Question

Find two numbers such that the sum of thrice the first number and twice the second number is 102. Also, five times the first number exceeds thrice the second number by 18. Find the numbers.

Answer 1

Here, let the two numbers be:

First number = x

Second number = y

According to the given conditions:

(1) Thrice the first + twice the second = 102,

3x + 2y = 102     ...(1)

(2) Five times the first exceeds thrice the second by 18,

5x = 3y + 18     ...(2)

`x = (3y + 18)/5`      ...(3)

Now, substituting equation (3) into equation (1):

`3((3y + 18)/5) + 2y = 102`

`(9y + 54)/5 + 2y = 102`

Multiplying the entire equation by 5:

`5((9y + 54)/5) + 5(2y) = 5(102)`

9y + 54 + 10y = 510

9y + 10y = 510 − 54

19y = 456

y = `456/19`

∴ y = 24

Thus, substituting y = 24 into equation (3):

`x = (3(24) + 18)/5`

`x = (72 + 18)/5`

`x = 90/5`

∴ x = 18

Hence, the numbers are 18 and 24.