MATHEMATICS IN EVERYDAY LIFE 8

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Kenneth Tyler
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1 MATHEMATICS IN EVERYDAY LIFE 8 Chapter : Playing with Numbers ANSWER KEYS. Generalised form: (i) 8 = EXERCISE. (ii) 98 = (iii) 7 = (iv) = Usual form: (i) = = 7 (ii) = = 9 (iii) 00 x + 0xy + z = 00x + 0y + z = xyz. 9 and 9 are the numbers formed by the digits (i) We know that, ab ba when divided by 9, the quotient is (a b). (rule ) 9 9 when divided by 9, the quotient is 9 =. (ii) ab ba when divided by (a b) the quotient is when divided by (9 ) i.e.,, the quotient is and 97 are the numbers formed by the digits 7 (i) If the sum of these numbers ( ) is divided by, we get = 7 = as the quotient. (ii) If the sum of the numbers ( ) is divided by the sum of digits (7 + 9) =, we get = 7. The given number is 78. = as quotient. Other two numbers obtained by arranging the digits of 78 in cyclic order are 87 and 78. The sum of the numbers = = 09 The sum of the digits 7, 8 and = = 9. (i) When divided by, gives the quotient 9. (ii) When divided by 7, gives the quotient sum of the digits 09 7 = 9 = 7 = 7 (iii) When divided by 9 (sum of the digits), gives the quotient =.. Let abc = 9 and cba = 9 abc cba = 99 (a c) abc cba 9 = 99 (a c) = 9 (a c) = 9 (a c) = 9 ( a c ) 9 = (a c) = (9 ) = 7 = 77. Hence, the required quotient is 77. EXERCISE.. Sum of the numbers in each row, each column, and each diagonal is.. C C C C C C = 9 Mathematics In Everyday Life-8

2 . C C C The given numbers are 0,,,,, and. Since, each side of triangle add up to 9. (i) = 9 (ii) + + = 9 (iii) + + = 9 0. The given numbers are,,,, and. Since, each side of triangle add upto. (i) + + = (ii) + + = (iii) + + =. The given numbers are 0,,,,,,, 7 and 8. Since each side of a triangle adds up to. 8 (i) = (ii) = (iii) = 7 0 EXERCISE.. A number is divisible by, if the sum of digits of that number is (i) The sum of digits of number is + + =, which is not So, the number is not (ii) The sum of digits of number 7 is ( + + 7) =, which is So, the number 7 is (iii) The sum of digits of number 97 is ( ) = 8, which is So, the number 97 is Similarly (iii) 9, and (vi) also Hence, (ii), (iii), (iv) and (vi). If the sum of digits of the given number is divisible by 9, then the given number is divisible by 9. (i) The sum of digits of number 99 is ( ) = 9, which is not divisible by 9. So, the number 99 is not divisible by 9. (ii) The sum of digits of number 77 is ( ) = 8, which is divisible by 9. So, the number 77 is divisible by 9. Similarly, (iii) 9 and (iv) 989 also divisible by 9. Hence, (ii), (iii) and (iv) are divisible by 9.. For a number to be divisible, (sum of digits at odd places) (sum of digits at even places) = 0 or a multiple of. (i) (ii) 98 Sum of digits at odd places = + = Sum of digits at even places = Difference between two sums = = 0 So, the number is divisible by. Sum of digits at odd places = = 7 Sum of digits at even places = Difference between two sums = 7 = Which is multiple of. So, the number 98 is divisible by. (iii) 809 Sum of digits at odd places = = Sum of digits at even places = 0 + = Difference between two sums = = Which is a multiple of. So, the number 809 is divisible by. Similarly (iv) 7 also divisible by. Hence, (i), (ii), (iii) and (iv) are divisible by and (v), (vi) are not divisible by.. x is a multiple of. (x + ) = 0 or multiple of. x + can takes values from 0,,,,.... Since x is a digit, for the value of digit x. x + = x = = 9 x = 9. Answer Keys

3 . y is a multiple of y + = 8 + y is a multiple of y can take values 0, 9, 8, 7,.... Since, y is a digit. y =. 8 + y = 9 y = 9 8. A number is divisible by, if the sum of digit of number is The sum of the digits of number 99 is = 9 = (7 + ). If is subtracted from the number the rest number is exactly Hence, remainder =. 7. 8y is divisible by. ( + 8) (y + ) is 0 or multiples of. 7 y can take values 0,,,.... Since y is a digit. 7 y = 0 y 7 8. y8 is divisible by i.e., it is divisible by and. + + y + 8 is divisible by,, 9,, y can take value,, 9,,.... Since y is a digit. + y =,, 8, y = 0,,, The sum of digits of number 87 is ( ) = 9 which is not 9 = 7 +, if is subtract from the numbers the rest number is exactly Hence, required remainder =.. Q + Q 0 EXERCISE. If Q replaced with 8, then 8 + =, gives as units digit in the sum and = 0 gives 0 as tens place in the given sum. Hence, Q 8.. A B A B B 8 B + = 8 B = 7 and + 7 =, gives as tens place in the sum. Hence, A, B A + 9 A A For, as units digit in the given sum, + A should be. + A = A 8. A B C A B If we take B =. Then =, gives at units digit as B. And if we take A =, then = =, gives as A at tens place of the product. A =, B =, C =. MULTIPLE CHOICE QUESTION. Sum of number = 87 Sum of their digits = = 7 required quotient = 87 7 = Hence, option (a) is correct. If a number is divisible by, the sum of digit of the number is Sum of digits of number is + + = is Sum of digit of number 99 is = is not Sum of digit of number 78 is = 0 is not Sum of digit 8 is = 8 is Thus, (b) 99 and (c) 78 is not Hence, option (b) and (c) is correct.. (ab + ba) is always a multiple of. Hence, option (d) is correct.. If a number is divisible by 9, if the sum of digits is divisible by is divisible of 9, since sum of digits = 8 is divisible by 9.. y is + + y + is a multiple of. 0 + y can take values from 0,,, 9,,.... Since, y is a digit, for least value of y. 0 + y = y = Hence, option (c) is correct. Mathematics In Everyday Life-8

4 . If the units digit of a number is 0, then the number is divisible by and 0. 0 is a number which is divisible by and abc cba is always a multiple of 9 and. Hence, option (a) and (b) is correct. 8. x is divisible by 9. ( + + x) is a multiple of 9. + x can take values from 9, 8, 7,.... Since, x is a digit, for value of x. + x = 9 x = Hence, option (c) is correct. 9. A + A + A = BA If + + = Thus, A =, B = A. Fill in the blanks: MENTAL MATHS CORNER. The quotient when the difference of is divided by 9 is.. If we subtract the ones digit of a number from that number, the remainder is always divisible by, and 0.. The three whole numbers whose product and sum are equal are, and. + + = and =. The two numbers whose product is a one-digit number and the sum is a two-digit number are B. True or False. If a number is divisible by, it must be divisible by 9. (False). If a number divides three numbers exactly, it must divided their sum exactly.. A number is divisible by 8, if it is divisible by both and. (False). The sum of two consecutive odd numbers is always divisible by.. If a number is divisible by any number p, then it will also be divisible by each of the factor of p.. (i) = 0 + (ii) 79 = REVIEW EXERCISE (iii) = 0 + (iv) 9 = (i) 9 = (ii) 78 = (iii) 8 = (iv) 99 = z is a multiple of. ( + + z + ) = 9 + is a multiple of. 9 + z can take values from 0,,, 9,,,.... Since, z is a digit. z = 0,,, 9.. (i) A + B 9 + z = 9,,, 8 If 7 + =, gives as units digit. A 7 Now, 7 + =, therefore, B =. Hence, A 7, B (ii) 9 A A 9 A Here, the ones digit of A A = A, it must be that A =, or. A =, satisfy A A = A, but it is not possible as the product is 9A. Now, A =, we have 9 Clearly, A =, satisfy the give product. Hence, A. Let ab be the two digit number and ba is the number obtaining reversing the digit. ab ba = 9 (a b) ab ba = 8 9 (a b) = 8 a b = 8 9 = a b = Hence, the difference between the digits of the number is. Answer Keys

5 . C C C y 7 is divisible by y = + y is a multiple of 9. + y can take values from 9, 8, 7,.... Since y is a digit. Therefore Hence, y + y = 7 y = HOTS QUESTIONS. abc and cab be the three digit numbers. Now, abc = a 00 + b 0 + c = 00a + 0b + c bca = b 00 + c 0 + a = 00b + 0c + a cab = c 00 + a 0 + b = 00c + 0a + b abc + bca + cab = a + b + c = (a + b + c) abc + bca + cab = (a + b + c) Hence, the sum of abc + bca + cab is divisible by.. A B C D D C B A If we put D = 8. 8 =, gives as units place. And = 8. Let the -digit number be ab. ab = 0a + b...(i) If we multiply ab by 9 we get a -digit number say dce. dce = 9(ab) 00d + 0c + e = 9(0a + b)...(ii) If is written before the -digit number and is written after the -digit number we get a -digit number. i.e., dce. As per given condition, dce = + dce d + 00c + 0e + = + 00d + 0c + e (00d + 0c + e) = + (00d + 0c + e) (0a + ) = + 9(0a + ) 90(0a + ) 9(0a + ) = 9 8(0a + b) = 9 0a + b = 9 from (ii) or ab = 9 from (i) My age is 9 years. Thus, A If C = 7. 7 = 8 and 8 + = gives as tens place. If B =. = and + = 7 as hundreds place. Hence, A =, B =, C = 7, D = 8. Mathematics In Everyday Life-8

MATHEMATICS IN EVERYDAY LIFE 8

MATHEMATICS IN EVERYDAY LIFE 8 MATHEMATICS IN EVERYDAY LIFE Chapter : Square and Square Roots ANSWER KEYS EXERCISE.. We know that the natural numbers ending with the digits,, or are not perfect squares. (i) ends with digit. ends with