Pankaj borrowed ₹ 10815, which is to be paid back in 3 equal half yearly instalments. If the interest is compounded half yearly at `(40)/(3)%` per annum, how much is each instalment ?

Options

A. 2048
B. 3150
C. 4096
D. 5052

Answer

To solve the problem of how much each instalment is for Pankaj who borrowed ₹ 10815 to be paid back in 3 equal half-yearly instalments with an interest rate of \( \frac{40}{3}\% \) per annum compounded half-yearly, we can follow these steps:

### Step 1: Convert the interest rate to a half-yearly rate
The annual interest rate is \( \frac{40}{3}\% \). To find the half-yearly interest rate, we divide this by 2.

\[ \text{Half-yearly interest rate} = \frac{40/3}{2} = \frac{40}{6} = \frac{20}{3}\% \]

### Step 2: Convert the percentage to a decimal
To use the interest rate in calculations, we convert it to a decimal by dividing by 100.

\[ \text{Half-yearly interest rate (decimal)} = \frac{20/3}{100} = \frac{20}{300} = \frac{1}{15} \approx 0.0667 \]

### Step 3: Use the formula for the present value of an annuity
The formula for the present value (PV) of an annuity is given by:

\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

Where:
- \( PV \) is the present value (the amount borrowed, ₹ 10815),
- \( P \) is the amount of each instalment,
- \( r \) is the half-yearly interest rate in decimal (0.0667),
- \( n \) is the number of instalments (3).

### Step 4: Rearrange the formula to solve for P
Rearranging the formula to solve for \( P \):

\[ P = PV \times \left( \frac{r}{1 - (1 + r)^{-n}} \right) \]

### Step 5: Substitute the values into the formula
Substituting the known values into the formula:

\[ P = 10815 \times \left( \frac{0.0667}{1 - (1 + 0.0667)^{-3}} \right) \]

### Step 6: Calculate \( (1 + r)^{-n} \)
Calculating \( (1 + 0.0667)^{-3} \):

\[ (1 + 0.0667)^{-3} \approx (1.0667)^{-3} \approx 0.8365 \]

### Step 7: Calculate \( 1 - (1 + r)^{-n} \)
Now calculate:

\[ 1 - 0.8365 = 0.1635 \]

### Step 8: Substitute back to find \( P \)
Now substitute back to find \( P \):

\[ P = 10815 \times \left( \frac{0.0667}{0.1635} \right) \approx 10815 \times 0.407 \]

### Step 9: Calculate the final value of \( P \)
Calculating the final value:

\[ P \approx 10815 \times 0.407 \approx 4395.99 \]

### Final Answer
Each instalment is approximately ₹ 4396 (rounded to the nearest rupee).

---