Question

Four years ago, a father's age was 6 times that of his son. Twelve years from now, the father's age will be twice that of the son. What is the ratio of father and son's present ages? Choose the correct option
A. 6 : 1
B. 7 : 1
C. 8 : 2
D. 7 : 2

Answer 1

Hint: First, take the present ages of father and son as some variables. Then form the equations from the given conditions. Then solve these equations for the variables and find the required ratio of father’s present age to that of son's present age.

Complete step by step solution:
In the problem, we have to find the ratio of father’s present age to that of son's present age.
Now, let the present age of the father is x and the present age of the son is y.
Now, it is given that four years ago, a father's age was 6 times that of his son’s age.
So, four years ago father was \[x-4\] years old and son was \[y-4\] years old.
Now, according to the question, we will have:
\[\begin{align}
  & \Rightarrow x-4=6(y-4) \\
 & \Rightarrow x=6y-24+4 \\
 & \Rightarrow x=6y-20\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1) \\
\end{align}\]
Next, it is also given that twelve years from now, father’s age will be twice that of the son’s age.
So, the father's age after 12 years will be \[x+12\] years and the son's age will be \[y+12\]. Thus, the equation formed will be:
\[\begin{align}
  & \Rightarrow x+12=2(y+12) \\
 & \Rightarrow x=2y+24-12 \\
 & \Rightarrow x=2y+12\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2) \\
\end{align}\]
So here, we will equate two equations i.e. (1) and (2), to get:
\[\begin{align}
  & \Rightarrow 6y-20=2y+12 \\
 & \Rightarrow 4y=32 \\
 & \Rightarrow y=8 \\
\end{align}\]
And putting y=8 in the equation (1), we will get:
\[\begin{align}
  & \Rightarrow x=6\times 8-20 \\
 & \Rightarrow x=28 \\
\end{align}\]
Hence, the present age of father is 28 years and the present age of son is 8 years.
Now, the ratio of the father’s present age to that of son's present age will be given as:
\[\begin{align}
  & \Rightarrow \dfrac{28}{8} \\
 & \Rightarrow \dfrac{7}{2} \\
\end{align}\]
Or we can write it as 7 : 2. So the correct answer is option D.

Note: When we are talking about the age in the past then we need to subtract the required years from the present age. Similarly, for the age in future we need to add the required years to the present age. Here the addition and subtraction is done to the present ages only. When we find the ratio of the father’s present age to that of son's present age, then the present age of father is in the numerator and the present age of son will be in the denominator of the fraction to obtain the correct ratio.