When Rajesh's age was same as the present age of Garima, the ratio of their ages was 3: 2. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become

When Rajesh's age was the same as Garima's present age, their age ratio was 3:2. What will be the ratio of their ages when Garima's age becomes equal to Rajesh's present age?

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Let's define:

$R$ = Rajesh's present age

$G$ = Garima's present age

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The key insight here is understanding what "When Rajesh's age was same as Garima's present age" means.

This means we need to go back in time to when Rajesh was $G$ years old. Since Rajesh is currently $R$ years old, we go back $(R - G)$ years.

$(R - G)$ years ago:

Rajesh's age was: $R - (R - G) = G$

Garima's age was: $G - (R - G) = 2G - R$

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We're told that $(R - G)$ years ago, their age ratio was 3:2.

$\dfrac{\text{Rajesh's age then}}{\text{Garima's age then}} = \dfrac{3}{2}$

$\dfrac{G}{2G - R} = \dfrac{3}{2}$

Cross multiplying: $2G = 3(2G - R)$

$2G = 6G - 3R$

$3R = 6G - 2G = 4G$

$R = \dfrac{4G}{3}$

This tells us that Rajesh is currently $\tfrac{4}{3}$ times as old as Garima.

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Now we need to find their ages when "Garima's age becomes the same as Rajesh's present age."

This means Garima will be $R$ years old. Since she's currently $G$ years old, this will happen in $(R - G)$ years.

In $(R - G)$ years:

Garima's age will be: $G + (R - G) = R$

Rajesh's age will be: $R + (R - G) = 2R - G$

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$\dfrac{\text{Rajesh's future age}}{\text{Garima's future age}} = \dfrac{2R - G}{R}$

Since we found $R = \dfrac{4G}{3}$, let's substitute this:

$\dfrac{2 \cdot \dfrac{4G}{3} - G}{\dfrac{4G}{3}}$

Simplifying the numerator:

$2 \cdot \dfrac{4G}{3} - G = \dfrac{8G}{3} - G$

$= \dfrac{8G}{3} - \dfrac{3G}{3} = \dfrac{5G}{3}$

So our ratio becomes:

$\dfrac{\dfrac{5G}{3}}{\dfrac{4G}{3}} = \dfrac{5G}{3} \times \dfrac{3}{4G} = \dfrac{5}{4}$

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The ratio of their ages when Garima's age becomes equal to Rajesh's present age will be 5:4.

This makes sense because Rajesh will always be older than Garima, and the ratio 5:4 = 1.25 is reasonable for their age difference pattern.