Question

The present age of A is twice to that of B. 30 years from now, the age of A will be $1\dfrac{1}{2}$ times that of B. The present ages of A and B (in years) respectively, are ____.
(a) 60, 30
(b) 30, 60
(c) 40, 50
(d) 50, 40

Answer 1

Hint: We will first take the present ages as respected variables and after that we will form equations according to the information given in the question. We can start by considering the present age of B as x and for A, we will get it as 2x. We will change the mixed fraction by using the formula $a\dfrac{b}{c}=\dfrac{\left( a\times c \right)+b}{c}$ and solve the question further.

Complete step-by-step solution -
To start with this question we will first consider their present ages. For that we will first suppose that the present age of B be x. According to the question we have that the present age of A is twice to that of B. Therefore the present age of A becomes 2x and the present age of B is x....(i).
Now we will see that after 30 years from now the ages will be raised by 30 years. Thus, we get that the age of A becomes 2x + 30 and the age of B becomes x + 30.
According to the question we have that after 30 years from now, the age of A will be $1\dfrac{1}{2}$ times that of B. Thus, we will have age of A = $1\dfrac{1}{2}$ the age of B. This can be numerically written as $\left( 2x+30 \right)=1\dfrac{1}{2}\left( x+30 \right)$.....(ii).
We can write the mixed fraction $1\dfrac{1}{2}$ as $\dfrac{3}{2}$. Therefore, the equation becomes $\left( 2x+30 \right)=\dfrac{3}{2}\left( x+30 \right)$. After cross multiplication we will get,
 $\begin{align}
  & 2\left( 2x+30 \right)=3\left( x+30 \right) \\
 & \Rightarrow 4x+60=3x+90 \\
\end{align}$
Now, we will take the x terms on one side of the equation and the constants on the other side of the equation. Thus, we will get
$\begin{align}
  & \Rightarrow 4x-3x=90-60 \\
 & \Rightarrow x=30 \\
\end{align}$
Now, we will substitute the value of x in equation (i).
Thus, we will get the present age of A as $2(30) = 60$ years and the present age of B is 30 years.
Hence the correct option is (a).

Note: Forming the equation by using the information given to us is the most important part of the question. This is because if we form the equation wrong then we will get the answer wrong. If suppose one gets age as a negative number then understand that there is some mistake done in the formation of the equation or while solving the question. Do not stop after getting the value of x. We will substitute the value of x in the present ages of A and B which will give the right answer. One can use options as an alternative solution. In this we will substitute the values given in the question in equation (ii) to get the answer or directly putting the values in equation (i) will also be helpful in getting the answer in a short time.