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

${\displaystyle x+10=2(y+10)}$

${\displaystyle \Rightarrow x+10=2y+20}$

${\displaystyle \Rightarrow x-2y-10=0}$

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

${\displaystyle x-5=3(y-5)}$

${\displaystyle \Rightarrow x-5=3(y-5)}$

${\displaystyle \Rightarrow x-3y+10=0}$

So, we have two equations

${\displaystyle x-2y-10=0}$

${\displaystyle x-3y+10=0}$

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

By using cross-multiplication, we have

${\displaystyle \frac{x}{(-2)\times 10-(-3)\times (-10)}=\frac{-y}{1\times 10-1\times (-10)}=\frac{1}{1\times (-3)-1\times (-2)}}$

${\displaystyle \Rightarrow \frac{x}{-20-30}=\frac{-y}{10+10}=\frac{1}{-3+2}}$

${\displaystyle \Rightarrow \frac{x}{-50}=\frac{-y}{20}=1}$

${\displaystyle \Rightarrow x=50,y=20}$

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