Direction: In the given question, read the given statement and compare the two given quantities on its basis.
Ram invested ₹P in scheme A and ₹2P in scheme B for two years each. Scheme A offers simple interest p.a. Scheme B offers compound interest (compounded annually) at the rate of `10%` p.a. The ratio of the interest earned from scheme A to that earned from scheme B was 8:21.
Quantity I. Rate of interest offered by scheme A.
Quantity II. Rate of interest offered by scheme C (simple interest p.a.), when ₹1600 invested for 3 years earns an interest of ₹384

Options

A. Quantity I = Quantity II
B. Quantity I `gt` Quantity II
C. Quantity I `lt` Quantity II
D. Quantity I `ge` Quantity II

Answer

To solve the problem, we need to compare two quantities based on the information given about Ram's investments in two schemes.

### Step-by-Step Solution:

1. **Understanding the Investments**:
- Ram invests ₹P in Scheme A (Simple Interest).
- Ram invests ₹2P in Scheme B (Compound Interest at 10% p.a.).
- Both investments are for 2 years.

2. **Calculating Simple Interest for Scheme A**:
- The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
- For Scheme A:
- Principal (P) = ₹P
- Rate (R) = ? (we need to find this)
- Time (T) = 2 years
- Therefore, the Simple Interest from Scheme A is:
\[ \text{SI}_A = \frac{P \times R \times 2}{100} = \frac{2PR}{100} \]

3. **Calculating Compound Interest for Scheme B**:
- The formula for Compound Interest (CI) for 2 years is:
\[ \text{CI} = P \left(1 + \frac{r}{100}\right)^n - P \]
- For Scheme B:
- Principal = ₹2P
- Rate = 10%
- Time = 2 years
- Therefore, the Compound Interest from Scheme B is:
\[ \text{CI}_B = 2P \left(1 + \frac{10}{100}\right)^2 - 2P \]
\[ = 2P \left(1.1^2\right) - 2P = 2P \left(1.21\right) - 2P = 2.42P - 2P = 0.42P \]

4. **Setting Up the Ratio**:
- According to the problem, the ratio of interest earned from Scheme A to Scheme B is given as:
\[ \frac{\text{SI}_A}{\text{CI}_B} = \frac{8}{21} \]
- Substituting the values we calculated:
\[ \frac{\frac{2PR}{100}}{0.42P} = \frac{8}{21} \]
- Simplifying, we can cancel out P:
\[ \frac{2R}{100 \times 0.42} = \frac{8}{21} \]
- Cross-multiplying gives:
\[ 2R \times 21 = 8 \times 42 \]
\[ 42R = 336 \]
\[ R = \frac{336}{42} = 8 \]
- Therefore, the rate of interest offered by Scheme A is **8%**.

5. **Calculating Rate of Interest for Scheme C**:
- For Scheme C, we know:
- Principal = ₹1600
- Time = 3 years
- Interest = ₹384
- Using the Simple Interest formula:
\[ 384 = \frac{1600 \times R \times 3}{100} \]
- Rearranging gives:
\[ R = \frac{384 \times 100}{1600 \times 3} \]
\[ = \frac{38400}{4800} = 8 \]
- Therefore, the rate of interest offered by Scheme C is also **8%**.

### Conclusion:
- **Quantity I** (Rate of interest offered by Scheme A) = 8%
- **Quantity II** (Rate of interest offered by Scheme C) = 8%

Thus, both quantities are equal.

### Final Answer:
**Both quantities are equal.**

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