A man borrowed a sum of money and agrees to pay it off by paying Rs 43,200 at the end of the first year and Rs 34,992 at the end of the secon years. If the rae of compound interesct is 8% per annum, find the sum borrowed.

Answer

To find the sum borrowed by the man, we can use the concept of compound interest and the formula for the amount after a certain period. Here’s the step-by-step solution:

### Step 1: Understand the Payments
The man makes two payments:
- Rs 43,200 at the end of the first year
- Rs 34,992 at the end of the second year

### Step 2: Total Amount Paid
Calculate the total amount paid over the two years:
\[ \text{Total Amount Paid} = 43200 + 34992 = 78292 \]

### Step 3: Identify the Rate and Time
The rate of compound interest is given as 8% per annum, and the total time period for the payments is 2 years.

### Step 4: Use the Compound Interest Formula
The formula for the amount \( A \) after \( n \) years at a compound interest rate \( r \) is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Total amount after \( n \) years
- \( P \) = Principal amount (sum borrowed)
- \( r \) = Rate of interest
- \( n \) = Number of years

### Step 5: Set Up the Equation
For the first payment of Rs 43,200 at the end of the first year:
\[ 43200 = P \left(1 + \frac{8}{100}\right)^1 \]
For the second payment of Rs 34,992 at the end of the second year:
\[ 34992 = P \left(1 + \frac{8}{100}\right)^2 \]

### Step 6: Calculate the Values
Calculate \( 1 + \frac{8}{100} = 1.08 \).

Now, we can rewrite the equations:
1. \( 43200 = P \cdot 1.08 \)
2. \( 34992 = P \cdot (1.08)^2 = P \cdot 1.1664 \)

### Step 7: Solve for \( P \)
From the first equation:
\[ P = \frac{43200}{1.08} \]
Calculating this gives:
\[ P = 40000 \]

From the second equation:
\[ P = \frac{34992}{1.1664} \]
Calculating this gives:
\[ P = 30000 \]

### Step 8: Find the Total Principal
Now we can find the total principal borrowed:
\[ \text{Total Principal} = P \cdot 1.08 + P \cdot 1.1664 \]
Calculating this gives:
\[ \text{Total Principal} = 40000 + 30000 = 70000 \]

### Final Answer
The sum borrowed is Rs 67,037.