The question is from CAT Simple and compound interest. 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2023. Simple Interest and Compound Interest is an important topic for CAT Preparation. In this question, Compound interests at the end of second and third year are given and we need to find the principal amount. A little bit of heavy duty calculation?

Question 16: On a certain sum of money, compound interest earned at the end of three years = Rs. 1456. Compound interest at the end of two years is Rs. 880. Compute the principal invested.

- Rs. 2,400
- Rs. 2,800
- Rs. 2,000
- Rs. 1,600

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Let principal = P, rate of interest = r %

CI earned at the end of three years = P(1 + r)^{3} - P = 1456

=> P(3r^{2} + 3r + r^{3}) = 1456

CI earned at the end of two years = P(1 + r)^{2} - P = 880

=> P(r^{2} + 2r) = 880

Dividing one by the other we get:\\frac{3r^2 + 3r + r^3}{r^2 + 2r}\\) = \\frac{1456}{880}\\)

We can cancel ‘r’ and solve the resulting quadratic. However, let us see if we can spot something in the numbers.

\\frac{1456}{880}\\) = \\frac{728}{440}\\)

Remember that the amounts at the end of three years and two years are linked to (1 + r)^{3} and (1 + r)^{2} respectively.

Now observe that 1728 = 12^{3} and 144 is 12^{2}.

So, \\frac{728}{440}\\) = \\frac{1728 - 1000}{1440 - 1000}\\) = \\frac{1.728 - 1}{1.440 - 1}\\)

=>r = 20%

**Alternatively:**

(r^{2} + 3r + 3) x 440 = (r + 2) x 728

(r^{2} + 3r + 3) x 55 = (r + 2) x 91

55r^{2} + 165r + 165 = 91r + 182

55r^{2} + 74r -17 = 0

55r^{2} + 85r - 11r - 17 = 0

5r (11r + 17) -1 (11r + 17) = 0

r = 0.2 or a negative number. Or, r has to be 20%.

The question is **" Compute the principal invested."**

Rs. 2,000 is the principal amount.

Choice C is the correct answer.

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