Ratio and Proportion. Quantitative Aptitude & Business Statistics

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1 Ratio and Proportion Statistics

2 Ratio and Proportion Ratio: A ratio is a comparison of the sizes of two or more quantities of the same kind of division. If a and b are two quantities of the same kind by division.

3 Ratios can be written, or expressed, three (3) different ways. 1. a to b 2. a:b 3. a b 3

4 a is called the first term or antecedent and b is called the second term or consequent. Because a ratio is a quotient (fraction), its denominator cannot be zero. 4

5 Inverse Ratio One ratio is the inverse of another if their product is 1.Thus a:b is the inverse of b:a and vice versa. 1. A ratio a:b is said to be greater inequality if a>b and less inequality if a<b. 2.The ratio compound of the two ratios a:b and c:d is ac:bd 5

6 3.A ratio is said to be compounded itself is called duplicate ratio. Thus a 2 :b 2 is the duplicate ratio of a:b Similarly,the triplicate ratio of a:b is a 3 :b 3 For example Duplicate ratio of 2:3 is 4:9 Triplicate ratio of 2:3 is 8:27 6

7 4.The sub duplicate ratio of a:b is a : b 5.The sub-triplicate ratio of a:b is 3 a : 3 b For example,duplicate ratio of 2:3 is Triplicate ratio of 8:27 is, 2: : 27 2 : 3 7

8 5.If the ratio of two similar quantities can be expressed as a ratio of two integers,the Quantities are said to be commensurable, otherwise, they are said to be 3 : 2 incommensurable cannot be expressed as the ratio of two integers. 8

9 6.Continued ratio is the relation (or comparison) between the two magnitudes of three magnitudes of three or more quantities of the same kind. the continued ratio of three similar Quantities a,b and c is a:b:c 9

10 For example Continued ratio of Rs.200,Rs.400 and Rs.600 is Rs200:Rs400:Rs.600.= 1:2:3 10

11 Example-1 The monthly incomes of two persons are in the ratio of 4:5 their monthly expenditure are in the ratio 7:9.If each saves Rs.50per month,find their monthly incomes. 11

12 Solution Let the monthly incomes are 4X and 5X If each saves Rs.50.Per month Then expenditures are Rs.(4x-50)and (5x-50) 4x 50 5x 50 = Then X=

13 Hence monthly incomes of the two persons are Rs.4X100(Rs.400)and Rs.5x100(Rs.500) 13

14 Example -2 Find in what ratio will the total wages of the workers of a factory be increased or decreased if there be a reduction in the number of workers in the ratio 15:11and increment in their wages in the ratio 22:25 14

15 Solution Let x be the original number of workers and Rs.Y the average wages per workers Then the total wages before changes=rs.xy After increment,the wages per workers=rs.(25y)/22 15

16 The total wages after changes =(11/15 X) Rs.(25y)/22= Rs.5xy/6. Hence the required ratio in which the total wages decrease is xy:5xy/6=6:5 16

17 Proportion An equality of two ratios is called Proportion. Four quantities a,b,c,d are said to be in proportion a:b=c:d (also written as a:b :: c:d a:b is as to c:d) if a/b =c/d i.e if ad=bc The quantities are a,b,c,d are terms of the proportion ;a,b,c and d are called its first,second,third and fourth terms respectively. 17

18 First and fourth terms called are called extremes. The second and third terms are called means (or middle terms) If a:b =c:d then d is called fourth proportional If a:b=c:d are in proportion then a/b =c/d i.e ad=bc i.e product of extremes =product of means This is called cross product rule. 18

19 Three quantities a,b,c are same kind (in same units) are said to be continuous proportion) if a:b=b:c i.e b 2 =ac If a,b,c are continuous proportion,then middle term b is called then the middle term b is called mean proportional between a and c,a is called the first proportional and c is third proportional. 19

20 Thus, b is the mean proportional between a and c,then b 2 =ac i.e b= ac 20

21 In a ratio a:b,both quantities must be of the same kind while in a proportion a:b=c:d,all the quantities need not be same type. The first two quantities of same kind and last two quantities should be same kind. 21

22 Properties of Proportion if a:b =c:d,then ad=bc If a:b=c:d then b :a=d :c (invertendo) if a:b=c:d then a :c=b :d (Alternendo) if a:b =c:d,then a + b: b=c+d :d (componendo) 22

23 if a:b =c:d then a - b: b=c - d :d (Dividendo) if a:b =c:d then a + b: a - b =c+d :c-d (componendo and Dividendo) 23

24 if a:b=c:d=e:f=.,then each of these ratios (Addendo) is equal to (a + c +e+.):(b +d+ f+.) if a:b=c:d=e :f=.,then each of these ratios (Subtrahendo) is equal to (a- c e-.):(b d- f-.) 24

25 Example -1 Find the value of x if 10/3:x:: 5/2:5/4 Using the cross product rule X*5/2=(10/3)5/4 Or X=(10/3)*5/4=5/3 25

26 Example2 Find the fourth proportional to 2/3,3/7,4 Solution: Let the fourth proportional be X then 2/3,3/7,4 and x are in proportion. Using the cross product rule, (2/3)*x=(3*4)/7 Or X=(3*4*3)/7=18/7 26

27 Example3 If a:b=c:d =2.5:1.5,what are the values of ad: bc and a +c : b+d Solution: we have a/b=c /d =2.5/1.5..(1) From (1) ad=bc or ad/ bc=1:1 Again from (1) a/b=c /d=a + c/ b+d a+c/b+d=2.5/1.5=5/3 =5:3 27

28 Example:4 If a/3 =b/4 =c/7,then prove that a+b+c/c =2 Solution : We have a/3=b/4=c/7=a+b+c/3+4+7 a+b+c/14=c/7 or a+ b +c /c=14/7=2 28

29 Indices If n is a positive integer, and a is a real number,i.e n N and a R (where n is the set of all positive numbers and R is the set of all real numbers), a is used to continue product of n factors each equal to a as shown as bellow: 29

30 a n =a X a X a.to n factors Here a n is a power of a whose base is a and index or power is n. 30

31 Law s of Indices Law.1: a m X a n =a m+n, where m and n are positive integers Law.2: a a m n =a m-n where m and n are positive integers 31

32 Law.3: ( m ) n mn a = a where m and n are positive integers Law.4: where n takes all positive values. n ( ) n n ab = a.b 32

33 Find x,if Solution X X = ( X X ) 1 3 ( 2 2 X X ) = ( X ) X X X = ( X ) = ( X ) 3. 2 x 33

34 (If bases are equal,then power is also equal) ie 3/2=3/2* x X =1 34

35 35 Example =1 a c a c c b c b b a b a x x x x x x

36 36 Example = l nl n l n n mn m n m m lm l m l x x x x x x

37 If X = Then 3X 3-9x=10 37

38 38 Solution ) 3 ( ) (3 ) (3 ) 3 (3 ) ( 3 ) ( = = + b a ab b a b a = + = + + = x X x X x X

39 Logarithms The logarithm of a number to a given base is the index or the power to which the base must be raised to produce the number,i.e to make it equal to the given number. If there are three quantities indicated by say a, X and n, they are related as follows: 39

40 If a x =n, then X is said to be the logarithm of the numbers to the base a', symbolically it can be expressed as follows log a n=x 40

41 Definition of Logarithms Suppose b>0 and b 1, there is a number p such that: log b n = p if and only if b p = n 41

42 Fundamental Laws of Logarithm 1. Logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers to the same base,i.e log a mn=log a m +log a n 42

43 Fundamental Laws of Logarithm 2.Logarithm of the Quotient of two numbers is equal to the difference of the logarithms of the numbers to the same base,i.e log a m n = log a m log a n 43

44 Fundamental Laws of Logarithm 3. Logarithm of the number is raised to the power equal to the index of the power raised by the logarithms of the number to the same base,i.e log m n log m n = a a 44

45 Why Logarithms Logarithms were originally developed to simplify complex arithmetic calculations. They were designed to transform multiplicative processes into additive ones. 45

46 Logarithm Tables The Logarithms of a number consists of two parts,the whole part or integral part is called the characteristic and the decimal part is called the mantissa. Where the former can be known by mere inspectiom,the later has to be obtained from logarithms tables. 46

47 Characteristic The Characteristic of the logarithmic of any number greater than 1 with positive and is one less than the number of digits to the left the decimal point in the given number. 47

48 Characteristic The Characteristic of the logarithm of any number less than one (1)is negative and numerically one more than the number of Zeros to the right of decimal point.if there is no Zero then obviously it will

49 Examples for Characteristic Number Characteristic 37 1(2-1) (4-1) (1-1) (number of Zeros on) 49

50 Examples for Characteristic Number Characteristic

51 Mantissa The mantissa is the fractional part of the logarithm of a given number Number Mantissa Logarithm Log 4597 =6625( (Mean Difference) =

52 Anti logarithms If X is the logarithms of a given number n with a given base then n is called the antilogarithm (anti log) of X to that base. This can be expressed as follows If log a n =X Then n = anti log X 52

53 For Example If log 61720= Then 61720=anti log

54 Example-1 3 Write 2 Solution: = 8 in logarithmic form. log 2 8 = 3 We read this as: the log base 2 of 8 is equal to 3. 54

55 Example-2 Write 4 2 = 16 in logarithmic form. Solution: log 4 16 = 2 Read as: the log base 4 of 16 is equal to 2. 55

56 Write 2 3 Solution: = 1 8 log = 3 in logarithmic form. 1 Read as: "the log base 2 of is equal to -3". 8 56

57 Solve: log 3 (4x +10) = log 3 (x +1) Since the bases are both 3 we simply set the arguments equal. 4x +10 = x +1 3x +10 = 1 3x = 9 57 x = 3

58 Example Solve: log 8 (x 2 14) = log 8 (5x) 58 Solution: x 2 14 = 5x x 2 5x 14 = 0 (x 7)(x + 2) = 0 Since the bases are both 8 we simply set the arguments equal. Factor (x 7) = 0 or (x + 2) = 0 x = 7 or x = 2 continued on the next page

59 Example continued Solve: log 8 (x 2 14) = log 8 (5x) Solution: x = 7 or x = 2 59

60 It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. 60

61 Example If log a bc=x, log b ca=y, log c ab=z prove that x y z = 1 61

62 X+1= log a bc+ log a a=log a abc Y+1= logb c ac+ log b b=log a abc Z+1= log c ab+log c c=log a abc Hence x + 1 y + 1 z

63 1 log abc a + 1 log abc b + 1 log abc c log abc a+ log abc b + log abc c =log abc abc =1 63

64 Multiple Choice Questions 64

65 1 is the mean proportional between 12x 2 and 27y 2. A) 18xy B) 81 xy C) 8 xy D) 19.5 xy 65

66 1 is the mean proportional between 12x 2 and 27y 2. A) 18xy B) 81 xy C) 8 xy D) 19.5 xy 66

67 2.log 32/4 is equal to A) log 32/log4 B) log 32 log4 C)2 3 D) None of these 67

68 2.log 32/4 is equal to A) log 32/log4 B) log 32 log4 C)2 3 D) None of these 68

69 3.The logarithm of a number consists of two parts, the whole part or the integral part is called the and the decimal part is called the. A) Characteristic, Number B) Characteristic, Mantissa C) Mantissa, Characteristic D) Number, Mantissa 69

70 3.The logarithm of a number consists of two parts, the whole part or the integral part is called the and the decimal part is called the. A) Characteristic, Number B) Characteristic, Mantissa C) Mantissa, Characteristic D) Number, Mantissa 70

71 4.The value of (8/27) 1/3 is A) 2/3 B) 3/2 C) 2/9 D) None of these 71

72 4.The value of (8/27) 1/3 is A) 2/3 B) 3/2 C) 2/9 D) None of these 72

73 5.The mean proportional between 1.4 gms and 5.6 gms is A) 28 gms. B) 2.8 gms C) 3.2 gms. D) None of these. 73

74 5.The mean proportional between 1.4 gms and 5.6 gms is A) 28 gms. B) 2.8 gms C) 3.2 gms. D) None of these. 74

75 6.The ratio compound of two ratios 4: 3 and 7: 3 is A) 12:21 B) 28:9 C) 9:28 D) None of these 75

76 6.The ratio compound of two ratios 4: 3 and 7: 3 is A) 12:21 B) 28:9 C) 9:28 D) None of these 76

77 7.The ratio of two quantities is 5: 9. If the antecedent is 25, the consequent is A) 9 B) 45 c) 40 D)None of these 77

78 7.The ratio of two quantities is 5: 9. If the antecedent is 25, the consequent is A) 9 B) 45 c) 40 D) None of these 78

79 8.If p: q = r: s, implies q: p = s: r, then the process is called A) Componendo B) Invertendo C) Alternendo. D) Dividendo 79

80 8.If p: q = r: s, implies q: p = s: r, then the process is called A) Componendo B) Invertendo C) Alternendo. D) Dividendo 80

81 9. log (3 5 7) 2 is equal to A) 2(log 3 + log 5 + log7) B) log ( ) C) 2(log 3 log 5 log 7) D) None of these 81

82 9. log (3 5 7) 2 is equal to A) 2(log 3 + log 5 + log7) B) log ( ) C) 2(log 3 log 5 log 7) D) None of these 82

83 10. The triplicate ratio of 4: 5 is. A) 125: 64 B)16:25 C)64:125 D) None of these 83

84 10. The triplicate ratio of 4: 5 is. A) 125: 64 B)16:25 C)64:125 D) None of these 84

85 Ratio and Proportion THE END

RATIO AND PROPORTION, INDICES, LOGARITHMS

CHAPTER RATIO AND PROPORTION, INDICES, LOGARITHMS UNIT I: RATIO LEARNING OBJECTIVES After reading this unit a student will learn How to compute and compare two ratios; Effect of increase or decrease of