Question

# Shamshad Ali buys a scooter for $Rs22000.$ He pays $4000$ cash and agrees to pay the balance in annual instalment of $Rs1000$ plus $10%$ interest on the unpaid amount. How much will the scooter cost him?

Hint: In order to solve this type of question, first we have to calculate the remaining balance and then we have to calculate the interest on the unpaid amount. After that we have to calculate the number of instalments.

Given-
Cost of the scooter = $Rs22000$
Down payment made by cash = $Rs4000$
Remaining balance =$Rs22000 - Rs4000 \\ = Rs18000 \\$
Annual instalment $= 1000 +$ Interest on unpaid amount $@10%$
${1^{st}}$ Instalment, unpaid amount $= 18000$
Interest on unpaid amount $= \dfrac{{10}}{{100}} \times 18000 = 1800$
Therefore, amount of instalment $= 1000 + 1800 = 2800 - - - - \left( 1 \right)$
${2^{nd}}$ Instalment, unpaid amount$= 18000 - 1000 = 17000$
Interest on unpaid amount $= = \dfrac{{10}}{{100}} \times 17000 = 1700$
Therefore, amount of instalment $= 1000 + 1700 = 2700 - - - - - \left( 2 \right)$
${3^{rd}}$ instalment, unpaid amount $= 17000 - 1000 = 16000$
Interest on unpaid amount $= = \dfrac{{10}}{{100}} \times 16000 = 1600$
Therefore, amount of instalment $= 1000 + 1600 = 2600 - - - - - \left( 3 \right)$
Thus, from (1), (2) and (3) our instalments are $2800,2700,2600.$
Number of instalments $= \dfrac{{\operatorname{Remaining} {\text{ }}balance{\text{ }}left}}{{balance{\text{ cleared per instalment}}}} = \dfrac{{18000}}{{1000}} = 18$
So, our instalments are $2800,2700,2600,......to{\text{ 18terms}}{\text{.}}$
We can observe that this is an $AP$ as difference between consecutive terms is an $AP.$
Hence,
First term $\left( a \right) = 2800$
Common difference= $\left( d \right) = 2700 - 2800 = - 100$
Number of terms$= n = 18$
We need to calculate total amount paid in $18$ instalments i.e.$\left( {2800 + 2700 + 2600......to{\text{ 18terms}}} \right)$
We have to apply the formula
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Where,
${S_n} = sum{\text{ of n terms of A}}{\text{.P}}{\text{.}} \\ {\text{n = number of terms}}{\text{.}} \\ {\text{a = first term and d = common difference}} \\$
Putting the value of $n = 18,a = 2800{\text{ }}and{\text{ d = - 100}}$
${S_n} = \dfrac{{18}}{2}\left( {2\left( {2800} \right) + \left( {18 - 1} \right)\left( { - 100} \right)} \right)$
Or ${S_n} = 9\left( {5600 + 17\left( { - 100} \right)} \right)$
Or ${S_n} = 9\left( {5600 - 1700} \right)$
Or ${S_n} = 9\left( {3900} \right)$
Or ${S_n} = 35100$
Hence, the amount paid in $18$ instalments$= Rs{\text{ }}35100$

Note: Whenever we face these type of question the key concept is that we have to calculate the annual instalment with annual interest and then total number of instalment and simply substituting the value of $\left( a \right)$ and $\left( d \right)$ in the equation ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ .