Question

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

Answer 1

Hint:
Let us assume the age of Vimal is x years and the age of Sarita is y years. Then write the ratio of their ages. Then find the value of x in terms of y. Then form another ratio using the second statement of the question. Then substitute the value of x in the obtained equation and solve to get the values of y. Then substituting this value in the value of x in terms of y, we get the value of x.

Complete step by step solution:
Given, the ages of Vimal and Sarita are in the ratio\[7:5\].
Let the ages of Vimal be x years and the age of Sarita be y years.
Then according to question,
$ \Rightarrow x:y = 7:5$
We can also write it as-
$ \Rightarrow \dfrac{x}{y} = \dfrac{7}{5}$
Then on transferring y on the right side, we get-
$ \Rightarrow $ x=$\dfrac{7}{5}y$ -- (i)
Now, it is given that four years later, their ages will be in the ratio$4:3$.
Now four years later, the age of Vimal will be=$x + 4$
And the age of Sarita after four years will be=$y + 4$
Then according to question,
$ \Rightarrow \dfrac{{x + 4}}{{y + 4}} = \dfrac{4}{3}$
On substituting the value of x from eq. (i) in the above equation, we get-
$ \Rightarrow \dfrac{{\dfrac{7}{5}y + 4}}{{y + 4}} = \dfrac{4}{3}$
On taking LCM of the numerator on the left side, we get-
$ \Rightarrow \dfrac{{\dfrac{{7y + 20}}{5}}}{{y + 4}} = \dfrac{4}{3}$
On simplifying, we get-
$ \Rightarrow \dfrac{{7y + 20}}{{5\left( {y + 4} \right)}} = \dfrac{4}{3}$
On cross-multiplication, we get-
$ \Rightarrow 3\left( {7y + 20} \right) = 4 \times 5\left( {y + 4} \right)$
On multiplication, we get-
$ \Rightarrow 21y + 60 = 20y + 80$ -- (ii)
Now, on taking variables on the left side and the constants on the right side, we get-
$ \Rightarrow 21y - 20y = 80 - 60$
On subtracting both sides, we get-
$ \Rightarrow y = 20$
On putting the value of y in eq. (i), we get-
$ \Rightarrow x = \dfrac{7}{5} \times 20$
On solving, we get-
$ \Rightarrow x = 7 \times 4 = 28$

Answer- Hence, the age of Vimal is $28$ years and age of Sarita is $20$ years.

Note:
Here students can also solve the question this way-
Instead of finding the value of x in terms of y, we can find the value of y in terms of x and solve in the same manner to get the answer then we get-
$ \Rightarrow y = \dfrac{5}{7}x$
Then we can write-
$ \Rightarrow \dfrac{{x + 4}}{{\dfrac{5}{7}x + 4}} = \dfrac{4}{3}$
On solving this, we get-
$ \Rightarrow 21x + 84 = 20x + 112$
On solving further we get-
$ \Rightarrow x = 28$
Then we can substitute value of x in $y = \dfrac{5}{7}x$
Then we get-
$ \Rightarrow y = \dfrac{5}{7} \times 28 = 20$
Hence we will get the same answer.