Question

4. Mr. Mehra buys a house for Rs. $2,20,000$ by taking loan from a finance company. He pays Rs. $40,000$ cash as down payment and agrees to pay the balance as annual installments of Rs. $10,000$ plus $10\%$ on the unpaid amount.

(i) In how many years will he clear the loan?

(ii) How much will the house cost him?

Answer 1

Step 1: Note the given data

Cost of house Rs. $2,20,000$

The down payment is Rs. $40,000$

yearly installment is Rs. $10,000$ plus $10\%$ on the unpaid amount

Step 2: Finding the number of years for clearing the total loan

Outstanding amount $=\mathrm{total}\mathrm{cost}\mathrm{of}\mathrm{House}-\mathrm{down}-\mathrm{payment}$

$$\begin{gathered} =220000-40000 \\ =180000 \end{gathered}$$

Repayment timing for paying the loan $=\frac{\mathrm{Outstanding}\mathrm{amount}}{\mathrm{installment}}$

Required years of clearing the loan $=\frac{180000}{10000}$

$=18$ years

Step 3: Finding the total installment of the first and second year

Unpaid amount $=180000$

Installment of the first year $t_1=10000+10\%$ of the unpaid amount

$=10000+10\%of180000$

$$\begin{gathered} =1000+\left(\frac{10}{100}\times 18000\right) \\ =10000+18000 \\ =28000 \end{gathered}$$

Unpaid amount after the first installment $=180000-10000$

$=170000$

Installment of the second year $t_2=10000+10\%$ of the unpaid amount

$$\begin{gathered} =10000+10\%of170000 \\ =10000+\left(\frac{10}{100}\times 170000\right) \end{gathered}$$

$$\begin{gathered} =10000+17000 \\ =27000 \end{gathered}$$

Step 4: Finding the common difference

The formula for finding the common difference $={(n+1)}^{th}term-n^{th}term$

Common difference $d=27000-28000$

$=-1000$

Step 5: Finding the paying amount of the last installment

The formula for finding the $n^{th}$ term of A.P. $=t_1+(n-1)d$

The amount of the last installment $=28000+(18-1)\times (-1000)$

$$\begin{gathered} =28000+17\times (-1000) \\ =28000-17000 \\ =11000 \end{gathered}$$

Step 6: Finding the sum of the total paying amount

The formula for finding the sum of the first $n$ terms of an A.P. $=\frac{n}{2}(t_1+t_n)$

Total cost $=\frac{18}{2}(28000+11000)$

$$\begin{gathered} =9\times 39000 \\ =351000 \end{gathered}$$

The total cost of house $=351000+40000$

$=391000$

Hence, in $18$ years, Mr. Mehra will clear the loan and the total cost of the house Mr. Mehra will pay Rs. $391000$.