At what rate percent per year will a sum double itself in `6""1/4` years ?

Answer

To find the rate percent per year at which a sum will double itself in `6 1/4` years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to determine the rate of interest (r) at which a principal amount (P) will double in `6 1/4` years.

2. **Convert Mixed Fraction to Improper Fraction**:
- `6 1/4` years can be converted to an improper fraction:
\[ 6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4} \text{ years} \]

3. **Define the Amount**:
- If the principal is P, then the amount (A) after doubling will be:
\[ A = 2P \]

4. **Calculate Simple Interest (SI)**:
- The simple interest (SI) can be calculated as:
\[ SI = A - P = 2P - P = P \]

5. **Use the Simple Interest Formula**:
- The formula for simple interest is:
\[ SI = \frac{P \times r \times t}{100} \]
- Substitute SI, P, and t into the formula:
\[ P = \frac{P \times r \times \frac{25}{4}}{100} \]

6. **Simplify the Equation**:
- Cancel P from both sides (assuming P ≠ 0):
\[ 1 = \frac{r \times \frac{25}{4}}{100} \]

7. **Rearrange to Solve for r**:
- Multiply both sides by 100:
\[ 100 = r \times \frac{25}{4} \]
- Multiply both sides by 4:
\[ 400 = 25r \]
- Divide both sides by 25:
\[ r = \frac{400}{25} = 16 \]

8. **Conclusion**:
- Therefore, the rate percent per year at which the sum will double itself in `6 1/4` years is:
\[ r = 16\% \]