At what rate the simple interest will be `2/5` times to te original sum in 10 yr ?

Options

A. `4%`
B. `5(2)/(3)%`
C. `6%`
D. `6(2)/(3)%`

Answer

To solve the problem of finding the rate at which the simple interest will be \( \frac{2}{5} \) times the original sum in 10 years, we can follow these steps:

### 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. **Set up the equation based on the problem statement:**
According to the problem, the Simple Interest (SI) is \( \frac{2}{5} \) times the Principal (P). Therefore, we can express this as:
\[ SI = \frac{2}{5} P \]

3. **Substitute the values into the SI formula:**
We know that the time (T) is 10 years. Substituting SI and T into the formula gives us:
\[ \frac{2}{5} P = \frac{P \times R \times 10}{100} \]

4. **Simplify the equation:**
We can eliminate \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{2}{5} = \frac{R \times 10}{100} \]

5. **Cross-multiply to solve for R:**
Cross-multiplying gives us:
\[ 2 \times 100 = 5 \times R \times 10 \]
Simplifying this:
\[ 200 = 50R \]

6. **Solve for R:**
Dividing both sides by 50:
\[ R = \frac{200}{50} = 4 \]

7. **Conclusion:**
The rate of interest (R) is \( 4\% \) per annum.

### Final Answer:
The rate at which the simple interest will be \( \frac{2}{5} \) times the original sum in 10 years is **4% per annum**.