Question

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

Answer 1

Step $1$. Solving given data

Given,
The farmer pays in cash $=6000$
So, the unpaid amount $=12000-6000=6000$
According to the question, the interest paid annually will be
$=12\%\text{of}6000,12\%\text{of}5500,12\%\text{of}5000,...,12\%\text{of}500$

Hence, the total interest to be paid $=12\%\text{of}6000+12\%\text{of}5500+12\%\text{of}5000+...+12\%\text{of}500$
$=12\%\text{of}(6000+5500+5000+\dots +500)$

$=12\%\text{of}(500+1000+1500+\dots +6000)...\left(1\right)$

Step $2$. Finding number of instalments.

The series $500,1000,1500\dots 6000$ is an A.P. with the first term and common difference both equal to $500$.

Let's take the number of terms of the A.P. to be $n$
$6000=500+(n-1)500$
$6000-500=500n-500$
$6000=500n$
$\frac{6000}{500}=n$
$n=12$

Step $3$. Finding total cost of tractor.
Now, $a=500,d=500\&n=12$
The sum of the A.P $=\frac{12}{2}\left[2\right(500)+(12-1\left)\right(500\left)\right]$
$=6[1000+5500]$
$=6\left(6500\right)$
$=39000$
So, total interest to be paid $=12\%\text{of}(500+1000+1500+\dots +6000)$ {from $\left(1\right)$}
$=12\%\text{of}39000$
$=\frac{12}{100}\times 39000$
$=12\times 390$
$=4680$
Therefore, the tractor will cost the farmer
$=(\text{Rs}12000+\text{Rs}4680)=\text{Rs}16680$