Question

If $z$ varies directly with $x$ and inversely with $y$ such the constant of proportionality is equal to 1. Find the percentage change in $z$ if $x$ is increased by 5% and $y$ is increased by 20%.

• A. 13.75%
• B. 15%
• C. 10%
• D. 12.5%

Answer 1

The correct option is D 12.5%

Given,
$z\propto x$
$z\propto \frac{1}{y}$
$\text{or,}z\propto \frac{x}{y}$
Since the constant of proportionality is equal to 1,
$\text{So,}z=\frac{x}{y}$

$x$ increases by 5%
$\therefore x=x+\frac{5}{100}\times x=\frac{105x}{100}$

$y$ increases by 20%
$\therefore y=y+\frac{20}{100}\times y=\frac{120y}{100}$

$\text{or,}z=\frac{\frac{105x}{100}}{\frac{120y}{100}}=\frac{105x}{120y}$

$\therefore \text{Change in}z$
$=\frac{x}{y}-\frac{105x}{120y}=\frac{15x}{120y}$

$\text{and \% change in}z$
$=\frac{\frac{15x}{120y}}{\frac{x}{y}}\times 100$
$=\frac{15x}{120y}\times \frac{y}{x}\times 100=12.5\%$