Question

Given below are two statements: Assertion (A): Population of a city increases by \(10\%\) every year. If its present population is \(2\) Lacs, its population after \(3\) years will be \(2,66,200\). Reason (R): If population of a city is \(P\) and it increases by \(R\%\) annually, then population after \(n\) years is \(P\left(1+\frac{R}{100}\right)^n\).

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For repeated percentage increase, use compound growth formula \(P\left(1+\frac{R}{100}\right)^n\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The goal is to verify if the annual compound growth calculation of a city's population in Assertion (A) is correct and whether the standard compound interest model in Reason (R) is the correct mathematical justification.

Step 2: Key Formula or Approach:
The formula for compounding population growth annually is: \[ A = P \left(1 + \frac{R}{100}\right)^n \] where $P$ is the initial population, $R$ is the annual rate of increase, and $n$ is the number of years.

Step 3: Detailed Explanation:

• Verification of Assertion (A):
Given initial population $P = 2,00,000$, growth rate $R = 10\%$, and time period $n = 3$ years. Apply the growth formula: \[ A = 200000 \left(1 + \frac{10}{100}\right)^3 \] \[ A = 200000 \times (1.1)^3 \] Since $(1.1)^3 = 1.331$: \[ A = 200000 \times 1.331 = 2,66,200 \] Therefore, Assertion (A) is correct.
• Verification of Reason (R):
The reason states the general mathematical formula for geometric compound growth. This formula is indeed correct and represents the exact model used to compute the population in Assertion (A). Thus, Reason (R) is the correct explanation of Assertion (A).

Step 4: Final Answer:
Both statements are correct, and (R) is the correct explanation of (A). This corresponds to option (A).