EXERCISE 7.. PROBLEMS ON NUMBERS(SOLVED EXAMPLES IN 1 10)

. PROBLEMS ON NUMBERS

In this section, questions involving a set of numbers are put in the form of a puzzle. You have to analyze the given conditions, assume the unknown numbers and form equations accordingly, which on solving yield the unknown numbers.

Solved examples.

Ex. 1. A number is as much greater than 36 as is less than 86.find the number.

Sol. let the number be x. then, x -36 = 86 –X     2X =86 + 36 = 122    x = 61.

Hence, the required number is 61.

Ex.2. find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number.

Sol. let the number be x. then, 7X -15=2X + 10    5X =25     x=5.

Hence, the required number is 5.

Ex.3. the sum of a rational number and its reciprocal is 13/6. Find the number.

Sol. let the number be x.

Then, x + 1/X = 13/6    X² + 1 / x   = 13/6  6x² - 13X + 6 =0

6X² - 9X -4X + 6 = 0   (3X – 2) (2X -3)  =0.

X = 2/3 or X = 3/2

Hence, the required the number is 2/3 or 3/2

Ex. 4. 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.

Sol. Let the numbers be X and (184 –x) . Then,

X/3 – (184 –x)/7 = 8  7x – 3 (184 –X)  = 168   10X = 720     X =72.

So, the numbers are 72 and 112. Hence, smaller number = 72.

Ex.5. the difference to two numbers is 11 and one-fifth of their sum is 9. Find the numbers.

Sol. let the numbers be X and y. then,

X – Y =11  (I) and 1/5 (x +Y) =9   X +y =45(II)

Adding (I) and (II), we get: 2X = 56 are ?X = 28. Putting x =28 in (I), we get: y=17.

Hence, the numbers are 28 and 17.

Ex.6. if the sum of two numbers is 42 and their product is 437, then find the absolute difference between the numbers.

Sol. let the numbers be X and y. then, x + Y = 42 and XY = 437.

X – y = (x + y)²- 4xy = (42)² - 4 x 437 = 1764 – 1748 =16 =4.

Required difference =4.

Ex. 7. The sum of two numbers is 15 and the sum of their squares is 113. Fin the numbers.

Sol. let the numbers be X and (15 –X).

Then, x²+ (15 –X)² = 113  x² - 30X = 113

2X² -30X + 112 = 0  X² - 15X + 56 = 0

(X – 7) (X – 8)= 0  X = 7 or X = 8.

Ex. 8. The average of four consecutive even numbers is 27. Find the largest of these numbers.

Sol. let the four consecutive even numbers be X, X + 2, X + 4 and X + 6.

Then, sum of these numbers = (27 x 4) = 108.

So, X + (X + 2) +(X +4) + (X + 6) = 108 or 4X = 96 or = 24.

Largest number = (X + 6) = 30.

Ex. 9. the sum of the squares of three consecutive odd numbers is 2531. Find the numbers.

Sol. let the numbers be x, X +2 and X + 4.

Then, x² + (X + 4)²=2531   3X² + 12X – 2511 =0

X² + 4X – 837 = 0   (X -27) (X + 31) = 0   X = 27.

Hence, the required numbers are 27, 29 and 31.

Ex.10. Of two numbers, 4 times the smaller one is less than 3 times the larger one by 5. If the sum of the numbers is larger than 6 times their difference by 6, find the two numbers.

Sol. let the numbers be X an Y, such that X>y.

Then, 3X – 4y = 5…(I) an (X + y) – 6 (X – y) = 6  -5X + 7y = 6

Solving (I) an (II), we get: X = 59 and y + 43.

Hence, the required numbers are 59 and 43.