A person had Rs. 21900. He lent a part of it at 5% and the remaining at `1 (2)/(3)%` simple interest for one year. If the total interest (annually) was Rs. 584, find the sum lent at 5%.

Options

A. Rs. 15330
B. Rs. 6750
C. Rs. 6570
D. Rs. 13530

Answer

To solve the problem step by step, we will follow these instructions:

1. **Identify the total amount and the interest rates**:
- Total amount = Rs. 21,900
- Interest rate at 5% = 5%
- Interest rate at \(1 \frac{2}{3}\)% = \( \frac{5}{3}\)% (which is equivalent to \(1.6667\)%)

2. **Let \(X\) be the amount lent at 5%**:
- Therefore, the amount lent at \(1 \frac{2}{3}\)% will be \(21,900 - X\).

3. **Set up the equation for total interest**:
- The interest from the amount lent at 5% for one year is given by:
\[ \text{Interest}_1 = \frac{X \times 5 \times 1}{100} = \frac{5X}{100} = \frac{X}{20} \]
- The interest from the amount lent at \(1 \frac{2}{3}\)% for one year is:
\[ \text{Interest}_2 = \frac{(21,900 - X) \times \frac{5}{3} \times 1}{100} = \frac{(21,900 - X) \times 5}{300} = \frac{5(21,900 - X)}{300} = \frac{21,900 - X}{60} \]

4. **Combine the interests and set equal to total interest**:
- The total interest earned is Rs. 584:
\[ \frac{X}{20} + \frac{21,900 - X}{60} = 584 \]

5. **Find a common denominator to solve the equation**:
- The common denominator for 20 and 60 is 60. Multiply the entire equation by 60 to eliminate the fractions:
\[ 60 \left(\frac{X}{20}\right) + 60 \left(\frac{21,900 - X}{60}\right) = 60 \times 584 \]
- This simplifies to:
\[ 3X + (21,900 - X) = 35,040 \]

6. **Combine like terms**:
- Rearranging gives:
\[ 3X - X + 21,900 = 35,040 \]
\[ 2X + 21,900 = 35,040 \]

7. **Isolate \(X\)**:
- Subtract 21,900 from both sides:
\[ 2X = 35,040 - 21,900 \]
\[ 2X = 13,140 \]
- Divide by 2:
\[ X = \frac{13,140}{2} = 6,570 \]

8. **Conclusion**:
- The sum lent at 5% is Rs. 6,570.