Ratios and proportional relationships, explained with examples

What a ratio is

A ratio describes how two amounts relate. If a recipe uses 3 cups of flour for every 2 cups of sugar, the ratio of flour to sugar is 3 to 2, written 3:2 or 3/2. Order matters: the ratio of sugar to flour is 2:3.

Ratios can compare part to part (flour to sugar) or part to whole (flour to all 5 cups of flour and sugar together). Many mistakes come from mixing these up, so say the ratio in words before you write it.

Equivalent ratios describe the same mix. 3:2, 6:4, and 12:8 all make batter that tastes the same, just more of it. A ratio table lays these out in rows and makes the pattern easy to see: whatever you multiply one quantity by, you multiply the other by too.

Unit rates: how much for one

A unit rate tells you how much of one quantity goes with 1 unit of the other: 60 miles per hour, 30 cents per ounce, 14.50 dollars per hour. Unit rates turn many comparison problems into a single division, which is why shoppers, drivers, and cooks use them without thinking of them as math.

Percent is a rate as well: 25 percent means 25 per 100, or one quarter. A 40-dollar jacket at 25 percent off loses a quarter of its price, 10 dollars, so it costs 30 dollars.

What makes a relationship proportional

Two quantities are proportional when their ratio never changes. If you earn 14.50 dollars an hour, then pay divided by hours is always 14.50, whether you work 2 hours or 40. That fixed number is the constant of proportionality, often called k, and the relationship can be written y = kx; here, pay = 14.5 times hours.

Compare a taxi that charges 3 dollars to start plus 2 dollars a mile. The cost rises steadily, but it is not proportional: a 1-mile ride costs 5 dollars (5 dollars per mile) and a 2-mile ride costs 7 dollars (3.50 per mile), so the ratio changes from trip to trip.

Three quick tests tell proportional relationships apart from the rest:

• In a table, every y divided by its x gives the same value.
• On a graph, the points lie on a straight line that passes through the origin, (0, 0).
• As an equation, it can be written as y = kx, with nothing added on.

Solving a proportion: a map scale example

Once you know a relationship is proportional, you can set two equal ratios side by side and solve for the missing value. Keep the units in the same positions in both ratios so you compare like with like.

How ratios are taught in California, grades 6 and 7

In the California Common Core State Standards for Mathematics, ratios and proportional relationships get their own domain in grades 6 and 7, and they lead straight into slope and linear functions in grade 8.

• Grade 6 (6.RP): ratio language, unit rates, and using ratio tables, tape diagrams, double number lines, and graphs to solve problems, including percent as a rate per 100 and converting measurement units.
• Grade 7 (7.RP): unit rates with fractions (walking 1/2 mile every 1/4 hour is 2 miles per hour), deciding whether two quantities are proportional, finding the constant of proportionality, and multistep percent problems such as tax, tips, discounts, markups, and percent change.
• Grade 8 and beyond: the constant of proportionality becomes the slope of a line, and proportional relationships become the simplest linear functions.

Common mistakes with ratios

These are the errors that show up most often in grades 6 and 7:

• Adding instead of multiplying. 3:2 and 5:4 are not the same mix just because both numbers went up by 2; equivalent ratios scale by multiplying.
• Flipping the order partway through a proportion, so centimeters over kilometers on one side becomes kilometers over centimeters on the other.
• Mixing part-to-part and part-to-whole. With 12 girls and 18 boys in a class, the ratio of girls to boys is 12:18, or 2:3, but the fraction of the class that is girls is 12/30, or 2/5.
• Assuming every straight-line graph is proportional. It must also pass through the origin.
• Taking a percent of the wrong amount, such as working out a discount from the sale price instead of the original price.

Practicing ratios at home

Ratios are everywhere once you look for them: doubling a recipe, comparing price per ounce at the store, reading a map scale, working out how long a trip takes at a steady speed. Ask which quantity goes on top and why, and estimate before calculating.