Question

Four years ago, the ages of Ritu and Reena were in the ratio 5:6 respectively. Eight years from now, the ratio of their respective ages will be 8:9. What is the sum of their ages at present?
A. 52 years
B. 50 years
C. 60 years
D. 62 years

Answer 1

Hint: We have to find the sum of the present ages of Ritu and Reena and we have been given the ratio of their ages 4 years ago and the ratio of their ages 8 years after. For this, we will assume it to be x and y respectively. Then, we will calculate both of their ages of 4 years ago and then make an equation by equating their ratio to the given ratio and form one equation. Then we will do the same for their ages 8 years from now and form a second equation. Then we will solve these two equations and hence we will get an answer.

Complete step-by-step solution
We have to find the sum of the present ages of Ritu and Reena. For this, let us assume their present ages to be x and y respectively. Thus, we have to find x+y.
Now, we have been given that, 4 years ago, their ages were in the ratio 5:6.
Now, 4 years ago, the age of Ritu is:
$x-4$
Also, 4 years ago, the age of Reena is:
$y-4$
These ages have been given to us in the ratio of 5:6.
Thus, we can say that:
$\dfrac{x-4}{y-4}=\dfrac{5}{6}$
Solving this, we will get:
$\begin{align}
  & \dfrac{x-4}{y-4}=\dfrac{5}{6} \\
 & \Rightarrow 6\left( x-4 \right)=5\left( y-4 \right) \\
 & \Rightarrow 6x-24=5y-20 \\
\end{align}$
$\Rightarrow 6x-5y=4$ …..(i)
This is our first equation in x and y.
Now, 8 years from now, the age of Ritu will be:
$x+8$
Also, 8 years from now, the age of Reena will be:
$y+8$
These ages have been given to us to be in the ratio 8:9.
Thus, we can say that:
$\begin{align}
  & \dfrac{x+8}{y+8}=\dfrac{8}{9} \\
 & \Rightarrow 9\left( x+8 \right)=8\left( y+8 \right) \\
 & \Rightarrow 9x+72=8y+64 \\
\end{align}$
$\Rightarrow 9x-8y=-8$ …..(ii)
This is our second equation in x and y.
Now, to solve this by elimination method, we will multiply equation (i) by 3 and equation (ii) by 2.
Thus, we get equation (i) as:
$\begin{align}
  & 3\left( 6x-5y=4 \right) \\
 & \Rightarrow 18x-15y=12 \\
\end{align}$
And equation (ii) as:
$\begin{align}
  & 2\left( 9x-8y=-8 \right) \\
 & \Rightarrow 18x-16y=-16 \\
\end{align}$
Now, solving these two equations by elimination method we get:
$\begin{align}
  & \text{ }18x-15y=12 \\
 & \underline{-\left( 18x-16y=-16 \right)} \\
 & \underline{\text{ }y=28\text{ }} \\
\end{align}$
Thus, the value of y is 28.
Now, putting y=28 in equation (ii) we get:
$\begin{align}
  & 9x-8y=-8 \\
 & \Rightarrow 9x-8\left( 28 \right)=-8 \\
 & \Rightarrow 9x-224=-8 \\
 & \Rightarrow 9x=216 \\
 & \therefore x=\dfrac{216}{9}=24 \\
\end{align}$
Thus, the value of x is 24.
Hence, the age of Ritu is 28 and that of Reena is 24.
Hence, the sum of their ages is:
$\begin{align}
  & x+y \\
 & \Rightarrow 28+24 \\
 & \therefore 52 \\
\end{align}$
Hence, option (A) is the correct option.

Note: We have here used the elimination method to solve these equations and to make the coefficient of x same in both these equations, we have multiplied them by 3 and 2 respectively. We could have done the same to make the coefficients of y same and we still would have gotten the same answer. We also could have solved these equations by any other method other than the elimination method.