A and B invested Rs (P + 2400) and (P + 4400) on CI at the rate of 10% and 20% respectively for two years and earned total interest of Rs 8680. Find the value of P?

Options

A. Rs 9200
B. Rs 6400
C. Rs 8000
D. Rs. 9600

Answer

To solve the problem step by step, we will calculate the compound interest earned by both A and B and set up an equation based on the total interest earned.

### Step 1: Identify the investments and rates
- A invests \( P + 2400 \) at a rate of 10% for 2 years.
- B invests \( P + 4400 \) at a rate of 20% for 2 years.

### Step 2: Calculate the effective interest rates
For compound interest, the effective rate for 2 years can be calculated using the formula:
\[ \text{Effective Rate} = 2R + \frac{R^2}{100} \]

- For A (10%):
\[ \text{Effective Rate for A} = 2 \times 10 + \frac{10^2}{100} = 20 + 1 = 21\% \]

- For B (20%):
\[ \text{Effective Rate for B} = 2 \times 20 + \frac{20^2}{100} = 40 + 4 = 44\% \]

### Step 3: Calculate the interest earned by A and B
Using the effective rates calculated:
- Interest earned by A:
\[ \text{Interest}_A = \frac{21}{100} \times (P + 2400) \]

- Interest earned by B:
\[ \text{Interest}_B = \frac{44}{100} \times (P + 4400) \]

### Step 4: Set up the equation for total interest
According to the problem, the total interest earned by A and B is Rs 8680. Therefore, we can write:
\[ \frac{21}{100} \times (P + 2400) + \frac{44}{100} \times (P + 4400) = 8680 \]

### Step 5: Clear the fractions by multiplying through by 100
\[ 21(P + 2400) + 44(P + 4400) = 868000 \]

### Step 6: Expand the equation
\[ 21P + 50400 + 44P + 193600 = 868000 \]

### Step 7: Combine like terms
\[ 65P + 244000 = 868000 \]

### Step 8: Isolate P
\[ 65P = 868000 - 244000 \]
\[ 65P = 624000 \]

### Step 9: Solve for P
\[ P = \frac{624000}{65} = 9600 \]

### Conclusion
The value of \( P \) is Rs 9600.

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