# The ratio of ages of Seema and Rajashree is 3 : 1. The ratio of ages of Rajashree and Atul is 2 : 3. Then find the ratio of ages of Seema, Rajashree and Atul.

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**Hint:**We must first convert the ratio of ages of Seema and Rajashree into exact ages with the help of variable $x$. Similarly, we will then convert the ratio of ages of Rajashree and Atul into exact ages with the help of variable $y$. We can then find the relation between $x\text{ and }y$ using the age of Rajashree. Hence, we can find the required ratio.

**Complete step-by-step solution:**

It is given that the ratio of ages of Seema and Rajashree is 3 : 1. We can write this mathematically as,

$\dfrac{\text{Age of Seema}}{\text{Age of Rajashree}}=\dfrac{3}{1}$.

So, let us assume that the age of Seema is $3x$ and the age of Rajashree is $x$.

We are also given that the ages of Rajashree and Atul are in the ratio 2 : 3. Thus, can be written mathematically as

$\dfrac{\text{Age of Rajashree}}{\text{Age of Atul}}=\dfrac{2}{3}$.

Now, we can also assume that the age of Rajashree is $2y$ and the age of Atul is $3y$.

Thus, from the above discussion, we can say that

$\text{Age of Rajashree}=x=2y$.

Hence, we get $x=2y...\left( i \right)$

Thus, the ratio of ages of Seema, Rajashree and Atul is

$\text{Required ratio}=3x:x:3y$.

We can easily substitute the value of $x$ from equation (i). Thus, we get

$\text{Required ratio}=6y:2y:3y$

We can see that the variable $y$ is common in every part of this ratio. So, we can cancel this term.

Thus, we get

$\text{Required ratio}=6:2:3$.

**Hence, the ratio of ages of Seema, Rajashree and Atul is 6 : 2 : 3.**

**Note:**While converting the ratios into exact ages, we must be very careful, not to assume the same variable for the two ratios. We could have also solved this question by multiplying the first ratio by 2, and then combining the two ratios into a single ratio.

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