Lesson 4: Comparing and Graphing Ratios

Property

To compare ratios systematically, create ratio tables with equivalent ratios and find a common value in one column to directly compare the corresponding values in the other column.
If ratio $a:b$ and ratio $c:d$ both have the same value for one quantity, compare the other quantities directly.

Examples

Property

When two rates are given, it can be difficult to determine which rate is higher or lower because they have different values.
It is not until both rates are converted to the same unit (a unit of one) that the comparison becomes easy.
This is useful for finding the better deal or determining which object is moving faster.

Examples

• Store A sells 10 pens for 2 dollars. Store B sells 12 pens for 3 dollars. Store A's rate is $\frac{2}{10} = 0.20$ dollars per pen. Store B's is $\frac{3}{12} = 0.25$ dollars per pen. Store A is cheaper.
• A train travels 210 miles in 3 hours. A bus travels 130 miles in 2 hours. The train's speed is $\frac{210}{3} = 70$ mph. The bus's speed is $\frac{130}{2} = 65$ mph. The train is faster.
• One faucet fills a 10-gallon tub in 4 minutes. Another fills a 12-gallon tub in 5 minutes. The first faucet's rate is $\frac{10}{4} = 2.5$ gal/min. The second is $\frac{12}{5} = 2.4$ gal/min. The first faucet is faster.

Explanation

To compare different deals or speeds, convert them to a common language: the unit rate. By finding the 'cost per one' or 'distance per one,' you can easily see which option is cheaper, faster, or better.

Property

If quantities $y$ and $x$ are in proportion then the graph of pairs $(x, y)$ in this relation will be a straight line through the origin.
That line is characterized by the assertion that $\frac{y}{x}$ is constant, and in fact, is the constant of proportionality.

Examples