A sum of `Rs. 13,360` was borrowed `8(3)/4%` per annum compound interest and paid back in two years in two equal annual instalments. What was the amount of each instalment?

Options

A. `Rs.5,769`
B. `Rs. 7,569 `
C. `Rs. 7,009 `
D. `Rs. 7,500 `

Answer

To solve the problem of finding the amount of each installment for a sum of Rs. 13,360 borrowed at an interest rate of 8(3/4)% per annum compounded annually, we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The interest rate given is 8(3/4)%. First, we convert this to an improper fraction:
\[ 8(3/4) = 8 + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{35}{4} \]
So, the interest rate is \( \frac{35}{4}\% \) per annum.

### Step 2: Calculate the effective interest rate
To work with the interest rate, we convert it to a decimal:
\[ \text{Rate} = \frac{35}{4} \div 100 = \frac{35}{400} = 0.0875 \]
Thus, the effective interest rate is 0.0875 per annum.

### Step 3: Calculate the total amount after 2 years
Using the formula for compound interest, the amount \( A \) after \( n \) years is given by:
\[ A = P(1 + r)^n \]
Where:
- \( P = 13,360 \) (the principal)
- \( r = 0.0875 \) (the rate)
- \( n = 2 \) (the time in years)

Substituting the values:
\[ A = 13360 \times (1 + 0.0875)^2 \]
Calculating \( (1 + 0.0875)^2 \):
\[ (1 + 0.0875)^2 = (1.0875)^2 = 1.18265625 \]
Now, calculating \( A \):
\[ A = 13360 \times 1.18265625 \approx 15770.50 \]

### Step 4: Determine the installment amount
Since the total amount after 2 years is to be paid back in 2 equal installments, we can let \( x \) be the amount of each installment. The present value of the two installments can be calculated as:
\[ x + \frac{x}{(1 + r)} = 15770.50 \]
Substituting \( r = 0.0875 \):
\[ x + \frac{x}{1.0875} = 15770.50 \]
Multiplying through by \( 1.0875 \) to eliminate the fraction:
\[ 1.0875x + x = 15770.50 \times 1.0875 \]
Calculating \( 15770.50 \times 1.0875 \):
\[ 15770.50 \times 1.0875 \approx 17100 \]
Thus, we have:
\[ 2.0875x = 17100 \]
Now, solving for \( x \):
\[ x = \frac{17100}{2.0875} \approx 8195.65 \]

### Step 5: Round to the nearest whole number
Since we typically deal with whole currency units, we round \( x \) to the nearest whole number:
\[ x \approx 8196 \]

### Conclusion
The amount of each installment is approximately **Rs. 8196**.

---