Question

Six years ago a man was three times as old as his son. In 6 years time, he will be twice as old as his son. Find their present ages
A) 30, 15
B) 40, 20
C) 42, 18
D) 41, 19

Answer 1

Hint:
Here we need to calculate the present age of a man and his son. For that, we will first assume the present age of man to be any variable and also the present age of his son to be any variable. Then, we will form the first equation using the first condition and then we will form the second equation using the second condition given in the question. After solving these two equations, we will get the value of the variable and hence their present ages.

Complete step by step solution:
Let the present age of a man be \[x\] and the present age of his son be \[y\] .
It is given that six years ago, the age of a man was three times the age of his son i.e.
Six years ago, the age of man was \[x - 6\] and the age of his son was \[y - 6\].
Using the first condition, we get
\[ \Rightarrow y - 6 = 3\left( {x - 6} \right)\]
On further simplification, we get
\[\begin{array}{l} \Rightarrow y - 6 = 3x - 18\\ \Rightarrow 3x - y = 12\end{array}\] ………… \[\left( 1 \right)\]
It is also given that after six years, the age of a man will be two times the age of his son.
Six year later, the age of man will be \[x + 6\] and the age of his son will be \[y + 6\].
Using the first condition, we get
\[ \Rightarrow y + 6 = 2\left( {x + 6} \right)\]
On further simplification, we get
\[\begin{array}{l} \Rightarrow y + 6 = 2x + 12\\ \Rightarrow 2x - y = - 6\end{array}\] ………… \[\left( 2 \right)\]
Subtracting equation 2 from equation 1, we get
\[\begin{array} \underline \begin{array} 3x - y = 12\\ - 2x \mp y = \mp 6\end{array} \\x = 18\end{array}\]
Now, we will substitute the value of \[x\] in equation 1.
\[ \Rightarrow 3 \times 18 - y = 12\]
On multiplying the terms, we get
\[\begin{array}{l} \Rightarrow 54 - y = 12\\ \Rightarrow y = 42\end{array}\]
Thus, the man is 42 years old and son is 18 years old.

Hence, the correct option is option C.

Note:
Here, we have formed two linear equations. Linear equation is a type of equation where the highest degree of the variable is 1. Linear equations can have only one solution. Apart from linear equations, the other types of equations are quadratic equation, cubic equation and so on. In the quadratic equation the highest degree is 2 whereas in cubic equations, the highest degree is 3.