Question

 The ages of Vimal and Sarita are in the ratio of \[7:5\]. Four years later, their ages will be in the ratio \[4:3\]. Find their ages.

Answer 1

Hint: We will first let the ages of Vimal be \[x\] and Sarita be \[y\]. As given in the problem, we will substitute the ratio of ages equal to \[7:5\] and find the value of \[x\] in terms of \[y\]. Next, the ratio is given for the ages after 4 years so, we will add 4 in both of their ages and put it equal to the given ratio and substitute the value of \[x\] in terms of \[y\] and find the value of \[y\] and then \[x\] and hence we will get the desired result.

Complete step-by-step answer:
We will first let the ages of Vimal be \[x\] and Sarita be \[y\].
Now, as given that the ratio of the ages of Vimal and Sarita is \[7:5\]. So, we will put the ratio given equally to \[x:y\].
Thus, we get,
\[ \Rightarrow x:y = 7:5\]
From this, we can evaluate the value of \[x\] in terms of \[y\],
\[ \Rightarrow x = \dfrac{7}{5}y\]
Next, we have the ratio of the ages given after 4 years so, we will add 4 in \[x\] and \[y\] and put it equal to \[4:3\].
Thus, we have,
\[ \Rightarrow \left( {x + 4} \right):\left( {y + 4} \right) = 4:3\]
Now, we will substitute the value of \[x\] that is \[x = \dfrac{7}{5}y\] in the above expression.
\[
   \Rightarrow 3\left( {\dfrac{7}{5}y + 4} \right) = 4\left( {y + 4} \right) \\
   \Rightarrow \left( {21y + 60} \right)\dfrac{1}{5} = 4y + 16 \\
   \Rightarrow 21y + 60 = 20y + 80 \\
   \Rightarrow 21y - 20y = 80 - 60 \\
   \Rightarrow y = 20 \\
 \]
Next, we will substitute the value of \[y\] in \[x = \dfrac{7}{5}y\]. Thus, we have,
\[
   \Rightarrow x = \dfrac{7}{5}\left( {20} \right) \\
   \Rightarrow x = 28 \\
 \]
Thus, we get the age of Vimal as 20 years and the age of Sarita as 28 years.
Note: It is necessary to let the ages in variables and then put it equal to the ratios given in the problem to evaluate the ages. As the ratio is given for the ages after 4 years, so, we have added 4 in the variables and get the ages. While solving, we need to write the one variable in terms of another variable. While simplifying the expression, we have taken the \[y\] variable on one side and the constants on the other side.