Proportional relationships, direct and inverse variation | Pre-algebra

When one quantity tracks another

Gas costs $$4$$ dollars a gallon. Here's what different amounts cost:

Gallons, $$x$$	$$1$$	$$2$$	$$5$$	$$10$$
Cost in dollars, $$y$$	$$4$$	$$8$$	$$20$$	$$40$$

Double the gallons and the cost doubles; multiply the gallons by $$5$$ and the cost is multiplied by $$5$$ . Every cost divided by its number of gallons gives the same result: $\frac{4}{1} = \frac{8}{2} = \frac{20}{5} = \frac{40}{10} = 4$ .

Proportional relationships

In the gas example, $$k = 4$$ dollars per gallon and the equation is $$y = 4x$$ .

Inverse variation

A $$240$$ -mile trip takes less time when you drive faster:

Speed (mph), $$x$$	$$30$$	$$40$$	$$60$$	$$80$$
Time (hours), $$y$$	$$8$$	$$6$$	$$4$$	$$3$$

Double the speed and the time is cut in half. This time it's the product that stays the same: $30 \cdot 8 = 40 \cdot 6 = 60 \cdot 4 = 80 \cdot 3 = 240$ .

The graph of inverse variation isn't a line. It's a curve that gets closer and closer to the axes without touching them.

Solving problems

First decide which kind of relationship you have. Ask yourself: if one quantity doubles, does the other double (direct) or get cut in half (inverse)?

Practice

Does this table show direct variation, inverse variation, or neither?

$$x$$	$$2$$	$$4$$	$$5$$
$$y$$	$$10$$	$$5$$	$$4$$

ShowHide the solution

Ratios: $$5$$ , $$1.25$$ , $$0.8$$ . Not constant, so not direct variation.

Products: $2 \cdot 10 = 20$ , $4 \cdot 5 = 20$ , $5 \cdot 4 = 20$ . Constant, so it's inverse variation with $$k = 20$$ : $y = \frac{20}{x}$ .

Quick check

Which equation shows a proportional relationship?

Quick check

$$y$$ varies inversely with $$x$$ , and $$y = 9$$ when $$x = 4$$ . What is $$y$$ when $$x = 6$$ ?