If a sum of money at simple interest doubles in 12 years, the rate of interest per annum is

Options

A. `16 "" (2)/(3) %`
B. `7.5 % `
C. `8 "" (1)/(3) %`
D. `10 %`

Answer

To solve the problem of finding the rate of interest per annum when a sum of money doubles in 12 years at simple interest, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Principal Amount**:
Let the principal amount (the initial sum of money) be \( P = x \) rupees.

2. **Determine the Amount After 12 Years**:
Since the sum doubles in 12 years, the total amount \( A \) after 12 years will be:
\[ A = 2P = 2x \text{ rupees} \]

3. **Calculate the Simple Interest (SI)**:
The simple interest earned over 12 years can be calculated as:
\[ SI = A - P = 2x - x = x \text{ rupees} \]

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

5. **Substitute Known Values into the Formula**:
We know:
- \( SI = x \)
- \( P = x \)
- \( T = 12 \) years

Substituting these values into the formula gives:
\[ x = \frac{x \times R \times 12}{100} \]

6. **Simplify the Equation**:
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{R \times 12}{100} \]

7. **Solve for \( R \)**:
Rearranging the equation to isolate \( R \):
\[ R \times 12 = 100 \]
\[ R = \frac{100}{12} = \frac{25}{3} \]

8. **Convert to Percentage**:
To express \( R \) as a percentage:
\[ R = \frac{25}{3} \approx 8.33\% \]
This can also be written as \( 8 \frac{1}{3}\% \).

### Conclusion:
The rate of interest per annum is \( 8 \frac{1}{3}\% \).

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