If the true discount on Rs. 161 due `2 1/2%` years hence be Rs. 21. At what rate per cent is the interest calculated?

Options

A. `6%`
B. `4%`
C. `8%`
D. `12%`

Answer

To solve the problem, we need to find the rate of interest (R) given the true discount, principal amount, and time period.

### Step-by-Step Solution:

1. **Identify the given values:**
- True Discount (TD) = Rs. 21
- Principal (P) = Rs. 161
- Time (T) = 2 1/2 years = \( \frac{5}{2} \) years

2. **Use the True Discount formula:**
The formula for True Discount is:
\[ TD = \frac{P \times R \times T}{100 + R \times T} \]
where:
- TD = True Discount
- P = Principal
- R = Rate of interest (in percent)
- T = Time (in years)

3. **Substitute the known values into the formula:**
\[ 21 = \frac{161 \times R \times \frac{5}{2}}{100 + R \times \frac{5}{2}} \]

4. **Simplify the equation:**
- Multiply both sides by \( 100 + R \times \frac{5}{2} \):
\[ 21 \times (100 + R \times \frac{5}{2}) = 161 \times R \times \frac{5}{2} \]
- Expanding the left side:
\[ 2100 + 21 \times R \times \frac{5}{2} = 161 \times R \times \frac{5}{2} \]

5. **Rearranging the equation:**
- Move all terms involving R to one side:
\[ 2100 = 161 \times R \times \frac{5}{2} - 21 \times R \times \frac{5}{2} \]
- Factor out \( R \times \frac{5}{2} \):
\[ 2100 = R \times \frac{5}{2} (161 - 21) \]
- Simplifying:
\[ 2100 = R \times \frac{5}{2} \times 140 \]

6. **Isolate R:**
\[ R = \frac{2100 \times 2}{5 \times 140} \]
- Calculate:
\[ R = \frac{4200}{700} = 6 \]

7. **Conclusion:**
The rate of interest is \( R = 6\% \).

### Final Answer:
The rate at which the interest is calculated is **6%**.

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