Question

The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

Answer 1

Let the numbers are x and y. One of them must be greater than or equal to the other. Let us assume that x is greater than or equal to y.

The sum of the two numbers is 8. Thus, we have ` x+y =8`

The sum of the two numbers is four times their difference. Thus, we have

` x + y = 4( x -y)`

` ⇒  x + y = 4x - 4y = 0`

`⇒ 4x - 4y - x - y =0`

` ⇒ 3 x -5 y = 0`

So, we have two equations

` x + y = 8`

`3x -5y = 0`

Here x and y are unknowns. We have to solve the above equations for x and y.

Multiplying the first equation by 5 and then adding with the second equation, we have

`5(x + y)+ (3x - 5y)= 5 xx 8 + 0 `

` ⇒ 8 x  = 40 `

`⇒ x = 40/8` 

`⇒ x = 5`

Substituting the value of x in the first equation, we have

` 5 + y = 8`

`⇒ y = 8 - 5  `

`⇒ y =3`

Hence, the numbers are 5 and 3.