₹ 8454 is invested in two parts at the rate of 12% per annum compounded annually for 13 years and 15 years respectively. If amount received on both ivestment is equal. Then find the difference between both investment.

Options

A. ₹954
B. ₹ 894
C. ₹ 1014
D. ₹1272

Answer

To solve the problem, we need to find the difference between two investments made from a total of ₹ 8454, where one part is invested for 13 years and the other for 15 years at a rate of 12% per annum compounded annually. The amounts received from both investments are equal.

### Step-by-Step Solution:

1. **Define the Investments:**
Let the first investment (for 13 years) be ₹ X. Then the second investment (for 15 years) will be ₹ (8454 - X).

2. **Use the Compound Interest Formula:**
The amount \( A \) after \( n \) years with principal \( P \) and rate \( r \) compounded annually is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
For the first investment (X):
\[ A_1 = X \left(1 + \frac{12}{100}\right)^{13} = X \left(\frac{112}{100}\right)^{13} \]
For the second investment (8454 - X):
\[ A_2 = (8454 - X) \left(1 + \frac{12}{100}\right)^{15} = (8454 - X) \left(\frac{112}{100}\right)^{15} \]

3. **Set the Amounts Equal:**
Since the amounts are equal:
\[ X \left(\frac{112}{100}\right)^{13} = (8454 - X) \left(\frac{112}{100}\right)^{15} \]

4. **Simplify the Equation:**
Dividing both sides by \(\left(\frac{112}{100}\right)^{13}\):
\[ X = (8454 - X) \left(\frac{112}{100}\right)^{2} \]
This simplifies to:
\[ X = (8454 - X) \cdot \frac{112^2}{100^2} \]

5. **Calculate \(\frac{112^2}{100^2}\):**
\[ \frac{112^2}{100^2} = \frac{12544}{10000} = 1.2544 \]
Therefore, we have:
\[ X = (8454 - X) \cdot 1.2544 \]

6. **Distribute and Rearrange:**
\[ X = 8454 \cdot 1.2544 - X \cdot 1.2544 \]
\[ X + X \cdot 1.2544 = 8454 \cdot 1.2544 \]
\[ X(1 + 1.2544) = 8454 \cdot 1.2544 \]
\[ X \cdot 2.2544 = 10620.6336 \]
\[ X = \frac{10620.6336}{2.2544} \approx 4704 \]

7. **Find the Second Investment:**
The second investment is:
\[ 8454 - X = 8454 - 4704 = 3750 \]

8. **Calculate the Difference:**
The difference between the two investments is:
\[ 4704 - 3750 = 954 \]

### Final Answer:
The difference between both investments is ₹ 954.

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