Question

A man is four times as old as his son. After $ 16 $ years, he will be only twice as old as his son. Find their present ages.

Answer 1

Hint:
Whenever we have to solve the problems where we are given the relationship between the ages of the two persons, we need to make the equations and solve them. First we will let the age of the son as $ x{\text{ years}} $ then we can get the relation between his age and man’s age which will give us man’s age $ = 4x $
Now we can similarly get the equation from the second statement where we are given their ages after $ 16 $ years and now we can solve both equations to get their present ages.

Complete step by step solution:
Here we need to solve the problem where we are given the relation between the ages of the son and the man. We need to make the equations where we can relate them with the variable and then solve the two equations. Now we need to think of how we will get the two equations.
So if we read the first statement we are given that man is four times as old as his son.
So we can let that the present age of the son $ = x{\text{ years}} $
So we will get the man’s age as four times as his son’s age which will be $ = 4x{\text{ years}} $
So we get their present ages of son and man as $ x,4x{\text{ years}} $
Now after $ 16{\text{ years}} $ , their ages will be:
Son’s age $ = (x + 16){\text{years}} $
Man’s age $ = (4x + 16){\text{years}} $
We are told that man is twice old as his son after $ 16 $ years and hence we can write the equation as:
 $ 4x + 16 = 2(x + 16) $
Now we can solve this equation and we will get:
 $ 4x + 16 = 2(x + 16) $
 $
  4x + 16 = 2x + 32 \\
  4x - 2x = 32 - 16 \\
  2x = 16 \\
  x = 8 \\
  $
Hence we get that $ x = 8 $

Hence we can say that son’s present age is $ 8{\text{ years}} $
Man’s present age is $ 4x = 4(8) = 32{\text{ years}} $

Note:
Here in these types of problems where we are given the relation between the ages of various people, we just need to convert the statement form into the equation form and then solve that equation to get the desired ages of all the people.