Amit borrowed 20,000 at 12% per annum compound interest. If he pays 40% of the sum borrowed at the end of the first year and 40% of the sum borrowed at the end of the second year, find the amount of loan outstanding at the beginning of the third year.

Answer

To solve the problem step by step, we will calculate the amount of loan outstanding at the beginning of the third year after Amit makes his payments at the end of each year.

### Step 1: Calculate the amount after the first year
Amit borrowed ₹20,000 at a compound interest rate of 12% per annum.

**Interest for the first year:**
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} = 20000 \times \frac{12}{100} \times 1 = 2400 \]

**Total amount after the first year:**
\[ \text{Total Amount} = \text{Principal} + \text{Interest} = 20000 + 2400 = 22400 \]

### Step 2: Calculate the payment made at the end of the first year
Amit pays 40% of the sum borrowed at the end of the first year.

**Payment at the end of the first year:**
\[ \text{Payment} = 20000 \times \frac{40}{100} = 8000 \]

### Step 3: Calculate the outstanding amount at the beginning of the second year
**Outstanding Amount after the first year:**
\[ \text{Outstanding Amount} = \text{Total Amount} - \text{Payment} = 22400 - 8000 = 14400 \]

### Step 4: Calculate the amount after the second year
Now, we will calculate the interest on the outstanding amount for the second year.

**Interest for the second year:**
\[ \text{Interest} = 14400 \times \frac{12}{100} \times 1 = 1728 \]

**Total amount at the end of the second year:**
\[ \text{Total Amount} = \text{Outstanding Amount} + \text{Interest} = 14400 + 1728 = 16128 \]

### Step 5: Calculate the payment made at the end of the second year
Amit again pays 40% of the sum borrowed at the end of the second year.

**Payment at the end of the second year:**
\[ \text{Payment} = 20000 \times \frac{40}{100} = 8000 \]

### Step 6: Calculate the outstanding amount at the beginning of the third year
**Outstanding Amount after the second year:**
\[ \text{Outstanding Amount} = \text{Total Amount} - \text{Payment} = 16128 - 8000 = 8128 \]

### Final Answer
The amount of loan outstanding at the beginning of the third year is **₹8128**.

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