A man borrows रु 5,000 at 12 percent compound interest payable every six months. He repays रु 1,800 at the end of every six months. Calculate the third payment he has to make at the end of 18 months in order to clear the entire loan.

Explanation

To solve the problem step by step, we will calculate the compound interest and the remaining balance after each payment. Here’s how to do it:

### Step 1: Determine the initial conditions
- Principal (P) = रु 5,000
- Annual interest rate = 12%
- Interest is compounded every 6 months, so the semi-annual interest rate = 12% / 2 = 6%.

### Step 2: Calculate the compound interest for the first 6 months
Using the formula for compound interest for 6 months:
\[ \text{Compound Interest} = P \times \frac{R \times T}{100} \]
Where:
- \( R = 6\% \)
- \( T = 0.5 \) years

Calculating:
\[ \text{CI} = 5000 \times \frac{6 \times 0.5}{100} = 5000 \times \frac{3}{100} = 150 \]

### Step 3: Calculate the total amount after the first 6 months
Total amount after 6 months:
\[ \text{Total Amount} = P + \text{CI} = 5000 + 150 = 5150 \]

### Step 4: Subtract the first repayment
After the first repayment of रु 1,800:
\[ \text{Remaining Amount} = 5150 - 1800 = 3350 \]

### Step 5: Calculate the compound interest for the next 6 months
Using the remaining principal (3350):
\[ \text{CI} = 3350 \times \frac{6 \times 0.5}{100} = 3350 \times \frac{3}{100} = 100.5 \]

### Step 6: Calculate the total amount after the second 6 months
Total amount after the second 6 months:
\[ \text{Total Amount} = 3350 + 100.5 = 3450.5 \]

### Step 7: Subtract the second repayment
After the second repayment of रु 1,800:
\[ \text{Remaining Amount} = 3450.5 - 1800 = 1650.5 \]

### Step 8: Calculate the compound interest for the next 6 months
Using the remaining principal (1650.5):
\[ \text{CI} = 1650.5 \times \frac{6 \times 0.5}{100} = 1650.5 \times \frac{3}{100} = 49.515 \]

### Step 9: Calculate the total amount after the third 6 months
Total amount after the third 6 months:
\[ \text{Total Amount} = 1650.5 + 49.515 = 1700.015 \]

### Step 10: Determine the third payment required to clear the loan
To clear the loan, the man needs to pay off the remaining amount:
\[ \text{Third Payment} = 1700.015 \]

### Final Answer:
The third payment he has to make at the end of 18 months to clear the entire loan is approximately **रु 1700.02**.

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