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 year is $P\left(1+\frac{R}{100}\right)^n$.
In the light of the above statements, choose the most appropriate answer from the options given below :
Verification of Assertion (A)
The Assertion (A) states that a city's population increases by 10% annually, starting from 2 Lacs (2,00,000), and will reach 2,66,200 after 3 years.
To verify this, we use the compound growth formula provided in Reason (R).
Given:
- Present Population ($P$): 2,00,000
- Annual Growth Rate ($R$): 10%
- Number of Years ($n$): 3
Using the formula: Population after $n$ years = $P\left(1+\frac{R}{100}\right)^n$
Population after 3 years = $2,00,000\left(1+\frac{10}{100}\right)^3$
Population after 3 years = $2,00,000\left(1+0.1\right)^3$
Population after 3 years = $2,00,000\left(1.1\right)^3$
Population after 3 years = $2,00,000 \times 1.331$
Population after 3 years = 2,66,200
The calculated population matches the value stated in Assertion (A). Therefore, Assertion (A) is correct.
Verification of Reason (R)
Reason (R) provides the standard formula for calculating population growth over time when the rate is constant annually: Population after $n$ year = $P\left(1+\frac{R}{100}\right)^n$. This is a well-established formula for compound growth. Therefore, Reason (R) is correct.
Relationship between Assertion and Reason
The calculation performed to verify Assertion (A) directly utilized the formula presented in Reason (R). This demonstrates that Reason (R) accurately explains the method used to arrive at the population figure in Assertion (A).
Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation for Assertion (A).
Important Questions from Percentage (Notes)
In an election between two candidates, a candidate who got $30\%$ of the total votes is defeated by $15000$ votes. The number of votes obtained by the winning candidate is:-
- Which of the following multipliers will increase a given number by 25.3 % ?
If A earns \(33\frac{1}{3}%\) more than B, then how much percent does B earn less than A?
- दो उम्मीदवारों के बीच एक कॉलेज चुनाव में, एक उम्मीदवार को कुल वैध मतों में से $55\%$ वोट मिले । $15\%$ वोट अवैध थे । यदि कुल वोट $15,200$ थे, तो दूसरे उम्मीदवार को कितने वैध वोट मिले ?
- एक शहर की जनसंख्या एक दशक में $1,75,000$ से बढ़कर $2,62,500$ हो गई । प्रति वर्ष जनसंख्या में औसत प्रतिशत वृद्धि है
