Question

A is 20 years older than B. 5 years ago, A was 3 times as old as B. Find their present ages.
(A) 35 years and 15 years
(B) 5 years and 25 years
(C) 18 years and 42 years
(D) 21 years and 44 years

Answer 1

Hint: Assume that the present age of B is x years. It is given that A is 20 years older than B. So, the present age of A is \[\left( x+20 \right)\] . We have to subtract 5 years from the present ages of A and B to get the ages of A and B, 5 years ago. So, 5 years ago, the ages of A and B were \[\left( x+15 \right)\] and \[\left( x-5 \right)\] respectively. Since the age of A is three times more than that B so, to more their ages equal we have to multiply the age of B by 3.So, \[\text{the age of A = 3 }\!\!~\!\!\text{ }\!\!\times\!\!\text{ the}\,\text{age}\,\text{of}\,\text{B}\] . Use this and get the value of x. With the help of x get the present ages of A and B.

Complete step-by-step answer:
First of all, let us assume the present age of B be x years …………………………….(1)
Now, we have two cases, \[{{1}^{st}}\] case , and \[{{2}^{nd}}\] case . In \[{{1}^{st}}\] case , we are dealing with the present ages and in \[{{2}^{nd}}\] case, we are dealing with the ages that were 5 years ago.
In \[{{1}^{st}}\] case, it is given that
A is 20 years older than B. It means that the present age of A is 20 years more than the age of B.
From equation (1), we have the present age of b.
The present age of A = \[\left( x+20 \right)\] ………………………….(2)
If we have to get the ages of A and B, 5 years ago, then we have to subtract 5 years from their present ages.
Now, in \[{{2}^{nd}}\] case, we have
The age of B, five years earlier = \[x-5\] ………………………………..(3)
The age of A, five years earlier = \[\left( x+20 \right)-5=x+15\] ……………………..(4)
It is given that 5 years ago, A was 3 times as old as B. It means that the age of A was three times more than the age of B.
From equation (3) and equation (4), we have the ages of A and B that was 5 years ago.
Since the age of A is three times more than that B so, to more their ages equal we have to multiply the age of B by 3.
\[\text{The age of A = 3 }\!\!~\!\!\text{ }\!\!\times\!\!\text{ The}\,\text{age}\,\text{of}\,\text{B}\] …………………….(5)
Now, putting the values of the age of A from equation (3) and the age of B from equation (4), in equation (5), we get
\[\Rightarrow \left( x+15 \right)\text{= 3 }\!\!~\!\!\text{ }\!\!\times\!\!\text{ }\left( x-5 \right)\]
\[\begin{align}
  & \Rightarrow \left( x+15 \right)\text{=}3x-15 \\
 & \Rightarrow 15+15=3x-x \\
 & \Rightarrow 30=2x \\
 & \Rightarrow \dfrac{30}{2}=x \\
 & \Rightarrow 15=x \\
\end{align}\]
Putting the value of x in equation (1) and equation (2), we get
The present age of B = 15 years.
The present age of a = 15 + 20 = 35 years.
Therefore, the present ages of A and B are 35 years and 15 years respectively.
Hence, the correct option is (A).

Note: We can also solve this question by assuming the present age of A as x years.
The present age of A = x years ………………………(1)
It is given that A is 20 years older than B. It means that B is 20 years younger than A
The present age of B = \[\left( x-20 \right)\] years ………………………..(2)
Now, 5 years ago,
The age of A = \[\left( x-5 \right)\] ………………….(3)
The age of B = \[\left( x-25 \right)\] ……………………………..(4)
It is given that 5 years ago, A was 3 times as old as B. It means that the age of A was three times more than the age of B.
Since the age of A is three times more than that of B, so to make their ages equal we have to multiply the age of B by 3.
\[\text{The age of A = 3 }\!\!~\!\!\text{ }\!\!\times\!\!\text{ The}\,\text{age}\,\text{of}\,\text{B}\] ………………………..(5)
Now, putting the values of the age of A from equation (3) and the age of B from equation (4), in equation (5), we get
\[\begin{align}
  & \Rightarrow \left( x-5 \right)\text{ = 3 }\!\!~\!\!\text{ }\!\!\times\!\!\text{ }\left( x-25 \right) \\
 & \Rightarrow x-5=3x-75 \\
 & \Rightarrow 75-5=3x-x \\
 & \Rightarrow 70=2x \\
 & \Rightarrow \dfrac{70}{2}=x \\
 & \Rightarrow 35=x \\
\end{align}\]
Now, putting the value of x in equation (1) and equation (2), we get
The present age of A = x = 35 years.
The present age of B = \[\left( x-20 \right)\] = 35 – 20 = 15 years.
Therefore, the present ages of A and B are 35 years and 15 years respectively.
Hence, the correct option is (A).