrs Average speed = $ $K $i $KP$i = $ $K $i i% = KK$ % = 22.4 km/hr Distance covered = S T = = 224 km Ex. 8. A man wants to cover 50 km on his bicycle. His speed is 12.5 km/hr. After every 12.5 km, he takes rest for 20 minutes. How long will he take to cover the whole distance? Sol. Speed = 12.5 km/hr Distance to be covered = 50 km Time required = %M K$.% 10 = KMM K$% = 4 hr Rest intervals = 20 min 3 = 60 min = 1 hr (After the third stoppage, he will reach his destination) Total time = = 5 hr Ex. 9. A car takes 5 hours to cover a distance of 300 km. What is the required speed of the car in km/hr to cover the same distance in 4/5 th of the previous time? Sol. The earlier speed = 300 km/5 hr or 60 km/hr The earlier time = 5 hr The new time = i % th of previous time = i % 5 = 4 hr Now he has to cover 300 km in 4 hr. The new speed = &MM = 75 km/hr. i Ex. 10. A man rode a certain distance at the speed of 25 km/hr and walked back at the speed of 4 km/hr. The whole journey took him 5 hours 48 minutes. What distance did he cover in all? Sol. Let the total distance be 2x km.

147 Then (x/25) + (x/4) = 5 48 / 60 = 5 4 / 5 = 29/5 29x/100 = 29/5 x = 20 km. Total distance covered = 2x = 40 km Ex. 11. A man travels a certain distance at the speed of 20 km/hr and returns to the same point at the speed of 30 km /hr. His average speed during the whole journey is Sol. Average speed = $ $M &M %M = K$M % = 24 km/hr Ex. 12. A started his journey on bicycle at 7.30 a.m. at a speed of 8 km/hr. B started from the same point half an hour later, but at a speed of 10 km/hr. At what time did B overtake A? Sol. When B starts, A would have covered = 8 ½ = 4 km in half an hour. Relative speed for B = 10 8 = 2 km/hr Time taken by B to overtake A = i = 2 hr. $ B overtook A at = 10 a.m. Ex. 13. A car travels a distance of 840 km at a uniform speed. If the speed of the car is increased by 10 km/hr, it takes two hours less to cover the same distance. What is the original speed of the car? Sol. Let the original speed of the car be x km/hr According to the given condition [840/(x+10) - (840/x)] = 2 As per given options, let x be 60 km/hr Time taken = eim = 14 hr hm Now, increased speed shall be = = 70 km/hr Time = eim = 12 hr. gm Difference in time = = 2 hr Ex. 14. The speed of A and B is in the ratio: 2: 3. A takes 10 minutes more than the time taken by B to reach the destination. If A had walked at double the speed, he would have covered the same distance in Sol. Given, both of them are travelling the same distance. Speed will be inversely proportional to time. A and B will take 3x min and 2x min respectively to cover the same distance. Also 3x 2x = 10 x = 10 A will take 30 min. to cover the distance. If A doubles his speed, then time taken will be15 min. Ex. 15. A train covers a certain distance in 50 minutes at the speed of 48 km/hr. What should be the speed of the train to complete the journey in 40 minutes? Sol. Given 40 : 50 : : 48 km/hr : x km/hr Where x is the speed of the train Time taken is 40 min. x = %M im = 60 km/hr im

148 Ex. 16. Excluding stoppages, the speed of a bus is 45 kmph and including stoppages it is 36 kmph. For how many minutes does the bus stop per hour? Sol. Bus will travel 45 km without stoppages and 36 km with stoppages in one hour Difference of 9 km s time is wasted in stoppages. Time wasted in stoppages per hour = j hours = 12 minutes. i% = K % Ex. 17. A ship went on a voyage. After it had travelled 180 miles, a plane started with 10 times the speed of the ship. Find the distance (in miles) when they will meet from starting point. Sol. Let speed of the ship = x and Time traveled after traveling 180 m but Now, according to the question, xt = 10xt On solving, we get xt = 20 Total distance traveled = = 200 m. Ex. 18. A ball moves at 120 meters per second and strikes an object in three seconds. If it moves at 100 meters per second, how long does it take to strike the same object? Sol. Total distance covered by ball = = 360 m Now, speed of ball = 100 m/s Time taken = &hm = Ke m/s. KMM % Ex. 19. Tony started for his office at a speed of 40 kmph. If he travels at this speed, he can reach his office exactly on time. But after travelling 20 km, he realized that he had forgotten an important document back at home. So, he went back home at a speed of 60 kmph, collected the document and again started for his office at a speed of 80 kmph and reached his office exactly on time. What is the distance between his home and his office? Sol. If the distance is d km, he needs c hrs to reach his office. im So, c = $M + $M + c im im hm em d = $MM = 66 $ km & & Ex. 20. Brown started 2 hours after John started and overtakes him in 4 hours. If the difference between the speed of John and Brown is 20 kmph, what is the distance travelled by Brown, before meeting John? Sol. Brown over takes John in 4 hours. This means that he can cover 4 20 = 80 km in this time. That means in 2 hours before Brown starts John has covered 80 km. So, John s speed is 40 kmph and that of Brown is 60kmph, So, the distance travelled by Brown = 60 4 = 240 km. Ex. 21. A man wants to cover 50 km on his bicycle. He covers 12.5 kmph. After every 12.5 km, he takes rest for 20 minutes. How long will he take to cover the whole distance? Sol. Speed = 12.5 km/hr

149 Distance to be covered = 50 km Time required = %M %MM 10 = = 4 hours K$.% K$% Rest intervals = 20 min 3 = 60 min = 1 hour (after third stoppage he will reach his destination and there will be no stoppage after that) Total time = = 5 hours. Ex. 22. A train starts from Delhi at 1 : 20 p.m. and reaches Indore at 5 : 50 p.m. Second train starts from Indore at 1 : 50 p.m. and reaches Delhi at 7 : 10 p.m. Find the ratio of speeds of the two trains. Sol. Time taken by the first train = 4 hr 30 min = 4.5 hr Time taken by the second train = 5 hr 20 min = Kh hr & Ratio of time taken = j : Kh $ & Ratio in their speeds = Kh = 32 : 27 & : j $ Problem on trains I. When train crosses a pole or any standing thing with negligible length II. III. IV. When train crosses a running man in same direction When train crosses a running man in opposite direction When train crosses a platform or any standing thing with some length L t S t L _ = length of train S _ = speed of train L t S t S m L _ = length of train S _ = speed of train S [ = speed of man L t S t + S m L _ = length of train S _ = speed of train S [ = speed of man L t + L p S t L _ = length of train S _ = speed of train L " = length of platform

150 V. When train crosses another running train in same direction VI. When train crosses another running train in opposite direction L t1 + L t2 S t1 S t2 L _K = length of train 1 L _$ = length of train 2 S _K = speed of train 1 S _$ = speed of train 2 L t1 + L t2 S t1 + S t2 L _K = length of train 1 L _$ = length of train 2 S _K = speed of train 1 S _$ = speed of train 2 Relative Speed Case1: Two bodies are moving in opposite directions at speed V K & V $ respectively. The relative speed is defined as V ] = V K + V $ Case2: Two bodies are moving in same directions at speed V K & V $ respectively. The relative speed is defined as V ] = V K + V $ Ex. 1. A train traveling at 72 kmph crosses a platform in 30 seconds and a man standing on the platform in 18 seconds. What is the length of the platform in meters? Sol. When the train crosses a man standing on a platform, the distance covered by the train is equal to the length of the train. However, when the same train crosses a platform, the distance covered by the train is equal to the length of the train plus the length of the platform. The extra time that the train takes when crossing the platform is on account of the extra distance that it has to cover = length of the platform. Therefore, length of the platform = speed of train extra time taken to cross the platform Length of platform = 72 kmph 12 seconds Converting 72 kmph into m/sec, we get 72 kmph = % 72 = 20 m/sec Therefore, length of the platform = = 240 meters. Ex. 2. A train traveling at 100 kmph overtakes a motorbike traveling at 64 kmph in 40 seconds. What is the length of the train in meters? Sol. When a train overtakes another object such as a motorbike, whose length is negligible compared to the length of the train, then the distance traveled by the train while overtaking the motorbike is the same as the length of the train. The length of the train = distance traveled by the train while overtaking the motorbike Ke

151 = relative speed between the train and the motorbike time taken In this case, as both the objects i.e., the train and the motorbike are moving in the same direction, the relative speed between them = difference between their respective speeds = = 36 kmph. Distance traveled by the train while overtaking the motorbike = 36 kmph 40 seconds. The final answer is given in meters and the speed is given in kmph and the time in seconds. So let us convert the given speed from kmph to m/sec. 1 kmph = 5/18 m/sec Therefore, 36 kmph = 36 5 /18 = 10 m/sec. Relative speed = 10 m/sec. Time taken = 40 seconds. Therefore, distance traveled = = 400 meters Ex. 3. A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is: Sol. Speed of the train relative to man = 125 m/sec 10 = 25 m/sec. 2 = 25 x 18 km/hr 2 5 = 45 km/hr. Let the speed of the train be x km/hr. Then, relative speed = (x - 5) km/hr. x - 5 = 45 x = 50 km/hr. Ex. 4. The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is: Sol. Speed = 45 x 5 m/sec = 25 m/sec Time = 30 sec. Let the length of bridge be x metres. Then, x = (130 + x) = 750 x = 245 m. Ex. 5. Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is: Sol. Let the speeds of the two trains be x m/sec and y m/sec respectively. Then, length of the first train = 27x metres, and length of the second train = 17y metres.

152 27x + 17y = 23 x+ y 27x + 17y = 23x + 23y 4x = 6y x = 3. y 2 Ex. 6. Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is: Sol. Let the length of each train be x metres. Then, distance covered = 2x metres. Relative speed = (46-36) km/hr = 10 x 5 m/sec 18 = 25 m/sec 9 2x = x = 100 x = 50. Ex. 7. Two trains are moving in opposite 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in seconds is: Sol. Relative speed = (60+ 90) km/hr 5 = 150 x m/sec 18 = 125 m/sec. 3 Distance covered = ( ) km = 2 km = 2000 m. 3 Required time = 2000 x sec = 48 sec. 125 Ex. 8. A jogger running at 9 kmph alongside a railway track in 240 metres ahead of the engine of a 120 metres long train running at 45 kmph in the same direction. In how much time will the train pass the jogger? Sol. Speed of train relative to jogger = (45-9) km/hr = 36 km/hr. = 36 x 5 m/sec 18 = 10 m/sec. Distance to be covered = ( ) m = 360 m.

153 Time taken = 360 sec = 36 sec. 10 Ex. 9. A 270 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train? Sol. Relative speed = ( ) km/hr = 200 x 5 m/sec 18 = 500 m/sec. 9 Let the length of the other train be x metres. Then, x = x = 500 x = 230. Ex. 10. Two trains, each 100 m long, moving in opposite directions, cross each other in 8 seconds. If one is moving twice as fast the other, then the speed of the faster train is: Sol. Let the speed of the slower train be x m/sec. Then, speed of the faster train = 2x m/sec. Relative speed = (x + 2x) m/sec = 3x m/sec. ( ) = 3x 8 24x = 200 x = So, speed of the faster train = 50 m/sec 3 = 50 x 18 km/hr 3 5 = 60 km/hr. Ex. 11. A man steals a car at 1: 30 pm and drives at 40 kmph. At 2 pm, the owner starts chasing his car at 50 kmph. At what time will he catch the man? Sol. Distance covered by the thief in 1 hour = 40 km Distance covered in K h = 20 km $ od`_^yq\ Now, time taken to catch the thief = ]\{^_dz\ `"\\c Relative speed = = 10 kmph (Both are moving in same direction) Time taken to catch the thief = $M = 2 hours. KM Time = 4 p.m ( 2 p.m. + 2 hours)

154 Ex. 12. A train is going at 48 kmph. A passenger counts the telegraph posts on the line as he passes them. If the posts are 50 metres apart, how many posts will he cross per minute? Sol. Speed of train = 48 kmph = 48 5/18 m/sec = ie % hm m/min.= 800 m/min Number of posts = 800/50 = 16 Ex. 13. Find the time taken by two trains, one 210 m long and the other 270 m long, to cross each other, if they are running at speeds of 46 kmph and 54 kmph respectively in the same direction. Sol. Relative speed = = 8 kmph. Distance to be covered = = 480 m Time = 0.480/8 = 0.06 hours. = 216 s. Ex. 14. Two trains start at the same time from station A and station B and proceed towards each other at the rate of 16 km and 21 km per hour respectively. When they meet, it is found that one train has travelled 60 km more than the other. The distance between the two stations is Sol. Distance between two stations = c `šp`5 = 60 $KPKh $KNKh &g = 60 = 444 km. % `š `5 Ke Ex. 15. Two persons start from A and B with the speed of 25 kmph and 49 kmph respectively towards each other. After they cross each other, the person from B covers 145 km to reach A. What is the distance (in km) covered by A & B? Sol. Let the total distance between A and B be x. Since both the persons are starting at the same time therefore time taken by them to meet will be same. Ratio of speed = ratios of distance 25 : 49 : : 145 : (x 145). Therefore, x = km. Ex. 16. Brown started 2 hours after John started and overtakes him in 4 hours. If the difference between the speed of John and Brown is 20 kmph, what is the distance travelled by Brown, before meeting John? Sol. Brown over takes John in 4 hours. This means that he can cover 4 20 = 80 km in this time. That means in 2 hours before Brown starts John has covered 80 km. So, John s speed is 40 kmph and that of Brown is 60kmph, So, the distance travelled by Brown = 60 4 = 240 km. Ex. 17. A train of length 120 metre overtakes another 100 meter long train in 8 seconds. If the speed of the slower train is 27 kmph, what is the speed of the faster train? Sol.

155 t = 8 sec --- (1) V 1 = 27 km/h = 27 x % m/s = 7.5 m/s Ke Speed of faster train =V 2 Net speed = V 2 - V 1 -- (2) From (1) & (2) V 2 - V 1 = $$M = 27.5 e V 2 = 35 m/s = 35 x Ke km/h = 126 km/hr % Ex. 18. A train crosses a platform in 60 seconds, travelling at the speed of 54 kmph. If the length of the platform is 500 metres, what is the length of the train? Sol. t = 60 sec Speed = 54 kmph = 15 m/s l 1 = 500 l 2 =? 15 = { šp{ 5 = %MMP{ 5 hm hm l 2 = = 400 m Ex. 19. A goods train runs at the speed of 72 kmph and crosses a 250 m long platform in 26 seconds. What is the length of the goods train? Sol. Speed = 72 x 5 m/sec = 20 m/sec. 18 Time = 26 sec. Let the length of the train be x metres. Then, x = x = 520 x = 270. Ex. 20. Two trains of length 115 m and 110 m respectively run on parallel rails. When running in the same direction the faster train passes the slower one in 25 seconds, but when they are running in opposite directions with the same speeds as earlier, they pass each other in 5 seconds. Find the speed of each train? Sol. Let the speed of trains be x m/sec and y m/sec respectively. When they move in same direction their relative speed is : x - y When they moves in opposite direction their relative speed is : x + y

156 On solving two equations x=27 m/s and y=18 m/sec Ex. 21. Two trains 110 m and 100 m in length respectively are running in same directions, one at the rate of 100 km / hr and the other at the rate of 64 km / hr. At what time faster train will clear other train from the moment they meet? Sol. Since trains are running in same direction, so relative speed = = 36 km / hr = [ 36 * ( 5 / 18 )] = 10 m / sec Total length to be travelled = = 210 m Ex. 22. A train passes a standing pole on the platform in 5 seconds and passes the platform completely in 20 seconds. If the length of the platform is 225 meters. Then find the length of the train? Sol. Let the length of the train is x meter So speed of train =( x / 5 ) m / sec Also speed of train = ( x ) / 20 m/sec Ex. 23. After travelling a certain distance, a train develops a snag and decreases its speed to half its original speed and reaches its destination 45 minutes late. Had the snag occurred 30 km further on, it would have reached its destination 15 minutes earlier. What is the speed of the train? Sol. Let the initial speed be S 1 and the later speed be S 2. When speed decreases by ½ the usual speed, then for any constant distance, time taken would be twice the usual. In condition (I), the usual time taken to travel CB = t CB = 45 minutes and in condition (II), the usual time taken to travel DB = t DB = = 30 minutes. Since in condition (I) and condition (II), the time taken to travel DB is the same, the usual time taken to travel CD = t CD = t CB t DB = = 15 minutes. The usual time taken to travel CD, t CD = 15 min. The usual speed = Uo = &M šž kmph = 120 kmph _ œ Ÿ} The usual speed is 120 kmph. Ex. 24. Two trains start at the same time from stations A and D and proceed towards each other at 36 kmph and 42 kmph respectively. When they meet, it is found that one train has travelled 48 km more than the other. The distance between the two stations (in km) is Sol. Let train from A cover a distance of x km and train from D cover x + 48 km. Time = O &h = OPie i$ 42x = 36x

157 x = Kg$e = 288 h Total distance = = 624 km Ex. 25. A train starts from X towards Y which is at a distance of 55 km at a speed of 40 km per hour. After running a certain distance, it increases its speed to 50 km per hour and reaches Y in 1 hour 15 minutes after leaving X. After what time did the train change its speed? Sol. Let the train increase its speed after covering the distance 'a' km. According to the question = a 55 a = 5a a a = 30 km Time to cover 'a' = &M = 0.75 hour = 45 min im Boats and streams Downstream/Upstream: In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream. Important facts: 1) If the speed of a boat in still water is u km/hr and the speed of the stream is vkm/hr, then: Speed downstream = (u + v) km/hr. Speed upstream = (u - v) km/h 2) If the speed downstream is a km/hr and the speed upstream is b km/hr, then: Speed in still water = K (a + b) km/hr $ Rate of stream = K (a b) km/hr $ Some Shortcut Methods Rule 1: A man can row certain distance downstream in t 1 hours and returns the same distance upstream in t 2 hours. If the speed of stream is y km/h, then the speed of man in still water is given by

158 Rule 2: A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by Rule 3: A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then the distance is given by Rule 4: A man can row in still water at x km/h. In a stream flowing at y km/h, if he rows the same distance up and down the stream, then his average speed is given by Ex. 1. A man can row a boat in upstream at 6 kmph and downstream at 12 kmph. Find the man's rate in still water and the rate of current. Sol. Ex. 2. A man can row 15 km /hr in still water. It takes him twice as long to row upstream as to row downstream. Find the rate of stream? Sol. Let man's rate in upstream be x km/hr, then his rate in downstream is = 2x kmph So rate in downstream= 2x= 20 km/hr Ex. 3. A man can swim 6 km / hr in still water. If river's rate is 2 km / hr, it takes the man 3 hours to row to a place and back. How far is the place? Sol. Man's rate in upstream=( 6-2 ) km/hr = 4 km/hr Man's stream rate in downstream= ( ) km/hr = 8 km/hr

159 Let the distance of the place be x km then, Ex. 4. A boat covers 24 km upstream and 36 km downstream in 6 hours another time it covers 36 km upstream and 24 km downstream in 13 / 2 hours. Find out the rate of current? Sol. Let the rate of upstream be x kmph, rate of downstream be y kmph, then On adding both equation : On subtracting both equation : On solving both final equations we get x=8 and y=12, now Ex. 5. A river flows at 1.5 kmph and a boatman who can row at 2.5 kmph on still water, takes 7.5hours to go a certain distance up the stream and return again to the starting point. Find the total distance covered by the boatman. Sol. The speeds of the boatman up and down the stream are 1 kmph and 4 kmph respectively. If the distance covered each way = x km, total time to go up and return = O + O = 7.5 hr. K i i.e. % i x = K% $ x = K% $ i % = 6 km The distance covered both ways is 12 km Ex. 6. A boy rows a boat upstream for a distance of 9 miles when the stream is flowing at 2 m/h. Then, he rows back with the current. The whole trip takes 6 hours. What will be the boy's rowing speed in still water? Sol. Let X m/h be the speed of the boat in still water. According to the given condition j + j = 6 ON$ OP$ 2X 2 6X 8 = 0 X 2 3X 4 = 0 (X 4) (X + 1) = 0 X = 4 or X = 1 (inadmissible) The required speed is 4 m/h Ex. 7. A man rows 5 km/hr in still water. If the river is flowing at 1 km/hr, it takes him 75 minutes to row to a place and back. How far is the place? Sol. Let the distance of the place be x km. According to the given condition:

160 O h + O i = 75/60 4(2x + 3x) = 60 5x = 15 x = 3 km. Ex. 8. A boatman takes his boat upstream from place A to place B, where AB = 21 km. Then, he returns to place A. He takes 10 hr in all. The time taken by him 7 km downstream is equal to the time taken by him 3 km upstream. Find the velocity of the river. Sol. Let the speeds upstream and downstream be S u and S d respectively. 21/S u + 21/S d = 10 7/S d = 3/S u Solving the two equations simultaneously, we get S d = 7 kmph S u = 3 kmph Let the speed of the boat and the current be x km/hr and y km/hr respectively. x + y = 7 x y = 3 Solving the two equations simultaneously, we get x = 5 km/hr and y = 2 km/hr. Ex. 9. The speed of a boat in still water is 10 m/s and the speed of the stream is 6 m/s. If the boat is moving upstream and back downstream, what is the ratio of the time taken to cover a particular stretch of distance in each direction? Sol. Given that the distance travelled in each direction is the same, and the effective speeds are in the ratio (10 6) : (10 + 6) = 1 : 4, The time taken to travel upstream and downstream is in the ratio (1/1): (1/4) = 4: 1. Ex. 10. A motorboat can travel at 10 km/hr in still water. It travelled 91 km downstream and then returned, taking 20 hours altogether. What is the rate of flow of the river? Sol. Let speed of the stream be x km/hr 91/(10 x) + 91/(10 + x) = 20 20/(100 x 2 ) = 20/ x 2 = 91 x 2 = 9 x = 3 km/h Ex. 11. A boat takes 6 hours to travel from A to B upstream. If the river current flows at the rate of 2 kmph, how long will the boat take to travel from B to A (downstream), the distance being 36 km? Sol. Ups terms t = 6 hr V 1 = x V 2 = 2 km /hr distance = 36 V 1 - V 2 = &h h = 6 V 1 = 8 km/hr

161 Down terms V 1 = 8 V 2 = 2 net = V 1 + V 2 = 10 km/hr Distance = 36, T = &h = 3.6 hrs KM Exercise 1. Raja has to leave office exactly after 20 minutes to catch a train which leaves station 6.40 p.m. and its 5.45 p.m. now. How much time does it take to travel from office the station? Ans: 35 minutes 2. Balu started in a car from Cochin to Trivandrum at 11 a.m. He travelled at 60 km per hour and covered half the distance in two hours At what uniform speed should he travel to reach Trivandrum at 4.00 p.m.? Ans: 40 km/hour 3. The charges of hired car are Rs. 3 per km for the first 60 km. Rs. 5 per km for the next 60 km and Rs. 10 per km for the further journey. If the balance amount left over with Atul is K KM less than what he paid towards the charges of the hired car for travelling 200 km. how much money did he have initially with him? Ans: Rs A travels a certain distance at a speed of x km per hour and an equal distance at the speed of y km per hour. Then the average speed of A for the whole journey (in km) per hour is Ans: 2xy/(x+y) 5. I ascend a mountain with a speed of 3 km per hour. Then I immediately return from the summit and run downwards with a speed of 6 km per hour. Then my average speed for the round trip is Ans: 4 km/hour 6. A and B are walking in the same direction. A who is an ahead walks at the rate of 3.5km an hour while B walks at the rate of 4 km per hour. The distance between then is 6 km. The B overtakes A in Ans: 12 hours 7. A man riding a cycle at 12 km per hour can reach a village in 4.5 hours. If he is delayed by 1.5 hour at the start, then in order to reach his destination in time, he should ride with a speed of Ans: 18 km/hour 8. A boy goes to school with a speed of 3 km/hour and returns with a speed of 2 km/hour. If he takes 5 hours in all, then the distance (in km/hour) between the village and the school is: Ans: 6

162 9. A boy walks to school at the rate of 3 km per hour and reaches school at 7.35 a.m. Then he walks at 5 km per hour and reaches at 7.25 a.m. If his school time is 7.30 a.m. how far is his school from his house? Ans: 1.25 km 10. A car is travelling is at the rate of 50km per hour whereas train is moving at 20 meters per second. Their respective speeds are in the ratio Ans: 25: A thief steals a scooter and drives of 60 km per hour. The policeman discovers the theft after 15 minutes and start chasing him at the rate of 75 km per hour his motor-cycle. After what time the thief caught by the police man? Ans: one hour 12. A car covers a distance of 690 km in 30 h. What is the average speed of the car? Ans: 23 km/h 13. A bus is running with a uniform speed of 37 km/h. What distance will be covered by bus in 8 h? Ans: 296 km 14. A person riding a bike crosses a bridge with a speed of 54 km/h. What is the length of the bridge if he takes 4 min to cross the bridge? Ans: 3600 m 15. A man covered a distance of 12 km in 90 min bycycle. How much distance will he cover in 3 h if he rides the cycle at a uniform speed? Ans: 24 km 16. A man completes 30 km of a journey at 6 km/h and the remaining 40 km of the journey in 5 h. Find the average speed for the whole journey Ans: 7 km/h 17. A certain distance is covered at a certain speed. If half of the distance is covered in double time, the ratio of the two speeds is Ans: 4:1 18. Moving h of its usual speed a train is 10 min late. Find its usual time to cover the journey g Ans: 60 min 19. A car covers a distance from town A to town B at the speed of 58 km/h and covers the distance from town B to town A at the speed of 52 km/h. What is the approximate average speed of the car? Ans: 55 km/h

163 20. A car reached Raipur from sonagarh in 35 min with an average speed of 69 km/h. If the average speed is increased by 36 km/h, how long will it take to cover the same distance? Ans: 23 min 21. A student walks from his house at 2.5 km/h and reaches his school late by 6 min. Next day, he increases his speed by 1 km/h and reaches 6 min before school time. How far is the school from his house? Ans: g i km 22. John started from A to B and vinod from B to A. If the distance between A and B is 125 km and they meet at 75 km from A. What is the ratio of John s speed to that of vinod s speed? Ans: 3:2 23. A man covers half on his journey at 6 km/h and the remaining half at 3 km/h. Find his average speed Ans: 4 km/h 24. The speeds of A and B are in the ratio of 3:4. A takes 20 min more than the time taken by B to reach a destination, in what time does A reach the destination? Ans: 1.33 h 25. A is twice as fast as B and B is thrice as fast as C. The journey covered by C in 56 min will be covered by A in Ans: 9.33 min 26. The ratio of speeds of a train and a car is 16:15 respectively and a bus covered a distance of 480 km in 8 h. The speed of the bus in & th of the speed of train. What distance will be covered i by car in 6 h? Ans: 450 km 27. The ratio between the speeds of two buses is 5:3. If the 1st bus runs 400 km in 8 h, then find the speed of the 2nd bus. Ans: 30 km/h 28. If sohail walks from his home to office at 16 km/h, he is late by 5min. If he walks at 20 km/h, he reaches 10 min before the office time. Find the distance of his office from his house Ans: 20 km 29. A car covers a distance of 200 km in 2 h 40 min whereas a jeep covers the same distance in 2 h. What is the ratio of their speeds? Ans: 3:4 30. A person goes from one point to another point with a speed of 5 km/h and comes back to

164 starting point with a speed for 3 km/h. Find the average speed for the whole journey. Ans: 3.75 km/h 31. A bus travels at the rate of 54 km/h without stoppages and it travels at 45 km/h with stoppages. How many minutes does the bus stop on an average per hour? Ans: 10 min 32. A car travels a distance of 75 km at the speed of 25 km/h. It covers the next 25 km of its journey at the speed of 5 km/h and the last 50 km of its journey at the speed of 25km/h. What is the average speed of the car? Ans: 15 km/h 33. A car runs at the speed of 40 km/h when not serviced and runs at 65 km/h when serviced. After servicing the car covers a certain distance in 5 h. How much approximate time will the car take to cover the same distance when not serviced? Ans: 8 h 34. An express train travelled at an average speed of 100 km/h, stopping for 3 min after every 75 km. A local train travelled at a speed of 50 km/h, stopping for 1 min after every 25 km. If the trains began travelling at the same time, how many kilometers did the local train travel in the time in which the express train travel 600 km? Ans: km 35. If you travel 39 km at a speed of 26 km/h, another 39 km at a speed of 39 km/h and again 39 km at a speed of 52 km/h. What is your average speed for the entire journey? Ans: 37 km/h 36. A and B are two stations 1000 km apart. A train starts from A and moves towards B at 40 km/h. Another train starts from B at the same time and moves towards A at 60 km/h. How far from A will they cross each other? Ans: 400 km 37. X and Y start walking towards each other at 8 am at the speeds of 3 km/h and 4 km/h, respectively. They were initially 17.5 km apart. At what time do they meet? Ans: am 38. A train leaves Manipur at 6 am and reaches Dispur at 10 am. Another train leaves Dispur at 8 am and reaches Manipur at 11:30 am. At what time do the two trains cross trains cross each other? Ans: 8:56 am 39. The average speed of a car is 75 km/h. The driver first decreases its average speed by 40% and then increases it by 50%. What is the new average speed now? Ans: 67.5 km

165 40. A car runs at the speed of 50 km/h when not serviced and runs at 60 km/h, when serviced. After servicing, the car covers a certain distance in 6 h. How much time will the car take to cover the same distance when not serviced? Ans: 7.2 h 41. A car driver covers a distance between two cities at a speed is 60 km/h and on the return his speed is 40 km/h. He goes again from the 1st to the 2nd city at twice that original speed and returns at half the original return speed. Find his average speed for the entire journey. Ans: 40 km/h 42. The distance between two stations X and Y is 450 km. A train L starts at 6 pm from x and moves towards y at an average speed of 60 km/h. Another train M starts from y at 5:20 pm and moves towards x at an average speed of 80 km/h. How far from x will the two trains meet and at what time? Ans: 170 km, 8:50 pm 43. Two cars A and B are running towards each other from two difference places 88 km apart. If the ration of the speeds of the cars A and B is 5:6 and the speed of the car B is 90 km/h, after what time will they meet each other? Ans: 32 min 44. If a 200 m long train crosses a platform of the same length as that of the train in 20 seconds, then the speed of the train is Ans: 72 km/h 45. A train 150 m long is running at a speed 90 kmph. Time taken by the train to cross a tree is Ans: 6 sec 46. A 100 m long train crosses a stationary man in 10 seconds. What is the speed of the train? Ans: 36 km/h 47. Two trains A and B, travelling in opposite directions, cross each other in six seconds. The speed of train A is 126 km/h, while that of train B is 90 km/h. If the length train A is 160 metres, what is the length of train B? Ans: 200 m 48. A train crosses a stationary person in 3 seconds while it takes 10 seconds to cross a platform, travelling at same speed. If the length of the platform is 210 meters, what is the length of the train? Ans: 90 m 49. A 225 m long train crosses another 150 m long train in 12.5 sec. If one of the trains 72 km/h, what is the speed of the other? Ans: 36 km/h

166 50. What is the sped of a train which overtakes a man walking at a speed of 5 km/h in 30 seconds, if the train is 274 metres long? Ans: km/h 51. Two trains (X, Y) start from A and B respectively, towards each other. After they meet at point C, the train X takes two hours to reach B while the train Y takes 4.5 hour to reach A. If the distance between places A and B is 450 kms, how far should C be from A? Ans: 270 km m long train takes 10 s to pass a man who is going in the same direction at 2 km/h. What is the speed of the train in km/h? Ans: A train starts from station X at the rate of 60 km/h and reaches station Y in 45 minutes. If the speed is reduced by 6 km/h. how much more time will the train take to return from station Y to station X? Ans: 5 min 54. A train takes 18 seconds to pass completely throught a station 162 m long and 15 seconds through another station 120 m long. The length of the train is Ans: 90 m 55. A train 100 metre long completely passes a man walking in the same direction at 6 km/h in 5 seconds and a car travelling in the same direction in 6 seconds. At what speed was the car travelling? Ans: 18 km/h 56. How many seconds will a 500 metres long train take to cross a man walking with a speed of 3 km/h in the direction of the moving train if the speed of the train is 63 km/h? Ans: A train runs at the rate of 120 km/h. The speed in metres per second is Ans: A train running at the speed of 72 km/h crosses a 260 metres long platform in 23 seconds. What is the length of the train in metres? Ans: Train A leaves Mumbai Central station for Lucknow at 11 a.m. running at the speed of 60 km/h. Train B leaves Mumbai Central station for Lucknow by the same route at 2 p.m. on the same day running at the speed of 72 km/h. At what time will the two trains meet each other? Ans: 5 am on the next day 60. A train running at speed of 120 km/h crosses a signal post in 15 seconds. What is the length

167 of the train in metres? Ans: Two trains start from P and Q respectively and travel towards each other at a speed of 50 km/h and 40 km/h respectively. By the time they meet, the first train has travelled 100 km more than the second. The distance between P and Q is Ans: 900 km 62. A train running at g KK saved if the train ran at its own speed? Ans: 8 h of its own speed reached a place in 22 hours. How much time could be 63. A train is 125 m long. If the train takes 30 seconds to cross a tree by the railway line, then the speed of the train is Ans: 15 km/h 64. A train passes two bridges of lengths 800 m and 400 m in 100 seconds and 60 seconds respectively. The length of the train is Ans: 200 m 65. A train 800 metres long is running at the speed of 78 km/h. If it crosses a tunnel in 1 minute, then the length of the tunnel (in metres) is Ans: Two trains one 160 m and the other 140 m long are running in opposite directions on parallel rails, the first at 77 km/h and the other at 67 km/h. How long will they take to cross each other? Ans: 7.5 s 67. Two trains start at the same time, one from New Delhi to Agra, the other from Agra to New Delhi, if they arrive at their respective destinations in one hour and four hours respectively after passing each other, how much faster is one train running than the other? Ans: 2 times 68. A and B are two stations 500 km apart. A train starts from A and moves towards B at the rate of 20 km/h. Another train starts from B at the same time and moves towards A at the rate of 30 km/h. How far from A will they cross each other? Ans: 200 m 69. A passenger train, moving at the rate of 90 km/h overtakes a goods train, 2 times as long and moving on parallel lines at the rate of 50 km/h and passes it completely in 54 seconds. How long would the passenger train take to pass completely through a station 400 meters in length? Ans: 24 s 70. Two trains for Bombay leave Delhi at 6 am and 6.45 am and travel at 100 km/h and 136

168 km/h respectively. How many kilometers from Delhi will the two trains be together? Ans: km 71. Two trains, Calcutta Mail and Bombay Mail, start at the same time from stations Calcutta and Bombay respectively towards each other. After passing each other, they take 12 hours and 3 hours to reach Bombay and Calcutta respectively. If the Calcutta Mail is moving at the speed of 48 km/h, the speed of the Bombay Mail is Ans: 96 km/h 72. Without stoppage, a train travels a certain distance with an average speed of 60 km/h, and with stoppage, it covers the same distance with an average speed of 40 km/h. On an average, how many minutes per hour does the train stop during the journey? Ans: 20 min/h 73. A railway passenger counts the telegraphs poles on the rail as he passes them. The telegraph poles are at a distance of 50 metres. What will be his count in 4 hours, if the speed of the train is 45 km/h? Ans: Two trains A and B start simultaneously in the opposite directions from two points A and B and arrive at their destinations in 9 and 4 hours respectively after their meeting each other. At what rate does the second train B travel if the first train travels at 80 km/h? Ans: 120 km/h 75. Two trains from Howrah leave Muzaffarpur at 8.30 am and 9.00 am respectively and travel at 60 km/h and 70 km/h respectively. How many kilometers from Muzaffarpur will the two trains meet? Ans: 180 km Directions for (Q. No 76 & 77): Answer the following questions on the basis of the information given below: (i) Trains A and B travelling on the same route heading towards the same train destination. Train B has already covered a distance of 220 kms before the train started (ii) The two trains meet each other after 11 hours after the start of train A (iii) Had the trains been travelling towards each other, (from a distance of 220 km) they would have met after one hour 76. What is the speed of train B in km/h? Ans: What is the speed of train A in km/h? Ans: A train meets an accident after moving 30 km. Its speed then comes down to i % th of the original one and consequently runs 45 mins late. Had the accident taken place 18 km farther

169 away, it would have been 36 min late. Find the original speed of the train and the distance between the two stations Ans: 30 km/h, 120 km Directions for (Q. No 79 & 80): Read the following and answer the questions that follow. The Kalinga Express started from Patna to Tata at 7 pm at a speed of 60 km/h. Another train, the Rajadhani Express, started form Tata to Patna at 4 am next morning at a speed of 90 km/h. The distance between Patna to Tata is 800 km 79. How far from Tata will the two trains meet? Ans: 156 km 80. At what time will the two trains meet? Ans: 5:44 am 81. A train 600 m long crosses a pole in 12 s. What is the speed of the train in km/h? Ans: 180 km/h 82. A train moving with a speed of 25 m/s crosses a standing man in 16 s. What is the length of the train? Ans: 400 m 83. A train 130 m long passes a bridge in 20 s moving with a speed of 90 km/h. Find the length of the bridge Ans: 370 m 84. A train 250 m long crosses a platform of equal length in 50 s. Find the speed of the train Ans: 10 m/s 85. A train speeds past a pole is 10 s and speeds past a platform of 300 m in 25 s. Its length in metres is Ans: 200 m 86. A train 100 m long completely passes a man walking in the same direction at 2 km/h in 9 s and a car travelling in opposite direction in 3 s. At what speed was the car travelling? Ans: 78 km/h 87. Two trains of equal length are running on parallel lines in the same direction at 45 km/h and 35 km/h. The faster train passes the slower train in 27 s. The length of each train is Ans: 300 m 88. A man in a train notices that he can count 21 telephone posts in 2 min. If they are known to be 50 m apart, then at what speed is the train travelling? Ans: 30 km/h

170 89. A train of length 150 m takes 40 s to cross a tunnel of length 300 m. The speed of the train is Ans: 40.5 km/h 90. If a train runs at 40 km/hour, it reaches its destination late by 11 minutes. But if it runs at 50 km/hour, it is late by 5 minutes only. The correct time (in minutes) for the train to complete the journey is Ans: A train overtakes two persons who are walking in the same direction in which the train is running, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train (in metres) Ans: Two trains of length 70 m and 80 m are running at speeds of 68 km/hr and 40 km/hr respectively on parallel tracks in opposite directions. In how many seconds will they pass each other? Ans: A man rows 6 km/h in still water. If the river is running at 3 km/h, it takes him 45 minutes to row to place and back. How far is the place? Ans: km 94. A boat goes 15 km upstream in 80 minutes. The speed of the stream is 5 km/h. The speed of boat in still water is Ans: km/h 95. In a stream, B lies in between A and C such that it is equidistant from both A and C. A boat can go from A to B and back in 6 h 30 minutes while it goes from A to C in 9 h. How long would it take to go from C to A? Ans: 4 h 96. In a stream that is running at 2 km/h, a man goes 10 km upstream and comes back to the starting point in 55 min. Find the speed of the man in still water Ans: 22 km/h 97. A motorcycle whose speed in still water is 15 km/h goes 30 km downstream and comes back in a total 4 hours 30 min. Determine the speed of the stream Ans: 5 km/h 98. Two swimmers, at opposite ends of a 90-foot pool, start to swim the length of the pool, one at the rate of 3 feet per second, the other at 2 feet per second. They swim back and forth for 12 minutes. Allowing no loss of time at the turns, find the number of times they pass each other Ans: 20

171 99. Two ports A and B are 300 km apart. Two ships leave A for B such that the second leaves 8 hours after the first. The ships arrive at B simultaneously. Find the time the slower ship spent on the trip if the speed of one of them is 10 km/h higher than of the other Ans: 20 h 100. In one hour, a boat goes 11 km along the stream and 5 km against the stream. The speed of the boat in still water (in km/hr) is Ans: A man can row upstream at 8 kmph and downstream at 13 kmph. The speed of the stream is Ans: 2.5 km/hr 102. A man rows downstream 32 km and 14 km upstream. If he takes 6 hours to cover each distance, then the velocity (in kmph) of the current Ans: A boat running downstream covers a distance of 16 km in 2 hours while for covering the same distance upstream, it takes 4 hours. What is the speed of the boat in still water? Ans: 6 km/hr 104. A boatman goes 2 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water? Ans: 1 hr 15 min 105. A man can row three-quarters of a kilometer against the stream in minutes. The speed (in km/hr) of the man in still water is Ans: A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is Ans: 3: A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively? Ans: cannot be determined 108. If a boat goes 7 km upstream in 42 minutes and the speed of the stream is 3 kmph, then the speed of the boat in still water is Ans: 13 km/hr 109. A man s speed with the current in 15 km/hr and the speed of the current is 2.5 km/hr. The

172 man s speed against the current is Ans: 10 km/h 110. If a man rows at the rate of 5 kmph in still water and his rate against the current is 3.5 kmph, then the man rate along the current is: Ans: 6.5 kmph 111. A boat can travel with a speed of 13 km/hr in still water. If the speed of the stream is 4 km/hr, find the time taken by the boat to go 68 km downstream Ans: 4 hours 112. Speed of a boat in standing water is 9 kmph and the speed of the stream is 1.5 kmph. A man rows to a place at a distance of 105 km and comes back to the starting point. The total time taken by him is Ans: 24 hours 113. The speed of a boat in still water is 15 km/hr and the rate of current is 3 km/hr. The distance travelled downstream in 12 minutes is Ans: 3.6 km 114. A man can row at 5 kmph in still water. If the velocity of current is 1 kmph and it takes him 1 hour to row to a place and come back, how far is the place? Ans: 2.4 km 115. A boat takes 19 hours for travelling downstream from point A to point B and coming back to a point C midway between A and B. If the velocity of the stream is 4 kmph and the speed of the boat in still water is 14 kmph. What is the distance between A and B? Ans: 180 km 116. A man can row 9.33 kmph in still water and finds that it takes him thrice as must time to row up than as to row down the same distance in the river. The speed of the current is Ans: 4.66 km/hr 117. A boat covers a certain distance downstream in 1 hour, while it comes back in 1 hours. If the speed of the stream be 3 kmph, what is the speed of the boat in still water? Ans: 15 kmph 118. A motorboat, whose speed is 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is Ans: The speed of a boat in still water is 10 km/hr. If it can travel 26 km downstream and 14 km upstream in the same time, the speed of the stream is Ans: 3 km/hr

173 120. A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is Ans: 2 mph 121. A man rows to a place 48 km distance in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. The rate of the stream is Ans: 1 km/hr 122. A boat covers 24 km upstream and 36 km downstream in 6 hours while it covers 36 km upstream and 24 km downstream in 6.5 hours. The velocity of the current is Ans: 2 mph 123. At his usual rowing rate, Rahul can travel 12 miles downstream in a certain river in 6 hours less than it takes him to travel the same distance upstream. But if he could double his usual rowing rate for his 24-mile round trip, the downstream 12 miles would then take only one hour less than the upstream 12 miles. What is the speed of the current in miles per hour? Ans: A man can row upstream at 7 km/h and downstream at 11 km/h. What is the man s rate in still water? Ans: 9 km/h 125. A man can row up stream at 4 km/hr and downstream at 7 km/h. What is the rate of the current? Ans: 1.5 km/h 126. A man rows 25 km downstream and 20 km upstream taking 5 h each time. What is the velocity of the current? Ans: 0.5 km/h 127. A man can row 9 km/h in still water. It takes him twice as long to row up as to row down the river. Find the rate of the stream Ans: 3 km/h

174 Time&Work In ratio and proportion, we learnt that there are two types of proportions: Direct proportion and Indirect proportion. If four numbers a, b, c and d are in direct proportion, then we write it as a: b :: c: d and from the rule: Q) If 4 pencils cost $10, then what will 6 pencils cost? Solution: Number of pencils and their Cost are quantities that are directly proportional. So, we can frame the expression 4 is to 10 is same as 6 is to What Price P? i.e. 4: 10 : : 6: P, then 4 P = 10 6, so P = (10 6)/4 = 15 So, 6 pencils will cost $15 Now, consider the following example: Q) 2 men complete a work in 1 day, then 1 man alone will finish the same work in how many days? Solution: Between Number of men and Amount of work, there is not a direct variation, rather there is an indirect proportion. To finish a same amount of work, More men will take Less time, while Fewer men take More time Now, from the Ratio and Proportion lesson, we know that if four quantities a, b, c and d are in indirect proportion, then we can set the relation as: a varies to b indirectly as c varies to d denoted as below: a: b : : d: c, and from the rule product of means is equal to product of extremes, we write: