Question

The sum of the ages of a father and his son is 48 years. Six years ago, the father’s age was five times the age of his son. Find their present ages.
1) Present age of father is 30 years and present age of son is 10 years
2) Present age of father is 48 years and present age of son is 16 years
3) Present age of father is 45 years and present age of son is 15 years
4) Present age of father is 36 years and present age of son is 12 years

Answer 1

Hint: Here, we will first find the equation from the given conditions. Then we will simplify the value of one variable in one of the equations to find the age. Then the other age can be calculated from the substitution.

Complete step-by-step answer:

Given that the sum of the ages of a father and his son is 48 years.

Let us assume that the present age of the father is \[x\] years old and the present age of the son is \[y\] years old.

Using the condition of the sum of the ages of a father and his son is 48 years, we get

\[x + y = 48{\text{ ......}}\left( 1 \right)\]

We will find the age of the father before 6 years.

\[x - 6\] years

We will now find the age of the son before 6 years.

\[y - 6\] years

Using the condition of the age of the father was five times the age of his son, we get

\[x - 6 = 5\left( {y - 6} \right)\]

Multiplying the right-hand side in the above equation to simplify it, we get

\[x - 6 = 5y - 30\]

Adding the above equation with 30 on each side, we get

\[
   \Rightarrow x - 6 + 30 = 5y - 30 + 30 \\
   \Rightarrow x + 24 = 5y \\
\]

Dividing the above equation by 5 into both sides, we get

\[
   \Rightarrow \dfrac{{x + 24}}{5} = \dfrac{{5y}}{5} \\
   \Rightarrow y = \dfrac{{x + 24}}{5} \\
\]

Substituting this value of \[y\] in the equation \[\left( 1 \right)\], we get

\[
   \Rightarrow x + \dfrac{{x + 24}}{5} = 48 \\
   \Rightarrow \dfrac{{5x + x + 24}}{5} = 48 \\
   \Rightarrow \dfrac{{6x + 24}}{5} = 48 \\
\]

Multiplying the above equation by 5 on both sides, we get

\[
   \Rightarrow 6x + 24 = 48 \times 5 \\
   \Rightarrow 6x + 24 = 240 \\
\]

Subtracting the above equation by 24 on each side, we get

\[
   \Rightarrow 6x + 24 - 24 = 240 - 24 \\
   \Rightarrow 6x = 216 \\
\]

Dividing the above equation by 6 into both sides, we get

\[
   \Rightarrow \dfrac{{6x}}{6} = \dfrac{{216}}{6} \\
   \Rightarrow x = 36 \\
\]

Substituting the above value of \[x\] in the equation \[\left( 1 \right)\], we get

\[36 + y = 48\]

Subtracting both sides by 36 in the above equation, we get

\[
   \Rightarrow 36 + y - 36 = 48 - 36 \\
   \Rightarrow y = 12 \\
\]

Thus, the present age of the father is 36 years and the present age of the son is 12 years.


Hence, option D is correct.

Note: In these types of questions, when there are more than two variables and to know whether we will be able to get the values of the variable by solving an equation or directly. When the number of the equation and the number of the variable is equal then we can say that every variable will have a unique value. We can also solve this using the elimination method.