A sum of 8000 is borrowed at 5% p.a. compound interest and paid back in 3 equal annual instalments. What is the amount of each instalment?

Options

A. 2937.67
B. 3000
C. 2037.67
D. 2739.76

Answer

To solve the problem of finding the amount of each installment when a sum of 8000 is borrowed at 5% p.a. compound interest and paid back in 3 equal annual installments, we can follow these steps:

### Step 1: Define the Variables
Let \( x \) be the amount of each installment.

### Step 2: Set Up the Equation
Since the loan is paid back in 3 equal installments, we can express the total amount paid back in terms of \( x \):
\[ 8000 = \frac{x}{(1 + \frac{5}{100})^1} + \frac{x}{(1 + \frac{5}{100})^2} + \frac{x}{(1 + \frac{5}{100})^3} \]

### Step 3: Simplify the Terms
Calculate \( 1 + \frac{5}{100} \):
\[ 1 + \frac{5}{100} = 1.05 \]
Now, substitute this back into the equation:
\[ 8000 = \frac{x}{1.05} + \frac{x}{(1.05)^2} + \frac{x}{(1.05)^3} \]

### Step 4: Calculate Each Term
Calculate \( (1.05)^2 \) and \( (1.05)^3 \):
\[ (1.05)^2 = 1.1025 \]
\[ (1.05)^3 = 1.157625 \]
Now substitute these values into the equation:
\[ 8000 = \frac{x}{1.05} + \frac{x}{1.1025} + \frac{x}{1.157625} \]

### Step 5: Combine the Terms
To combine the terms, find a common denominator, which is \( 1.05 \times 1.1025 \times 1.157625 \). However, we can also calculate the individual fractions:
\[ \frac{x}{1.05} + \frac{x}{1.1025} + \frac{x}{1.157625} = x \left( \frac{1}{1.05} + \frac{1}{1.1025} + \frac{1}{1.157625} \right) \]

### Step 6: Calculate the Sum of the Fractions
Calculate each fraction:
\[ \frac{1}{1.05} \approx 0.95238 \]
\[ \frac{1}{1.1025} \approx 0.90703 \]
\[ \frac{1}{1.157625} \approx 0.86383 \]
Now sum these values:
\[ 0.95238 + 0.90703 + 0.86383 \approx 2.72324 \]

### Step 7: Substitute Back into the Equation
Now substitute back into the equation:
\[ 8000 = x \times 2.72324 \]

### Step 8: Solve for \( x \)
To find \( x \):
\[ x = \frac{8000}{2.72324} \approx 2937.67 \]

### Conclusion
The amount of each installment is approximately:
\[ \text{Each installment} \approx 2937.67 \]