Two friends A and B jointly lent out 81,600 at 4% per annum compound interest. After 2 years A gets the same amount as B gets after 3 years. The investment made by B was

Options

A. 40000
B. 30000
C. 45000
D. 38000

Answer

To solve the problem step by step, we will use the formula for compound interest and set up equations based on the information given.

### Step 1: Understand the problem
We have two friends, A and B, who have jointly lent out a total of ₹81,600 at a compound interest rate of 4% per annum. After 2 years, the amount A receives is equal to the amount B receives after 3 years. We need to find out how much B invested.

### Step 2: Define variables
Let:
- \( x \) = the amount invested by A
- \( 81,600 - x \) = the amount invested by B (since the total investment is ₹81,600)

### Step 3: Write the formula for compound interest
The formula for the amount \( A \) after \( t \) years at a principal \( P \) and rate \( r \) is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]

### Step 4: Calculate the amounts for A and B
For A, who invests \( x \) for 2 years:
\[ A_A = x \left(1 + \frac{4}{100}\right)^2 = x \left(1.04\right)^2 = x \times 1.0816 \]

For B, who invests \( 81,600 - x \) for 3 years:
\[ A_B = (81,600 - x) \left(1 + \frac{4}{100}\right)^3 = (81,600 - x) \left(1.04\right)^3 = (81,600 - x) \times 1.124864 \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the amount A receives after 2 years is equal to the amount B receives after 3 years:
\[ x \times 1.0816 = (81,600 - x) \times 1.124864 \]

### Step 6: Solve the equation
Expanding both sides:
\[ 1.0816x = 81,600 \times 1.124864 - 1.124864x \]
Combine like terms:
\[ 1.0816x + 1.124864x = 81,600 \times 1.124864 \]
\[ 2.206464x = 81,600 \times 1.124864 \]
Now calculate \( 81,600 \times 1.124864 \):
\[ 81,600 \times 1.124864 \approx 91,800 \]
So we have:
\[ 2.206464x = 91,800 \]
Now, divide both sides by 2.206464 to find \( x \):
\[ x \approx \frac{91,800}{2.206464} \approx 41,600 \]

### Step 7: Find the investment made by B
Since \( x \) is the investment made by A:
\[ \text{Investment by B} = 81,600 - x = 81,600 - 41,600 = 40,000 \]

### Final Answer
The investment made by B is ₹40,000.

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