At what rate percent per annum will the simple interest on a sum of money be 2/5 of the amount in 10 years?

Options

A. `6(2)/(3)%`
B. `13(1)/(3)%`
C. `8(1)/(3)%`
D. `5%`

Answer

To solve the problem, we need to find the rate percent per annum at which the simple interest (SI) on a sum of money will be \( \frac{2}{5} \) of the amount in 10 years.

### Step-by-Step Solution:

1. **Understand the relationship between Simple Interest (SI), Principal (P), Rate (R), and Time (T)**:
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]

2. **Define the variables**:
Let the Principal amount be \( P \). According to the problem, the Simple Interest is \( \frac{2}{5} \) of the amount (A). The amount \( A \) can be expressed as:
\[ A = P + SI \]
Therefore, we can write:
\[ SI = \frac{2}{5} A \]

3. **Express the Amount (A)**:
Since \( A = P + SI \), substituting for SI gives:
\[ A = P + \frac{2}{5} A \]
Rearranging this, we have:
\[ A - \frac{2}{5} A = P \]
\[ \frac{3}{5} A = P \]
Thus, we can express \( A \) in terms of \( P \):
\[ A = \frac{5}{3} P \]

4. **Substituting into the SI formula**:
Now substituting \( SI = \frac{2}{5} A \):
\[ SI = \frac{2}{5} \left( \frac{5}{3} P \right) = \frac{2}{3} P \]

5. **Use the SI formula**:
Now we can substitute SI into the SI formula:
\[ \frac{2}{3} P = \frac{P \times R \times T}{100} \]
Given \( T = 10 \) years, we substitute:
\[ \frac{2}{3} P = \frac{P \times R \times 10}{100} \]

6. **Cancel \( P \) from both sides** (assuming \( P \neq 0 \)):
\[ \frac{2}{3} = \frac{R \times 10}{100} \]

7. **Solve for R**:
Rearranging gives:
\[ R \times 10 = \frac{2}{3} \times 100 \]
\[ R \times 10 = \frac{200}{3} \]
Dividing both sides by 10:
\[ R = \frac{200}{30} = \frac{20}{3} \]

8. **Convert to percentage**:
\[ R = \frac{20}{3} \approx 6.67\% \]

### Final Answer:
The rate percent per annum will be approximately \( 6.67\% \).