Exercise 3 Page 359

Suppose that we are in a store where any item is worth $\$0.50.$ This means that if we buy $x$ items the total amount of money $y$ that we will have to pay is $0.5x$ dollars. Let's make a table with some values of $x$ and $y.$

$x$	$y=0.5x$
$1$	$0.5$
$2$	$1$
$3$	$1.5$
$4$	$2$
$5$	$2.5$

Let's add one more column where we find the ratio between $y$ and $x.$

$x$	$y=0.5x$	$\frac{y}{x}$
$1$	$0.5$	$\frac{0.5}{1}=0.5$
$2$	$1$	$\frac{1}{2}=0.5$
$3$	$1.5$	$\frac{1.5}{3}=0.5$
$4$	$2$	$\frac{2}{4}=0.5$
$5$	$2.5$	$\frac{2.5}{5}=0.5$

Notice that the ratio between $y$ and $x$ is constant. When this happens we say that $y$ varies directly with $x.$

Two quantities (variables) vary directly when their ratio is constant. $$\begin{array}{c}\frac{y}{x}=k\end{array}$$

On the other hand, let's consider a rectangle with an area of $36{\text{cm}}^2.$ Below, we show a list of possible dimensions for this rectangle.

Length $(\ell )$	Width $(w)$
$36$	$1$
$18$	$2$
$12$	$3$
$6$	$6$

Notice that the product between $\ell$ and $w$ is always equal to $36.$ When this situation happens, we say that $\ell$ and $w$ show inverse variation.

Two quantities (variables) vary inversely when their product is constant. $$\begin{array}{c}xy=k\end{array}$$

Conclusion

• To determine if two quantities vary directly, we check if their ratio is constant.
• To determine if two quantities vary inversely, we check if their product is constant.