Let Rajesh's current age be R and Garima's current age be G. The problem states that when Rajesh was as old as Garima is now, their age ratio was 3:2.
Let's assume this was x years ago. At that time, Rajesh's age was R - x, which was equal to Garima's current age G (so, R - x = G). Garima's age at that time was G - x.
The ratio of their ages then was:
\( \frac{R-x}{G-x} = \frac{3}{2} \)
Cross-multiplying yields:
\(2(R-x) = 3(G-x)\)
Expanding and simplifying this equation gives:
\(2R - 2x = 3G - 3x\)
\(2R + x = 3G\) … (Equation 1)
Now, let's consider the future. Suppose in y years, Garima's age will be equal to Rajesh's current age R. This means Garima's future age will be G + y = R, and Rajesh's future age will be R + y.
The problem gives the future age ratio as 5:4:
\( \frac{R+y}{R} = \frac{5}{4} \)
Cross-multiplying this equation gives:
\(4(R+y) = 5R\)
Expanding and simplifying:
\(4R + 4y = 5R\)
\(4y = R\) … (Equation 2)
From Equation 2, we know that y = R/4. Substituting this value of y into Equation 1:
\(2R + (R/4) = 3G\)
To solve for G in terms of R, multiply both sides by 4:
\(8R + R = 12G\)
\(9R = 12G\)
Simplifying this gives the ratio of their current ages:
\(3R = 4G\)
The ratio of Rajesh's future age to Garima's current age, when Garima's future age equals Rajesh's current age, is stated to be 5:4. The calculation confirms this:
\(\frac{R+R/4}{R} = \frac{5}{4}\)
Therefore, the ratio of Rajesh's future age to Garima's current age is indeed 5:4.