The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is :

Question

Accepted Answer

To solve the problem, we need to find the third proportional to 12 and 30, and the mean proportional between 9 and 25, and then determine the ratio between these two values.

1. **Third Proportional:**
   - If 'b' is the third proportional to 'a' and 'c', then the relationship is given by: $ \frac{a}{c} = \frac{c}{b} $.
   - Here, $ a = 12 $ and $ c = 30 $.
   - Using the formula: $ \frac{12}{30} = \frac{30}{b} $.
   - Cross-multiplying gives: $ 12b = 30 \times 30 $.
   - Solving for 'b': $ 12b = 900 $ → $ b = \frac{900}{12} = 75 $.

2. **Mean Proportional:**
   - The mean proportional between two numbers 'a' and 'b' is given by the square root of their product: $ \sqrt{a \times b} $.
   - Here, $ a = 9 $ and $ b = 25 $.
   - Calculating the mean proportional: $ \sqrt{9 \times 25} = \sqrt{225} = 15 $.

3. **Ratio Calculation:**
   - We now have the third proportional as 75 and the mean proportional as 15.
   - The ratio of the third proportional to the mean proportional is: $ \frac{75}{15} = 5 $.

Thus, the ratio is 5:1, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (2:1):** This option suggests that the ratio is 2:1, which is incorrect because the calculated ratio is 5:1.
- **Option 2 (4:3):** This option suggests a ratio of 4:3, which does not match the calculated ratio of 5:1.
- **Option 4 (15:7):** This option suggests a ratio of 15:7, which is also incorrect as the actual ratio is 5:1.

Therefore, the correct answer is Option 3, which is supported by the calculations and logical reasoning provided.