The circumferences of two circles are in the ratio 3 : 4.

Let the the radii of the two circles be r and R, the circumferences of the circles be c and C and the areas of the two circles be a and A.
Now,

$\frac{c}{C}=\frac{3}{4}$

$\Rightarrow \frac{2 \pi r}{2 \pi R}=\frac{3}{4}$

$\Rightarrow \frac{r}{R}=\frac{3}{4}$

Now, the ratio between their areas is given by

$\frac{a}{A}=\frac{\pi r^{2}}{\pi R^{2}}$

$=\left(\frac{r}{R}\right)^{2}$

$=\left(\frac{3}{4}\right)^{2}$

$=\frac{9}{16}$

Hence, the correct answer is option (c).