# A machine was sold on a hire-purchase system on ${{1}^{st}}$ March, 2015, Rs. 9000 was paid at spot and rest was paid by four equal quarterly installments of 11000 each. The cash price of the machine was Rs. 50000. Find out the amount of interest included in each installment?

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Hint: We first find the total amount to purchase the machine by adding all installments and spot cash. We then find the total interest by subtracting cash price from total amount to purchase. We use the fact that the ratio of amount of interest included in ‘n’ each installment (if all the installments are equal) is n:n-1:….:3:2:1.

Given that we have a machine that was sold on a hire-purchase system on ${{1}^{st}}$ March, 2015. The cash price of the machine is given as Rs. 50000. Rs. 9000 was paid at spot and rest was paid by four quarterly installments of 11000 each. We need to find the amount of interest included in each installment.
Let us first find the purchase amount on the hire-purchase system.
Purchase amount = total amount paid to buy a machine.
So, purchase amount = $RS.\left( 9000+11000+11000+11000+11000 \right)$.
Purchase amount = Rs. 53000.
We can calculate the interest by subtracting cash price from purchase amount.
Interest = purchase amount – cash price.
Interest = Rs. 53000 – Rs. 50000.
Interest = Rs. 3000.
This interest is to be paid in 4 installments. It will be paid in the ratio of 4:3:2:1 for the ${{1}^{st}}$, ${{2}^{nd}}$, ${{3}^{rd}}$ and ${{4}^{th}}$ installments.
Share of the ${{1}^{st}}$ installment in interest = $3000\times \dfrac{4}{4+3+2+1}$.
Share of the ${{1}^{st}}$ installment in interest = $3000\times \dfrac{4}{10}$.
Share of the ${{1}^{st}}$ installment in interest = Rs. 1200.
Share of the ${{2}^{nd}}$ installment in interest = $3000\times \dfrac{3}{4+3+2+1}$.
Share of the ${{2}^{nd}}$ installment in interest = $3000\times \dfrac{3}{10}$.
Share of the ${{2}^{nd}}$ installment in interest = Rs. 900.
Share of the ${{3}^{rd}}$ installment in interest = $3000\times \dfrac{2}{4+3+2+1}$.
Share of the ${{3}^{rd}}$ installment in interest = $3000\times \dfrac{2}{10}$.
Share of the ${{3}^{rd}}$ installment in interest = Rs. 600.
Share of the ${{4}^{th}}$ installment in interest = $3000\times \dfrac{1}{4+3+2+1}$.
Share of the ${{4}^{th}}$ installment in interest = $3000\times \dfrac{1}{10}$.
Share of the ${{4}^{th}}$ installment in interest = Rs. 300.

∴ The amount included in ${{1}^{st}}$, ${{2}^{nd}}$, ${{3}^{rd}}$ and ${{4}^{th}}$ installments are Rs. 1200, Rs. 900, Rs. 600, Rs. 300.

Note: We can solve amount of interest included alternatively as follows:
Unpaid amount at ${{1}^{st}}$ installment = Rs. 53000 – Rs. 9000.
Unpaid amount at ${{1}^{st}}$ installment = Rs. 44000.
Unpaid amount at ${{2}^{nd}}$ installment = Rs. 44000 – Rs. 11000.
Unpaid amount at ${{2}^{nd}}$ installment = Rs. 33000.
Unpaid amount at ${{3}^{rd}}$ installment = Rs. 33000 – Rs. 11000.
Unpaid amount at ${{3}^{rd}}$ installment = Rs. 22000.
Unpaid amount at ${{4}^{th}}$ installment = Rs. 22000 – Rs. 11000.
Unpaid amount at ${{4}^{th}}$ installment = Rs. 11000.
Sum of unpaid amounts in all installments = Rs. 44000 + Rs. 33000 + Rs. 22000+ Rs. 11000.
Sum of unpaid amounts in all installments = Rs. 110000.
Total interest = Rs. 3000.
We know that share of interest = $\dfrac{\text{Total interest }\!\!\times\!\!\text{ unpaid amount at that installment}}{\text{total sum of unpaid amount in installments}}$.
Share of the ${{1}^{st}}$ installment in interest = $\dfrac{3000\times 44000}{110000}$.
Share of the ${{1}^{st}}$ installment in interest = $\dfrac{3000\times 4}{10}$.
Share of the ${{1}^{st}}$ installment in interest = Rs. 1200.
Share of the ${{2}^{nd}}$ installment in interest = $\dfrac{3000\times 33000}{110000}$.
Share of the ${{2}^{nd}}$ installment in interest = $3000\times \dfrac{3}{10}$.
Share of the ${{2}^{nd}}$ installment in interest = Rs. 900.
Share of the ${{3}^{rd}}$ installment in interest = $\dfrac{3000\times 22000}{110000}$.
Share of the ${{3}^{rd}}$ installment in interest = $3000\times \dfrac{2}{10}$.
Share of the ${{3}^{rd}}$ installment in interest = Rs. 600.
Share of the ${{4}^{th}}$ installment in interest = $\dfrac{3000\times 11000}{110000}$.
Share of the ${{4}^{th}}$ installment in interest = $3000\times \dfrac{1}{10}$.
Share of the ${{4}^{th}}$ installment in interest = Rs. 300.