Question

Five years ago, A’s age was four times the age of B. Five years hence, A’s age will be twice the age of B. Find their present ages.
A. Present age of A=29 years and that of B =11 years
B. Present age of A=25 years and that of B =10 years
C. Present age of A=65 years and that of B =20 years
D. Present age of A=40 years and that of B =10 years

Answer 1

Hint: In order to find the ages of A and B assume some unknown for their present ages and then try to find some algebraic equation and try to solve them in order to find the value of ages.

Complete step-by-step answer:

Let A’s present age = x years

And B’s present age = y years

So, five years ago,

A’s age was = (x-5) years

And B’s age was = (y-5) years

According to the question, five years ago, A’s age was four times the age of B.
\[
   \Rightarrow \left( {x - 5} \right) = 4\left( {y - 5} \right) \\
   \Rightarrow x - 5 = 4y - 20 \\
   \Rightarrow x = 4y - 20 + 5 \\
   \Rightarrow x = 4y - 15..................(1) \\
 \]

So, we have got one algebraic equation.
Five years later,

A’s age will be = (x+5) years

And B’s age will be = (y+5) years

Also according to the question, Five years hence, A’s age will be twice the age of B.
\[
   \Rightarrow \left( {x + 5} \right) = 2\left( {y + 5} \right) \\
   \Rightarrow x + 5 = 2y + 10 \\
   \Rightarrow x = 2y + 10 - 5 \\
   \Rightarrow x = 2y + 5..................(2) \\
 \]

And, we have got two algebraic equations. Now in order to solve them let us substitute the value of x from equation (1) into equation (2)
So, from equation (1) and equation (2) we have
\[
   \Rightarrow 4y - 15 = 2y + 5 \\
   \Rightarrow 4y - 2y = 5 + 15 \\
   \Rightarrow 2y = 20 \\
   \Rightarrow y = 10 \\
 \]

Now, substituting the value of y obtained back to equation (2)
\[
   \Rightarrow x = 2y + 5 \\
   \Rightarrow x = 2\left( {10} \right) + 5 \\
   \Rightarrow x = 20 + 5 \\
   \Rightarrow x = 25 \\
 \]

Hence the present age of $A = 25{\text{ and }}B = 10$

So, option B is the correct option.

Note: In order to solve such a type of question always try to convert the age related problem into an algebraic problem. Always consider the present age of the given persons in terms of some unknown variable and not the age 5 years ago or later. For solving such problems in competitive exams using the values in the options and finding the relation may be an easier and time saving method.