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)
- 20% of a number is more than 20% of 620 by 210. The number is:
- Girish scored 572 marks in an examination and was 30 marks short of 28% of maximum marks. In the same examination, his friend scored 473 marks. What is the percentage of marks scored by his friend?
- Anubhav spent 14% of his income on electricity bills, 28% on rent and 18% on shopping. If $\frac{1}{4}$ th of the remaining amount is ₹5120, how much did he spend on electricity bills ?
- The total population of a town is 50,000. The number of males and females increases by 10% and 15% respectively and consequently the population of the town becomes 56,000. What was the number of males in the town?
- Gokul donates $17\%$ of his savings to old age home, $17\%$ of the savings to orphanage and $17\%$ of his savings to foundations for medical help. The remaining amount of Rs.$9800$ of savings is been deposited in bank. Find the salary of Gokul, if $50\%$ of the salary is his savings amount. (In Rs.)
