Question

Ten years later, A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B?

Answer 1

Let the present age of A be x years and the present age of B be y years.

After 10 years, A’s age will be (x + 10) years and B’s age will be (y + 10) years. Thus using the given information, we have

`x + 10 =2(y +10)`

`⇒ x+ 10 = 2y +20`

`⇒ x- 2y -10 =0`

Before 5 years, the age of A was (x – y) years and the age of B was (y – 5) years. Thus using the given information, we have

`x-5 =3(y-5)`

`⇒ x - 5=3(y-5)`

`⇒ x- 3y +10=0`

So, we have two equations

`x- 2y -10 =0`

`x-3y +10=0`

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

By using cross-multiplication, we have

`x/((-2)xx10-(-3)xx(-10))=(-y)/(1xx10-1xx(-10))=1/(1xx(-3)-1xx(-2))`

`⇒ x/(-20-30)=(-y)/(10+10)=1/(-3+2)`

`⇒ x/(-50)=(-y)/20=1`

`⇒ x=50,y=20`

Hence, the present age of A is 50 years and the present age of B is 20 years.