If the interest on Rs. 50 at `4 1/2%` be equal to the discount on Rs. 59 for the same time and at the same rate when is the latter sum due?

Options

A. 2 years
B. 4 years
C. 6 years
D. 3 years

Answer

To solve the problem, we need to find the time when the interest on Rs. 50 at a rate of 4.5% is equal to the discount on Rs. 59 for the same time and rate.

### Step-by-Step Solution:

1. **Identify the given values:**
- Principal (P) = Rs. 50
- Rate (R) = 4.5% = 4.5/100 = 9/200
- Amount (A) = Rs. 59

2. **Calculate the Simple Interest (SI) on Rs. 50:**
- The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
- Here, we need to express SI in terms of time (T):
\[ SI = \frac{50 \times \frac{9}{200} \times T}{100} \]
- Simplifying this gives:
\[ SI = \frac{50 \times 9 \times T}{20000} = \frac{450T}{20000} = \frac{9T}{400} \]

3. **Calculate the Discount on Rs. 59:**
- The discount (D) is given by:
\[ D = A - P = 59 - 50 = 9 \]

4. **Set the Simple Interest equal to the Discount:**
- According to the problem, the interest on Rs. 50 equals the discount on Rs. 59:
\[ \frac{9T}{400} = 9 \]

5. **Solve for T:**
- To isolate T, we can multiply both sides by 400:
\[ 9T = 9 \times 400 \]
- Dividing both sides by 9:
\[ T = 400 \]
- Thus, we find:
\[ T = 44.44 \text{ years} \]

6. **Conclusion:**
- The time when the latter sum (Rs. 59) is due is approximately 4 years.