Question

The ratio of $a$ to $b$ is $2$ to $3$, where $a$ and $b$ are positive. If $x$ equals $a$ increased by $50$ percent of $a$ and $y$ equals $b$ decreased by $50$ percent of $b$, what is the value of $\frac{x}{y}$?

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $2$

Answer 1

Answer: (D)

$2$

The ratio of a $a$ to $b$ is $2$ to $3$ that is

$\frac{a}{b}=\frac{2}{3}...............1$

$x$ equals $a$ increased by $50$ percent of $a$ (given)

that is,

$x=a+\frac{50}{100}\times a$

$\Rightarrow x=a+\frac{a}{2}$ (Reducing the fraction)

$\Rightarrow x=\frac{2a+a}{2}$

$\Rightarrow x=\frac{3a}{2}$

$\Rightarrow a=\frac{2x}{3}...................2$

$y$ equals $b$ decreased by $50$ percent of $b$ (given)

that is,

$y=b-\frac{50}{100}\times b$

$\Rightarrow y=b-\frac{b}{2}$ (Reducing the fraction)

$\Rightarrow y=\frac{2b-b}{2}$

$\Rightarrow y=\frac{b}{2}$

$\Rightarrow b=2y............3$

Substituting equation $2$ and $3$ in equation $1$ we get,

$\Rightarrow \frac{2x}{3\times 2y}=\frac{2}{3}$

$\Rightarrow \frac{x}{3y}=\frac{2}{3}$

$\Rightarrow \frac{x}{3y}=\frac{2}{3}$

$\Rightarrow \frac{x}{y}=2$