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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?
Solution
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.
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