Q.A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?

The key idea is that interest is charged each year on the outstanding balance after the down payment, and each instalment of Rs 500 reduces that balance.

Step 1: Initial setup

Down payment = Rs 6000, so unpaid amount = $12000 - 6000 = 6000$.

He pays Rs 500 each year, so the number of instalments = $6000 \div 500 = 12$.

Step 2: Interest calculation

Interest is 12% per year on the unpaid amount at the start of each year.

Unpaid amounts at the beginning of years 1 to 12 are:

$6000, 5500, 5000, \dots, 500$ (an arithmetic progression with $a=6000$, $d=-500$, $n=12$).

Total interest = $12\%$ of the sum of these unpaid amounts. …

The key insight here is that the interest is not charged on the original loan amount, but on the unpaid amount at the start of each year. As the farmer pays off Rs 500 each year, the unpaid balance shrinks, and so does the interest charged. This makes the interest component of each instalment a decreasing arithmetic progression.

Let’s break it down.

1. Understand the payment structure. The tractor costs Rs 12000. The farmer pays Rs 6000 cash immediately. So, the unpaid amount (the loan) is:

This Rs 6000 is to be repaid in annual instalments of Rs 500 each.  
Number of instalments = $\frac{6000}{500} = 12$.  
So there will be 12 annual instalments.

2. What does each instalment contain?

Each instalment has two parts:

- A fixed principal repayment of Rs 500.

- An interest charge of 12% per annum on the unpaid amount at the beginning of that year.

The interest is calculated on the balance *before* that year’s Rs 500 is paid.

3. Calculate the interest for each year.

Let’s list the unpaid amount at the start of each year:

- **Year 1:** Unpaid amount = Rs 6000.  
  Interest = $12\% \text{ of } 6000 = \frac{12}{100} \times 6000 = 720$ rupees.  
  Instalment 1 = Rs 500 (principal) + Rs 720 (interest) = Rs 1220.

- **Year 2:** After paying Rs 500 in Year 1, unpaid amount = $6000 - 500 = 5500$.  
  Interest = $12\% \text{ of } 5500 = \frac{12}{100} \times 5500 = 660$ rupees.  
  Instalment 2 = Rs 500 + Rs 660 = Rs 1160.

- **Year 3:** Unpaid amount = $5500 - 500 = 5000$.  
  Interest = $12\% \text{ of } 5000 = 600$ rupees.  
  Instalment 3 = Rs 500 + Rs 600 = Rs 1100.

You can see the pattern: the interest decreases by Rs 60 each year (because the principal reduces by Rs 500, and 12% of Rs 500 is Rs 60).

4. Recognise the arithmetic progression.

The interest amounts form an AP:

First term $a = 720$, common difference $d = -60$, number of terms $n = 12$.

The total interest paid over 12 years is the sum of this AP:  
$$S_n = \frac{n}{2} [2a + (n-1)d]$$ …