A scooty is sold by an automobile agency for ? 19200 cash or for ? 4800 cash down payment together with five equal monthly instalments. If the rate of interest charged by the company is 12% per annum, then find each instalment.

Answer

To solve the problem of finding the monthly installment for the scooty, we can follow these steps:

### Step 1: Identify the Cash Price and Down Payment
The cash price of the scooty is given as Rs. 19,200. The down payment is Rs. 4,800.

### Step 2: Calculate the Remaining Amount
To find the remaining amount that needs to be financed through installments, we subtract the down payment from the cash price:
\[ \text{Remaining Amount} = \text{Cash Price} - \text{Down Payment} = 19200 - 4800 = 14400 \]

### Step 3: Determine the Rate of Interest
The rate of interest charged by the company is 12% per annum. Since the installments are monthly, we need to convert this annual rate to a monthly rate:
\[ \text{Monthly Rate} = \frac{12\%}{12} = 1\% \]

### Step 4: Set Up the Equation for Installments
Let \( X \) be the amount of each monthly installment. The total amount paid over 5 months will include the principal amount and the interest on the remaining balance. The interest for each month will be calculated on the remaining balance after each installment is paid.

The total amount paid can be represented as:
\[ \text{Total Amount Paid} = 5X \]

### Step 5: Calculate the Interest on the Remaining Amount
The interest for the first month will be calculated on the entire remaining amount of Rs. 14,400. For subsequent months, the remaining balance will decrease as installments are paid. The interest for each month can be calculated as follows:

- **Month 1:** Interest = \( 1\% \times 14400 = 144 \)
- **Month 2:** Remaining balance = \( 14400 - X \), Interest = \( 1\% \times (14400 - X) \)
- **Month 3:** Remaining balance = \( 14400 - 2X \), Interest = \( 1\% \times (14400 - 2X) \)
- **Month 4:** Remaining balance = \( 14400 - 3X \), Interest = \( 1\% \times (14400 - 3X) \)
- **Month 5:** Remaining balance = \( 14400 - 4X \), Interest = \( 1\% \times (14400 - 4X) \)

### Step 6: Total Interest Calculation
The total interest over the 5 months can be summed up:
\[ \text{Total Interest} = 144 + 1\% \times (14400 - X) + 1\% \times (14400 - 2X) + 1\% \times (14400 - 3X) + 1\% \times (14400 - 4X) \]
This simplifies to:
\[ \text{Total Interest} = 144 + 14400 \times 5\% - 10\% \times X = 144 + 720 - 0.1X \]

### Step 7: Set Up the Final Equation
Now, we can set up the equation:
\[ 5X = 14400 + (144 + 720 - 0.1X) \]
This simplifies to:
\[ 5X + 0.1X = 14400 + 864 \]
\[ 5.1X = 15264 \]
\[ X = \frac{15264}{5.1} \approx 2992.16 \]

### Step 8: Final Answer
Each monthly installment is approximately Rs. 2,992.16.

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