Compound Interest — Assertion-Reason Type Questions

Assertion (A): The population of a town in 2015 was 10,000. It grew by 10% every year. So, in the year 2020, the population was 15,000.

Reason (R): Formula used in such problems is V = V_o $\Big(1 + \dfrac{r}{100}\Big)^n$

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

The formula, V = V_o $\Big(1 + \dfrac{r}{100}\Big)^n$ is indeed the correct formula used for calculating future value in problems involving compound growth (like population growth, compound interest, etc.), where:

V is the final value.

V_o is the initial value.

r is the rate of growth (or interest) per period.

n is the number of periods.

This formula accurately reflects how a quantity increases by a certain percentage over multiple periods, with the growth compounding on the new total each period.

∴ Reason (R) is true.

Given,

V_o = 10,000

r = 10%.

n = 5 years

Substituting values we get :

$V = 10,000 \times \Big(1 + \dfrac{10}{100}\Big)^5\\[1em] = 10,000 \times \Big(1 + \dfrac{1}{10}\Big)^5\\[1em] = 10,000 \times \Big(\dfrac{10 + 1}{10}\Big)^5\\[1em] = 10,000 \times \Big(\dfrac{11}{10}\Big)^5\\[1em] = 10,000 \times \dfrac{1,61,051}{1,00,000}\\[1em] = 16,105.10$

Since population cannot be in fraction, the population in 2020 would be approximately 16,105.

∴ Assertion (A) is false.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): Two friends invest the same amount of money for the same time (> 2 years) at the same rate of interest. One earns simple interest, but the other earns compound interest. Then both will get the same amount of money back at the end of investment.

Reason (R): The principal for each conversion period increases for the compound interest calculation.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Simple interest is calculated only on the original principal amount. The interest earned each period remains constant.

Compound interest is calculated on the original principal amount + all accumulated interest from previous periods. This means the principal for calculating interest increases with each conversion period.

The person earning compound interest will always have a higher amount back at the end of the investment period, if the conversion period is greater than one.

∴ Assertion (A) is false.

The principal for each conversion period increases for the compound interest calculation. This is true and is the fundamental principle of compound interest.

∴ Reason (R) is true.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.