Divide Rs. 15,600 into two parts such that the interest on one at 5 percent for 5 years may be equal to that on the other at `4""1/2` per cent for 6 years.

Answer

To solve the problem of dividing Rs. 15,600 into two parts such that the interest on one part at 5% for 5 years is equal to the interest on the other part at 4.5% for 6 years, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
Let the first part be Rs. X. Therefore, the second part will be Rs. (15,600 - X).

2. **Identify the Interest Rates and Time Periods:**
- For the first part (X):
- Rate (R1) = 5%
- Time (T1) = 5 years
- For the second part (15,600 - X):
- Rate (R2) = 4.5% (which is 4.5/100 or 9/20)
- Time (T2) = 6 years

3. **Set Up the Interest Formula:**
The formula for simple interest (I) is given by:
\[ I = \frac{P \times R \times T}{100} \]
- Interest on the first part (I1):
\[ I1 = \frac{X \times 5 \times 5}{100} = \frac{25X}{100} = \frac{X}{4} \]
- Interest on the second part (I2):
\[ I2 = \frac{(15,600 - X) \times 4.5 \times 6}{100} = \frac{(15,600 - X) \times 27}{100} \]

4. **Set the Interests Equal:**
Since the interest on both parts is equal, we can set up the equation:
\[ \frac{X}{4} = \frac{(15,600 - X) \times 27}{100} \]

5. **Cross-Multiply to Solve for X:**
Cross-multiplying gives:
\[ 100X = 4 \times 27(15,600 - X) \]
Simplifying this:
\[ 100X = 108(15,600 - X) \]
\[ 100X = 1,684,800 - 108X \]
\[ 100X + 108X = 1,684,800 \]
\[ 208X = 1,684,800 \]
\[ X = \frac{1,684,800}{208} = 8,100 \]

6. **Calculate the Second Part:**
The second part will be:
\[ 15,600 - X = 15,600 - 8,100 = 7,500 \]

### Final Result:
- The first part is Rs. 8,100.
- The second part is Rs. 7,500.