Simple interst on Rs 500 for 4 years at 6.25% per annum is equal to the simple interst on Rs 400 at 5% per annum for a certain period of time. The period of time is

Options

A. 4 years
B. 5 years
C. `6 (1)/(4)` years
D. `8 (2)/(3)` years

Answer

To solve the problem, we need to find the time period for which the simple interest on Rs 400 at 5% per annum is equal to the simple interest on Rs 500 for 4 years at 6.25% per annum.

### Step-by-Step Solution:

1. **Identify the given values:**
- For the first case:
- Principal (P1) = Rs 500
- Rate (R1) = 6.25% per annum
- Time (T1) = 4 years
- For the second case:
- Principal (P2) = Rs 400
- Rate (R2) = 5% per annum
- Time (T2) = T (unknown)

2. **Use the formula for Simple Interest (SI):**
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]

3. **Calculate the simple interest for the first case:**
\[ SI_1 = \frac{500 \times 6.25 \times 4}{100} \]
- Calculate the numerator:
\[ 500 \times 6.25 = 3125 \]
\[ 3125 \times 4 = 12500 \]
- Now divide by 100:
\[ SI_1 = \frac{12500}{100} = 125 \]

4. **Set up the equation for the second case:**
We know that the simple interest for the second case (SI_2) is equal to SI_1:
\[ SI_2 = SI_1 \]
\[ \frac{400 \times 5 \times T}{100} = 125 \]

5. **Calculate the simple interest for the second case:**
\[ SI_2 = \frac{400 \times 5 \times T}{100} \]
- Calculate the numerator:
\[ 400 \times 5 = 2000 \]
- Thus, we have:
\[ \frac{2000 \times T}{100} = 125 \]

6. **Simplify the equation:**
\[ 20T = 125 \]

7. **Solve for T:**
\[ T = \frac{125}{20} = 6.25 \]

### Final Answer:
The period of time (T) is **6.25 years**.