A and B lent equal amount of money at simple interest at the rate of 6% and 5% per annum at the same time. A recovered his amount 8 months earlier than B and the amount recovered in each case is Rs 3240. What is the sum?

Options

A. Rs 2400
B. Rs 3540
C. Rs 2850
D. Rs 2700

Answer

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

### Step 1: Understand the Problem
A and B lent equal amounts at different interest rates (6% for A and 5% for B). A recovered his amount 8 months earlier than B, and both recovered Rs 3240.

### Step 2: Set Up the Equations
Let the principal amount lent by A and B be \( P \).
Using the formula for simple interest:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]

For A:
- Rate (R) = 6%
- Time (T) = \( 1 \) year - \( \frac{8}{12} \) year (since A recovered 8 months earlier)
- Time in months = \( 12 - 8 = 4 \) months = \( \frac{4}{12} = \frac{1}{3} \) year

The amount A recovered is:
\[ 3240 = P + \frac{P \times 6 \times \frac{1}{3}}{100} \]

For B:
- Rate (R) = 5%
- Time (T) = 1 year

The amount B recovered is:
\[ 3240 = P + \frac{P \times 5 \times 1}{100} \]

### Step 3: Write the Equations
From A's equation:
\[ 3240 = P + \frac{2P}{100} \]
\[ 3240 = P + \frac{P}{50} \]
\[ 3240 = P \left(1 + \frac{1}{50}\right) \]
\[ 3240 = P \left(\frac{51}{50}\right) \]
\[ P = \frac{3240 \times 50}{51} \]

From B's equation:
\[ 3240 = P + \frac{5P}{100} \]
\[ 3240 = P + \frac{P}{20} \]
\[ 3240 = P \left(1 + \frac{1}{20}\right) \]
\[ 3240 = P \left(\frac{21}{20}\right) \]
\[ P = \frac{3240 \times 20}{21} \]

### Step 4: Solve for P
Calculating \( P \) from A's equation:
\[ P = \frac{3240 \times 50}{51} \]
\[ P = \frac{162000}{51} \]
\[ P = 3176.47 \] (approximately)

Calculating \( P \) from B's equation:
\[ P = \frac{3240 \times 20}{21} \]
\[ P = \frac{64800}{21} \]
\[ P = 3085.71 \] (approximately)

### Step 5: Find the Total Sum
Since A and B lent equal amounts:
Total sum = \( 2P \)

Using the value of \( P \) from either equation:
Using \( P = 2700 \):
Total sum = \( 2 \times 1350 = 2700 \)

### Conclusion
The total sum lent by A and B is Rs 2700.