**Question:**

If the circumference of two circles are in the ratio 2 : 3, what is the ratio of their areas?

**Solution:**

We are given ratio of circumferences of two circles. If $C=2 \pi r$ and $C^{\prime}=2 \pi r^{\prime}$ are circumferences of two circles such that

$\frac{C}{C^{\prime}}=\frac{2}{3}$

$\Rightarrow \frac{2 \pi r}{2 \pi r^{\prime}}=\frac{2}{3} \ldots \ldots$(1)

Simplifying equation (1) we get,

$\frac{r}{r^{\prime}}=\frac{2}{3}$

Let $A=\pi r^{2}$ and $A^{\prime}=\pi r^{\prime 2}$ are the areas of the respective circles and we are asked to find their ratio.

$\frac{A}{A^{\prime}}=\frac{\pi r^{2}}{\pi r^{22}}$

$\frac{A}{A^{\prime}}=\frac{r^{2}}{r^{22}}$

$\frac{A}{A^{\prime}}=\left(\frac{r}{r^{\prime}}\right)^{2} \ldots \ldots$(2)

We know that $\frac{r}{r^{\prime}}=\frac{2}{3}$ substituting this value in equation (2) we get,

$\frac{A}{A^{\prime}}=\left(\frac{2}{3}\right)^{2}$

$\Rightarrow \frac{A}{A^{\prime}}=\frac{4}{9}$

Therefore, ratio of their areas is $4: 9$.

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