Question

The radii of two circlesare in the ratio2:3.what is the ratio of their circumferences

Answer 1

It will be 2:3,
lets take first radii as 2x and second as 3x
=> formula of circumference = 2πr
=> then ratio= 2π2x/ 2π3x =
=> 2:3 (ans)

Answer 2

FULL ANSWER WITH STEPS AND EXPLANATION

Let the radii of the two circles be $r_{1}$ and $r_{2}$ respectively. 
Then, the ratio between the radii of the two circles should be $r_{1}:r_{2}$.  

Finding the circumferences of the two circles
Circumference of the circle with radius $r_{1}$ = $2 \pi r_{1}$
Circumference of the circle with radius $r_{2}$ = $2 \pi r_{2}$

Now, it is given in the question that the ratio between the radii of the two circles is $2:3$. 
∴ $r_{1}:r_{2}=2:3$
⇒ $\frac{r_{1}}{r_{2}}=\frac{2}{3}$                                 ...(1)

Now, 
The ratio between the circumferences of the two circles
$=2 \pi r_{1}:2 \pi r_{2}$
$=\frac{2 \pi r_{1}}{2 \pi r_{2}}$
$=\frac{r_{1}}{r_{2}}$
$=\frac{2}{3}$                                         [Using (1)]
$=2:3$

∴ The ratio between the circumferences of the two circles is $2:3$. 

Hope this may help you. 

If you have any doubt, then you can ask me in the comments.