A computer is available for Rs. 39300 cash or for Rs. 12820 cash down payment and three equal half-yearly instalments. If the dealer charges interest at the rate of `20%` per annum compound semi-annually, then each instalment is

Options

A. Rs. 10648
B. Rs. 26480
C. Rs. 3330
D. Rs. 3310

Answer

To find the amount of each installment for the computer, we can follow these steps:

### Step 1: Determine the Amount Financed
First, we need to determine how much money is being financed after the down payment.

**Calculation:**
Total cost of the computer = Rs. 39,300
Down payment = Rs. 12,820

Amount financed = Total cost - Down payment
Amount financed = 39,300 - 12,820 = Rs. 26,480

### Step 2: Calculate the Effective Interest Rate
The interest rate is given as 20% per annum, compounded semi-annually. Since we are dealing with half-yearly installments, we need to convert this annual rate to a semi-annual rate.

**Calculation:**
Semi-annual interest rate = 20% / 2 = 10% = 0.10

### Step 3: Determine the Number of Installments
The number of installments is given as 3 half-yearly installments.

### Step 4: Use the Formula for Present Value of Annuity
We can use the formula for the present value of an annuity to find the amount of each installment (A). The formula is:

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

Where:
- \( P \) = Present value (amount financed) = Rs. 26,480
- \( A \) = Amount of each installment
- \( r \) = Semi-annual interest rate = 0.10
- \( n \) = Number of installments = 3

Rearranging the formula to solve for \( A \):

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

### Step 5: Substitute the Values into the Formula
Now we can substitute the known values into the formula.

**Calculation:**
\[ A = 26,480 \times \left( \frac{0.10}{1 - (1 + 0.10)^{-3}} \right) \]

Calculating \( (1 + 0.10)^{-3} \):
\[ (1 + 0.10)^{-3} = (1.10)^{-3} \approx 0.7513 \]

Now substituting this back into the equation:
\[ A = 26,480 \times \left( \frac{0.10}{1 - 0.7513} \right) \]
\[ A = 26,480 \times \left( \frac{0.10}{0.2487} \right) \]
\[ A = 26,480 \times 0.4026 \approx 10,648.27 \]

### Step 6: Round the Installment Amount
Since we typically round to the nearest whole number in financial transactions, we can round Rs. 10,648.27 to Rs. 10,648.

### Final Answer:
Each installment is approximately Rs. 10,648.

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