In how much time, will a sum of money become double of itself at 15 % per annum simple interset ?

Options

A. `6(1)/(4)` years
B. `6(1)/(2)` years
C. `6(1)/(3)` years
D. `6(2)/(3)` years

Answer

To solve the problem of how much time it will take for a sum of money to double itself at a simple interest rate of 15% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Doubling**:
- When we say a sum of money doubles, it means that the final amount (A) is twice the principal amount (P).
- For example, if the principal is P, then the final amount A = 2P.

2. **Calculate the Interest Earned**:
- The interest earned (I) when the money doubles is given by:
\[ I = A - P = 2P - P = P \]
- This means the interest earned is equal to the principal amount.

3. **Use the Simple Interest Formula**:
- The formula for simple interest is:
\[ I = P \times r \times t \]
- Where:
- \( I \) is the interest earned,
- \( P \) is the principal amount,
- \( r \) is the rate of interest (in decimal),
- \( t \) is the time in years.

4. **Substitute the Known Values**:
- From the previous steps, we know:
- \( I = P \)
- \( r = 15\% = \frac{15}{100} = 0.15 \)
- Substituting these into the simple interest formula gives:
\[ P = P \times 0.15 \times t \]

5. **Cancel the Principal Amount**:
- Since \( P \) is common on both sides, we can cancel it (assuming \( P \neq 0 \)):
\[ 1 = 0.15 \times t \]

6. **Solve for Time (t)**:
- Rearranging the equation to find \( t \):
\[ t = \frac{1}{0.15} \]
- Calculating this gives:
\[ t = \frac{100}{15} = \frac{20}{3} \text{ years} \]
- Converting \( \frac{20}{3} \) into a mixed number:
\[ \frac{20}{3} = 6 \frac{2}{3} \text{ years} \]

### Final Answer:
The time it will take for the sum of money to double itself at 15% per annum simple interest is **6 years and 8 months** (or **6 2/3 years**).