Question

The sum of the ages of Anup and his father is 100. When Anup is as old as his father now, he will be five times as old as his son Anuj is now. Anuj will be eight years older than Anup is now, when Anup is as old as his father. What are their ages right now?

Answer 1

Hint: In this question, first of all consider the present ages and new ages of them as variables. According to the given data we will get four equations. Then solve the four equations by using a substitution method to get the final answer.

Complete step by step answer:
Let the present ages of Anup, his father and his son Anuj be \[x,y\& z\] respectively.
According to the question, sum of ages of Anup and his father is 100 i.e.,
\[
   \Rightarrow x + y = 100 \\
   \Rightarrow y = 100 - x............................................\left( 1 \right) \\
\]
Let the new ages of Anup, his father and his son Anuj after a few years be \[x',y'\] and \[z'\] respectively.
Given after those few years,
new age of Anup = present age of his father = 5 times of present age of his son Anuj
\[ \Rightarrow x' = y = 5z....................................\left( 2 \right)\]
Also, given that the new age of Anuj = 8 + present ag of Anuj
\[ \Rightarrow z' = 8 + x....................................\left( 3 \right)\]
Since, number of years = new age – present age for each person, we have
\[{\text{No}}{\text{. of years }} = x' - x = y' - y = z' - z\]
Taking the first and last term, we have
\[
   \Rightarrow x' - x = z' - z \\
   \Rightarrow y - x = z' - \dfrac{y}{5}{\text{ }}\left[ {{\text{Using equation}}\left( 2 \right)} \right] \\
   \Rightarrow y - x = 8 + x - \dfrac{y}{5}{\text{ }}\left[ {{\text{Using equation}}\left( 3 \right)} \right] \\
   \Rightarrow y + \dfrac{y}{5} = 8 + x + x \\
   \Rightarrow \dfrac{{5y + y}}{5} = 8 + 2x \\
\]
By using equation (1), we have
\[
   \Rightarrow \dfrac{{6y}}{5} = 8 + 2\left( {100 - y} \right) \\
   \Rightarrow \dfrac{{6y}}{5} = 8 + 200 - 2y \\
   \Rightarrow \dfrac{{6y}}{5} + 2y = 208 \\
   \Rightarrow \dfrac{{6y + 10y}}{5} = 208 \\
   \Rightarrow 16y = 208 \times 5 \\
   \Rightarrow 16y = 1040 \\
  \therefore y - \dfrac{{1040}}{{16}} = 65 \\
  {\text{ }} \\
\]
By substituting \[y = 65\] in equation (1), we get
\[
   \Rightarrow 65 = 100 - x \\
  \therefore x = 100 - 65 = 35 \\
\]
By substituting \[y = 65\] in equation (2), we get
\[
   \Rightarrow 65 = 5z \\
  \therefore z = \dfrac{{65}}{5} = 13 \\
\]
Thus, present age of Anup = 35 years
Present age of his father = 65 years
Present age of Anuj = 13 years

Note: Here, the age of Anuj should be greater than Anup and his father also the age of Anup should be greater than his father. If our obtained answer suits this statement, then our answer is correct otherwise wrong.