In certain years a sum of money is doubled itself at `6(1)/(4)%` simple interest per annum. then the required time will be

Options

A. `12""1/2` years
B. 8 years
C. `10""2/3` years
D. 16 years

Explanation

To solve the problem of finding the time required for a sum of money to double itself at a simple interest rate of \(6 \frac{1}{4}\%\) per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to determine the time (T) it takes for a principal amount (P) to double itself at a given simple interest rate.

2. **Define the Principal and Amount**:
- Let the principal amount be \(P = x\) rupees.
- Since the amount doubles, the final amount \(A = 2x\) rupees.

3. **Identify the Rate of Interest**:
- The rate of interest is given as \(6 \frac{1}{4}\%\).
- Convert this mixed fraction to an improper fraction:
\[ 6 \frac{1}{4} = \frac{25}{4}\% \]

4. **Use the Simple Interest Formula**:
- The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
- Where \(P\) is the principal, \(R\) is the rate of interest, and \(T\) is the time in years.

5. **Calculate the Simple Interest**:
- Since the amount doubles, the simple interest earned (SI) is:
\[ SI = A - P = 2x - x = x \]

6. **Set Up the Equation**:
- Substitute the values into the simple interest formula:
\[ x = \frac{x \times \frac{25}{4} \times T}{100} \]

7. **Simplify the Equation**:
- Cancel \(x\) from both sides (assuming \(x \neq 0\)):
\[ 1 = \frac{25}{4} \times \frac{T}{100} \]
- Rearranging gives:
\[ T = \frac{100 \times 4}{25} \]

8. **Calculate the Time**:
- Simplifying further:
\[ T = \frac{400}{25} = 16 \]

9. **Conclusion**:
- The time required for the sum to double is \(T = 16\) years.

### Final Answer:
The required time is **16 years**.

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