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CAT 2023 Slot 2 — QA Question 9

InstallmentsEasy

Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the end of the first year and closes the loan by paying the outstanding amount at the end of the third year. Then, the total interest, in rupees, paid over the three years is nearest to

Answer & solution

  • A

    40991

  • B

    45311

  • C

    33130

  • 51311

Solution

Easy

The loan grows at 4%4\% per half-year (since 8%8\% p.a. compounded half-yearly =82%=\tfrac{8}{2}\% per six months). Carry the principal forward to year 33, subtract the future value of the year-11 repayment, and whatever remains is the final settlement. Total interest == (everything Anil paid) −- (amount borrowed).

1

Fix the half-yearly rate and time. Three years =6=6 half-years, and the year-11 payment sits for the remaining 22 years =4=4 half-years:

r=82%=4% per half-year⇒ factor=1.043 years=6 periods,year-1 payment waits 4 periods\begin{aligned} &r=\tfrac{8}{2}\%=4\%\ \text{per half-year}\Rightarrow\ \text{factor}=1.04\\ &3\ \text{years}=6\ \text{periods},\quad \text{year-1 payment waits }4\ \text{periods} \end{aligned}
2

Balance the loan at the end of year 3. Let xx be the final payment. Grow the principal and the first repayment to that date:

200000×(1.04)6=10320×(1.04)4+x253063≈12072+x⇒ x≈240991\begin{aligned} &200000\times(1.04)^6=10320\times(1.04)^4+x\\ &253063\approx 12072+x\\ &\Rightarrow\ x\approx 240991 \end{aligned}
3

Total interest == total paid −- amount borrowed:

Total paid=10320+240991=251311Interest=251311−200000=51311\begin{aligned} &\text{Total paid}=10320+240991=251311\\ &\text{Interest}=251311-200000=51311 \end{aligned}

The total interest paid is nearest to 51311\mathbf{51311}.

You never need to know xx on its own. Since interest == (final balance ++ first payment) −- principal, just compute 200000 (1.04)6−10320 (1.04)4+10320−200000≈51311200000\,(1.04)^6-10320\,(1.04)^4+10320-200000 \approx 51311 in one sweep.

CAT 2023 Slot 2 QA Q9: Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the — Solution | TheCATExam