Question

The present age of the son is half the present age of his father. Ten years ago, the father was thrice as old as his son. What are their present ages?

Answer 1

Hint: In this question, we need to determine the present ages of the son and the father such that it should satisfy all the given conditions of the question. For this, we will create the equations according to the conditions given in the question with the variables as the age of the father and son and solve them consecutively.

Complete step-by-step answer:
Let the present age of the son and the father be ‘S’ and ‘F’ respectively.
According to the question, the present age of the son is half the present age of his father. So, we can write
$S = \dfrac{1}{2} \times F - - - - (i)$
Now, ten years ago, the ages of the son and the father were given as
Son’s age = (S-10) years and Father’s age = (F-10) years
Also, it is given in the question that ten years ago, the father was thrice as old as his son. So, we can write
$
  F - 10 = 3(S - 10) \\
\Rightarrow F - 10 = 3S - 30 \\
\Rightarrow 3S - F = 30 - 10 \\
\Rightarrow 3S - F = 20 - - - - (ii) \\
 $
Now, solving the equations (i) and (ii) for the present ages of the father and son.
Substituting the value from the equation (i) in the equation (ii), we get
$
\Rightarrow 3S - F = 20 \\
\Rightarrow 3\left( {\dfrac{1}{2} \times F} \right) - F = 20 \\
\Rightarrow \dfrac{1}{2} \times F = 20 \\
 $
On cross-multiplying the terms, we get
$
\Rightarrow \dfrac{1}{2} \times F = 20 \\
  F = 40 \\
 $
Hence, the present age of the father is 40 years.
Now, again substituting the value of the present age of father in the equation (i) to determine the present age of the son.
$
  S = \dfrac{1}{2} \times F \\
   = \dfrac{1}{2} \times 40 \\
   = 20 \\
 $
Hence, the present age of the son is 20 years.

Note: Students must be very careful while establishing the equations for the conditions given in the question. Moreover, here we have taken the present age of the father and son, and that’s why we have subtracted ten from both of them for ten years ago.