A man took some loan from a bank at the rate of 8% compound interest per annum and he repayed the whole amount of the loan by paying ₹50000 and ₹62640 at the end of first year and second year, respectively. The sum of the loan (In ₹) was :

Options

A. ₹ 150000
B. ₹ 50000
C. ₹112640
D. ₹ 10000

Answer

To find the sum of the loan (P) taken by the man, we can break down the problem step by step.

### Step 1: Understand the Payments
The man made two payments:
- ₹50,000 at the end of the first year
- ₹62,640 at the end of the second year

### Step 2: Set Up the Equation
Let the principal amount (the sum of the loan) be P. The interest rate is 8% per annum, which can be expressed as a fraction:
\[ \text{Interest Rate} = \frac{8}{100} = 0.08 \]

### Step 3: Calculate the Amount after the First Year
At the end of the first year, the amount owed after adding interest to the principal is:
\[ A_1 = P + 0.08P = 1.08P \]

After paying ₹50,000, the remaining amount is:
\[ \text{Remaining Amount after 1st year} = 1.08P - 50000 \]

### Step 4: Calculate the Amount after the Second Year
This remaining amount will also accrue interest for the second year:
\[ A_2 = (1.08P - 50000) \times 1.08 \]

At the end of the second year, the man pays ₹62,640, so we can set up the equation:
\[ A_2 = 62,640 \]

### Step 5: Expand the Equation
Expanding the equation from Step 4:
\[ (1.08P - 50000) \times 1.08 = 62640 \]
\[ 1.08^2P - 50000 \times 1.08 = 62640 \]

Calculating \(1.08^2\):
\[ 1.08^2 = 1.1664 \]
So, the equation becomes:
\[ 1.1664P - 54000 = 62640 \]

### Step 6: Solve for P
Now, rearranging the equation to isolate P:
\[ 1.1664P = 62640 + 54000 \]
\[ 1.1664P = 116640 \]
\[ P = \frac{116640}{1.1664} \]

### Step 7: Calculate P
Calculating the value:
\[ P = 100000 \]

Thus, the sum of the loan (P) was ₹100,000.

### Summary of Solution
The sum of the loan was ₹100,000.

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