The circumference of two circles are in the ratio 2 : 3. What is the ratio of their areas ?

Question

Accepted Answer

To determine the ratio of the areas of two circles given that their circumferences are in the ratio 2:3, we need to use the relationship between the circumference and area of a circle.

1. **Understanding the Relationship:**
   - The circumference $ C $ of a circle is given by the formula $ C = 2\pi r $, where $ r $ is the radius.
   - The area $ A $ of a circle is given by the formula $ A = \pi r^2 $.

2. **Given Ratio of Circumferences:**
   - Let the circumferences of the two circles be $ C_1 $ and $ C_2 $.
   - According to the problem, $ \frac{C_1}{C_2} = \frac{2}{3} $.

3. **Expressing Radii in Terms of Circumference:**
   - From the circumference formula, $ r_1 = \frac{C_1}{2\pi} $ and $ r_2 = \frac{C_2}{2\pi} $.
   - Therefore, the ratio of the radii is $ \frac{r_1}{r_2} = \frac{C_1}{C_2} = \frac{2}{3} $.

4. **Calculating the Ratio of Areas:**
   - The areas of the circles are $ A_1 = \pi r_1^2 $ and $ A_2 = \pi r_2^2 $.
   - The ratio of the areas is $ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2 $.
   - Substituting the ratio of the radii, $ \frac{A_1}{A_2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} $.

5. **Conclusion:**
   - The ratio of the areas of the two circles is $ \frac{4}{9} $, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (2:3):** This option incorrectly assumes that the ratio of the areas is the same as the ratio of the circumferences. However, since area is proportional to the square of the radius, the ratio of the areas is the square of the ratio of the radii, not the same as the ratio of the circumferences.

- **Option 3 (1:3):** This option does not correctly reflect the mathematical relationship between the circumferences and areas of circles. The ratio of the areas should be the square of the ratio of the radii, which is not 1:3.

- **Option 4 (8:27):** This option suggests a cube of the ratio, which is not applicable in this context. The relationship between circumference and area involves squaring the ratio of the radii, not cubing it.

Thus, the correct answer is Option 2, with the ratio of the areas being 4:9.