GATE2019 CE-2: GA-7

Question

Population of state $X$ increased by x$\%$  and the population of state $Y$ increased by y$\%$ from $2001$ to $2011.$ Assume that $x$ is greater than $y$. Let $P$ be the ratio of the population of state $X$ to state $Y$ in a given year. The percentage increase in $P$ from $2001$ to $2011$ is ________

• A. $\frac{x}{y}$
• B. $x-y$
• C. $\frac{100(x-y)}{100+x}$
• D. $\frac{100(x-y)}{100+y}$

Answer 1

$P$ in 2001 $P_{2001}= \frac{a}{b}$ where $a$ is the population of $X$ and $b$ that of $Y.$

$P$ in 2011 $P_{2011}=\frac{a+\frac{x}{100}a}{b+\frac{y}{100}b}=\frac{(1+0.01x)a}{(1+0.01y)b}$

Percentage Increase in $P = \frac{{P_{2011}} – {P_{2001}}}{P_{2001}} \times 100$

$=\frac{\frac{(1+0.01x)a}{(1+0.01y)b} – \frac{a}{b}}{\frac{a}{b}}\times 100= \left[\frac{1+0.01x}{1+0.01y}-1\right] \times 100 = \frac{x-y}{100+y}\times 100= \frac{100(x-y)}{100+y}$

Correct Answer: D