Question

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

Answer 1

Given farmer buys a tractor for Rs. $12,000$ and pays Rs $6,000$ in cash

$\Rightarrow$ Unpaid amount $=12000-6000=Rs6,000$.

At the end of the first year, interest paid $=12$% of $6000$

Amount paid as installment at the end of $1$ year =Rs$500$.

So, loan amount left $=Rs6000-500=Rs5,500$

Now, at the end of the second year, interest paid $12$% of $5500$

Amount paid as installment at the end of the second year = Rs$500$

So, loan amount left $=Rs5500-500=Rs5000$

So, total interest to be paid =12% of 6000+12% of 5500+12% of 5000+...+12% of 500.
$=12$% of $(6000+5500+5000+...+500)$
$=12$% of $(500+1000+1500+...+5500+6000)$ ....(1)

Here, $500,1000,1500,...6000$ form an A.P. with both the first term and common difference equals to $500$.

Let the number of terms of A.P.be $n$.
$\therefore 6000=500+(n-1)\left(500\right)$

$\Rightarrow n=12$

$\therefore S_{12}=500+1000+1500+...+6000$

we know, $Sum$ $\left(S_n\right)=\frac{n}{2}[2a+(n-1\left)d\right]$

$\therefore S_{12}=\frac{12}{2}\left[2\right(500)+(12-1\left)\right(500\left)\right]$

$=6[1000+5500]=39000$

So, by eqn(1),

Total interest paid $=12$% of $(500+1000+1500+...+6000)$

$=12$% of $Rs.39,000$

$=Rs.4,680$

So, the cost of the tractor to him is $Rs.12000+4680=Rs16,680$