The areas of two circular fields are in the ratio 16 : 49. If the radius of the latter is 14 m, then what is the radius of the former?

### Concept & Formula The ratio of the areas of two circles is equal to the square of the ratio of their radii. $$ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{r_1}{r_2}\right)^2 $$ ### Step-by-Step Solution 1. **Given:** The ratio of the areas of two circular fields is $16 : 49$. The radius of the second (latter) field ($r_2$) is $14 \text{ m}$. Let the radius of the first field be $r_1$. 2. **Find the ratio of radii:** $\left(\frac{r_1}{r_2}\right)^2 = \frac{16}{49}$ Taking the square root of both sides: $\frac{r_1}{r_2} = \frac{4}{7}$ 3. **Calculate the radius of the former ($r_1$):** Substitute $r_2 = 14$: $\frac{r_1}{14} = \frac{4}{7} \Rightarrow r_1 = \frac{4 \times 14}{7} = 4 \times 2 = 8 \text{ m}$. ### Exam Strategy & Shortcut Find the square root of the area ratio directly: $\sqrt{16/49} = 4/7$. Since $7$ units correspond to $14 \text{ m}$ (a multiplier of $2$), the $4$ units will correspond to $4 \times 2 = 8 \text{ m}$. ### Common Pitfall Failing to correctly identify which radius corresponds to which part of the ratio is a common mistake. Read "former" and "latter" carefully. ### Final Answer Therefore, the correct answer is **8 m**.