Rs. 20400 was divided in two parts and then invested. One part invested at 6.25% for 8 years yields the same interest as the other part invested at 7% for 5 years. What is the value of smaller part?

Options

A. a. 9600
B. b. 8400
C. c. 10100
D. d. 6500

Answer

To solve the problem, we need to find the two parts of the total amount of Rs. 20,400 that yields the same interest when invested at different rates for different time periods.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the first part of the money be \( P \). Then the second part will be \( 20,400 - P \).

2. **Set Up the Interest Formula:**
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( SI \) is the simple interest, \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

3. **Calculate the Interest for Both Parts:**
- For the first part invested at 6.25% for 8 years:
\[ SI_1 = \frac{P \times 6.25 \times 8}{100} \]
- For the second part invested at 7% for 5 years:
\[ SI_2 = \frac{(20,400 - P) \times 7 \times 5}{100} \]

4. **Set the Interests Equal:**
Since both interests are equal, we can set up the equation:
\[ \frac{P \times 6.25 \times 8}{100} = \frac{(20,400 - P) \times 7 \times 5}{100} \]

5. **Simplify the Equation:**
We can eliminate the denominator (100) from both sides:
\[ P \times 6.25 \times 8 = (20,400 - P) \times 7 \times 5 \]
Simplifying further:
\[ 50P = (20,400 - P) \times 35 \]

6. **Expand and Rearrange:**
Expanding the right side:
\[ 50P = 20,400 \times 35 - 35P \]
Rearranging gives:
\[ 50P + 35P = 20,400 \times 35 \]
\[ 85P = 20,400 \times 35 \]

7. **Calculate \( P \):**
Now, calculate \( 20,400 \times 35 \):
\[ 20,400 \times 35 = 714,000 \]
Now divide by 85:
\[ P = \frac{714,000}{85} = 8,400 \]

8. **Find the Second Part:**
The second part is:
\[ 20,400 - P = 20,400 - 8,400 = 12,000 \]

9. **Identify the Smaller Part:**
The smaller part is \( 8,400 \).

### Final Answer:
The value of the smaller part is **Rs. 8,400**.