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CAT 2023 Slot 1 — QA Question 10

Simple and Compound InterestEasy

Anil invests Rs. 22000 for 6 years in a certain scheme with 4% interest per annum, compounded half-yearly. Sunil invests in the same scheme for 5 years, and then reinvests the entire amount received at the end of 5 years for one year at 10% simple interest. If the amounts received by both at the end of 6 years are same, then the initial investment made by Sunil, in rupees, is

Answer & solution

Answer: 20808

Solution

Easy

Both end with the same amount after 6 years. Write each final amount as a growth factor times the principal, set them equal, and the long compound-interest chain cancels neatly — leaving Sunil's principal in two clean steps.

1

Anil: 4%4\% p.a. compounded half-yearly =2%=2\% per half-year, over 66 years =12=12 periods:

AAnil=22000 (1.02)12A_{\text{Anil}}=22000\,(1.02)^{12}
2

Sunil: principal SS grows for 55 years =10=10 periods, then a final year at 10%10\% simple interest (factor 1.11.1):

ASunil=S (1.02)10×1.1A_{\text{Sunil}}=S\,(1.02)^{10}\times 1.1
3

Equate and cancel (1.02)10(1.02)^{10}:

S (1.02)10⋅1.1=22000 (1.02)12⇒ S⋅1.1=22000 (1.02)2⇒ S=220001.1 (1.02)2=20000 (1.02)2\begin{aligned} &S\,(1.02)^{10}\cdot 1.1 = 22000\,(1.02)^{12}\\ &\Rightarrow\ S\cdot 1.1 = 22000\,(1.02)^{2}\\ &\Rightarrow\ S=\frac{22000}{1.1}\,(1.02)^2=20000\,(1.02)^2 \end{aligned}
4

Evaluate:

S=20000×1.0404=20808S=20000\times 1.0404=20808
S=20808 rupeesS=\mathbf{20808}\ \text{rupees}

Don't expand (1.02)12(1.02)^{12}. The two principals differ only by the extra 22 half-years of compounding versus the 1.11.1 simple-interest year, so everything collapses to S=22000 (1.02)21.1S=\dfrac{22000\,(1.02)^2}{1.1}.

CAT 2023 Slot 1 QA Q10: Anil invests Rs. 22000 for 6 years in a certain scheme with 4% interest per annum, compounded half-yearly. Sun — Solution | TheCATExam