Quantity I : A man invested Rs. 5900 for 3 years in a scheme offering `R%` p.a. at SI and received Rs. 3186 as interest after 3 years . If the man invested Rs. 7900 at (R + 5) % p.a. at SI for 3 years, then find interest received by man (in Rs. )
Quantity II: A man invested Rs. X at `13%` p.a. at CI for 2 years and interest received by him after 2 years is Rs. 2325.96 Find X (in Rs. )

Options

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

Answer

Let's solve the problem step by step.

### Quantity I:
1. **Identify the given values:**
- Principal (P) = Rs. 5900
- Time (T) = 3 years
- Simple Interest (SI) = Rs. 3186
- Rate of Interest (R) = ?

2. **Use the formula for Simple Interest:**
\[ SI = \frac{P \times R \times T}{100} \]
Plugging in the values:
\[ 3186 = \frac{5900 \times R \times 3}{100} \]

3. **Rearranging the equation to find R:**
\[ 3186 = \frac{17700R}{100} \]
\[ 3186 \times 100 = 17700R \]
\[ 318600 = 17700R \]
\[ R = \frac{318600}{17700} = 18\% \]

4. **Calculate the interest for the second investment:**
- New Principal (P') = Rs. 7900
- New Rate (R') = R + 5 = 18 + 5 = 23%
- Time (T') = 3 years

5. **Using the SI formula again:**
\[ SI' = \frac{P' \times R' \times T'}{100} \]
Plugging in the values:
\[ SI' = \frac{7900 \times 23 \times 3}{100} \]
\[ SI' = \frac{543900}{100} = 5439 \]

### Quantity I Result:
The interest received by the man from the second investment is Rs. 5439.

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### Quantity II:
1. **Identify the given values:**
- Principal (X) = ?
- Rate of Interest = 13%
- Time = 2 years
- Compound Interest (CI) = Rs. 2325.96

2. **Use the formula for Compound Interest:**
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Rearranging gives:
\[ P \left(1 + \frac{R}{100}\right)^T = P + CI \]
Plugging in the values:
\[ X \left(1 + \frac{13}{100}\right)^2 = X + 2325.96 \]
\[ X \left(1.13\right)^2 = X + 2325.96 \]
\[ X \times 1.2769 = X + 2325.96 \]

3. **Rearranging to isolate X:**
\[ 1.2769X - X = 2325.96 \]
\[ 0.2769X = 2325.96 \]
\[ X = \frac{2325.96}{0.2769} \approx 8400 \]

### Quantity II Result:
The principal amount X is approximately Rs. 8400.

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### Final Comparison:
- Quantity I: Interest = Rs. 5439
- Quantity II: Principal = Rs. 8400

### Conclusion:
Since Rs. 5439 < Rs. 8400, we conclude that Quantity I is less than Quantity II.

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