Question

Ram is half of his father’s age. 20 years ago, the age of his father was 6 times the age of Ram. Find the age of Ram and his father.
(a) Ram = 30 years, Father = 60 years
(b) Ram = 25 years, Father = 50 years
(c) Ram = 23 years, Father = 44 years
(d) Ram = 20 years, Father = 40 years

Answer 1

Hint: We will take Ram’s present age as a variable and form linear equations using the data given in questions. Then we will solve these equations to find the value of the variable.
Complete step-by-step answer:
Let us consider the Ram's present age to be $x$.
Then, since Ram's age is half of his father’s age. His father’s age will be double the age of Ram.
Therefore, his father’s age will be $2x$.
Now, let us consider their ages 20 years ago.
20 years ago, their ages will be 20 years less than their present age.
Therefore, Ram’s age 20 years ago will be \[=x-20\] years.
And Ram’s father’s age 20 years ago will be \[~=2x-20\] years.
Now, according to question, 20 years ago Ram’s father’s age was 6 times that of Ram’s age.
Therefore, using above data, we get,
\[2x-20=6\left( x-20 \right)\]
Applying distributive law, we get,
\[2x-20=6x-120\]
Subtracting \[~2x\] from both sides of the equation, we get,
\[\begin{align}
  & 2x-20-2x=6x-120-2x \\
 & \Rightarrow 4x-120=-20 \\
\end{align}\]
Adding 120 both sides of the equation, we get,
$\begin{align}
  & 4x-120+120=-20+120 \\
 & \Rightarrow 4x=100 \\
\end{align}$
Dividing 4 from both sides of the equation, we get,
$x=\dfrac{100}{4}=25$
Hence, we get that, Ram’s present age is 25 years.
Therefore, his father’s present age will be $2\times 25=50$ years.
Therefore, the correct answer is option (b).

Note: This question can also be solved taking fathers present age to be $x$. Also, you can take Ram’s or his father’s age 20 years ago to be $x$ and then solve it. Answers will be the same with all approaches.