CAT 2020

Slot 3

QA

Question & Solution

1. Concept Used

• Topic: Compound Interest (Half-Yearly Compounding)
• Formula: $$A = P\left(1 + \frac{R}{100}\right)^n$$

When interest is compounded half-yearly, the annual rate is halved and the number of periods is doubled. So for an annual rate of 10% over 1.5 years: $$R_{\text{half-yearly}} = \frac{10}{2} = 5%$$ $$n = 1.5 \times 2 = 3 \text{ half-yearly periods}$$

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2. Calculation

We are given that the final amount ( A = 18522 ), the half-yearly rate ( R = 5% ), and the number of compounding periods ( n = 3 ).

Substituting into the compound interest formula: $$18522 = P \left(1 + \frac{5}{100}\right)^3$$

$$18522 = P \left(1.05\right)^3$$

Now compute ( (1.05)^3 ): $$(1.05)^2 = 1.1025$$ $$(1.05)^3 = 1.1025 \times 1.05 = 1.157625$$

So: $$18522 = P \times 1.157625$$

$$P = \frac{18522}{1.157625}$$

To simplify, express ( 1.157625 = \frac{9261}{8000} ) (since ( 21^3 = 9261 ) and ( 20^3 = 8000 )): $$P = 18522 \times \frac{8000}{9261}$$

Notice that ( 18522 = 2 \times 9261 ): $$P = 2 \times 9261 \times \frac{8000}{9261} = 2 \times 8000 = 16000$$

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3. Solution

Answer = 16000 ✅

The final calculated value is Rs. 16000. The person had originally invested ₹16,000.