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

Present population is: \[ P=2,00,000. \] Annual increase rate is: \[ R=10\%. \] Number of years: \[ n=3. \] For annual percentage increase, the formula is: \[ P\left(1+\frac{R}{100}\right)^n. \] Substitute the values: \[ 2,00,000\left(1+\frac{10}{100}\right)^3. \] \[ =2,00,000(1.10)^3. \] \[ =2,00,000(1.331). \] \[ =2,66,200. \] So Assertion (A) is correct. Reason (R) gives the correct compound growth formula. The reason also explains how the population after three years is calculated. Therefore, both Assertion and Reason are correct and Reason is the correct explanation of Assertion.