The ratio between the diameters of two circles is 5 to 7. The ratio between their areas is: A. 5 to 7 B. 25 to 49 C. 7 to 5 D. 49 to 25

---

The correct option is B, which is: $$25:49$$

Given that the ratio of the diameters of two circles is $5:7$, let's denote the diameters as $5x$ and $7x$.

Consequently, the radii of the circles will be: $$r_1 = \frac{5x}{2} \quad \text{and} \quad r_2 = \frac{7x}{2}$$

To determine the ratio of their areas, we use the area formula for a circle, $\pi r^2$.

Therefore, the area of the first circle is: $$\text{Area}_1 = \pi \left(\frac{5x}{2}\right)^2$$

And the area of the second circle is: $$\text{Area}_2 = \pi \left(\frac{7x}{2}\right)^2$$

Now, the ratio of the areas of the two circles is calculated as follows: $$\frac{\text{Area}_1}{\text{Area}_2} = \frac{\pi \left(\frac{5x}{2}\right)^2}{\pi \left(\frac{7x}{2}\right)^2}$$ The $\pi$ terms cancel out: $$\frac{\left(\frac{5x}{2}\right)^2}{\left(\frac{7x}{2}\right)^2} = \frac{\frac{25x^2}{4}}{\frac{49x^2}{4}}$$ The $\frac{x^2}{4}$ terms also cancel out: $$\frac{25}{49}$$

Hence, the ratio of their areas is: $$\boxed{25:49}$$