Chapter 3: Ratios, Rates & Proportions
Every time you compare two quantities — the ratio of wins to losses, the price per ounce of a product, or the scale of a map — you are working with ratios, rates, and proportions. These concepts are among the most practically useful in all of mathematics, forming the foundation of percent calculations, unit conversions, similar geometry, and algebraic direct variation.
3.1 Ratios
Definition: Ratio
A ratio is a comparison of two quantities $a$ and $b$ by division. It can be written in three equivalent ways: $$a \text{ to } b \qquad a:b \qquad \frac{a}{b}$$ A ratio should always be expressed in simplest form by dividing both terms by their GCF.
Unlike a fraction, a ratio compares quantities that may or may not have the same units. When the units are the same, they cancel and the ratio is dimensionless. When units differ, the ratio becomes a rate.
Example 3.1 — Writing and Simplifying Ratios
A class has 18 girls and 12 boys. Write the ratio of girls to boys in simplest form.
This means for every 3 girls there are 2 boys. The total is divided into $3 + 2 = 5$ parts, so girls make up $\frac{3}{5}$ of the class.
3.2 Rates and Unit Rates
Definition: Rate and Unit Rate
A rate is a ratio that compares two quantities measured in different units. A unit rate is a rate with a denominator of 1 — it tells you how much of one quantity corresponds to exactly one unit of the other.
Rate Example
$\dfrac{240 \text{ miles}}{4 \text{ hours}}$
Compares miles to hours.
Unit Rate Example
$\dfrac{60 \text{ miles}}{1 \text{ hour}} = 60$ mph
Miles per one hour.
Rate Example
$\dfrac{\$7.50}{3 \text{ lb}}$
Cost compared to weight.
Unit Rate Example
$\dfrac{\$2.50}{1 \text{ lb}} = \$2.50$/lb
Cost per one pound.
Example 3.2 — Finding and Comparing Unit Rates
Store A sells 5 pounds of apples for $\$6.75$. Store B sells 8 pounds for $\$10.40$. Which store has the better unit price?
3.3 Proportions
Definition: Proportion
A proportion is a statement that two ratios are equal: $$\frac{a}{b} = \frac{c}{d}$$ We say "$a$ is to $b$ as $c$ is to $d$." The values $b$ and $c$ are the means; $a$ and $d$ are the extremes.
Cross-Multiplication Property
In any true proportion $\dfrac{a}{b} = \dfrac{c}{d}$, the cross products are equal: $$a \times d = b \times c$$ This is equivalent to multiplying both sides by $bd$. We use this to solve for an unknown.
Example 3.3 — Solving a Proportion
Solve for $x$: $\dfrac{x}{15} = \dfrac{8}{12}$
Example 3.4 — Real-World Proportion
A car travels 156 miles on 6 gallons of gasoline. At the same rate, how far can it travel on 10 gallons?
3.4 Scale Drawings and Maps
A scale drawing represents a real object with all measurements reduced or enlarged by the same ratio, called the scale factor. Maps, blueprints, and model trains all use scale drawings.
Example 3.5 — Using a Map Scale
On a map, 1 inch represents 50 miles. Two cities are $3\dfrac{1}{2}$ inches apart on the map. What is the actual distance between the cities?
Example 3.6 — Finding a Scale Factor
A model car is 8 inches long. The actual car is 15 feet long. Express the scale as a ratio (model : actual) using the same units.
3.5 Similar Figures
Definition: Similar Figures
Two figures are similar (~) if they have the same shape but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional. $$\frac{\text{side}_1}{\text{side}_1'} = \frac{\text{side}_2}{\text{side}_2'} = \frac{\text{side}_3}{\text{side}_3'} = k$$ where $k$ is the scale factor.
Example 3.7 — Finding Missing Sides in Similar Triangles
Triangle $ABC \sim$ Triangle $DEF$. The sides of $\triangle ABC$ are 6, 8, and 10. The shortest side of $\triangle DEF$ is 9. Find the other two sides of $\triangle DEF$.
Note: 9-12-15 is a multiple of the 3-4-5 Pythagorean triple. ✓
3.6 Direct Variation
Definition: Direct Variation
Two quantities $x$ and $y$ are in direct variation (or are directly proportional) if there is a nonzero constant $k$ such that: $$y = kx$$ The constant $k$ is called the constant of variation or constant of proportionality. The graph is always a straight line through the origin.
The interactive graph below shows three direct variation relationships with different constants of proportionality $k$. Notice that a larger $k$ produces a steeper line.
Example 3.8 — Identifying and Using Direct Variation
The table below shows the relationship between hours worked $x$ and pay $y$. Determine whether $y$ varies directly with $x$, and if so, write the equation.
| Hours ($x$) | Pay ($y$) | $y/x$ |
|---|---|---|
| 2 | $\$28$ | $14$ |
| 5 | $\$70$ | $14$ |
| 8 | $\$112$ | $14$ |
3.7 Percent Proportions
Many percent problems can be solved efficiently using a proportion. Every percent problem involves three quantities: the percent (P), the whole (or base, W), and the part (A). They satisfy:
Percent Proportion Formula
$$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$$Equivalently: $\dfrac{A}{W} = \dfrac{P}{100}$, which gives $A = \dfrac{P \times W}{100}$.
You can solve for any one of the three quantities given the other two.
Example 3.9 — Three Types of Percent Problems
Type 1: Find the Part. What is 35% of 80?
Type 2: Find the Whole. 18 is 45% of what number?
Type 3: Find the Percent. 21 is what percent of 84?
Setting Up Proportions Correctly
- Label your ratios: Always write units or labels next to each quantity to make sure corresponding parts are in the same position.
- Check your proportion: After solving, substitute back and verify the cross products are equal.
- Unit consistency: Both ratios in a proportion must have the same units in the numerator and the same units in the denominator.
- Part-to-whole vs. part-to-part: In similarity, use corresponding side to corresponding side, not mixed ratios.
Example 3.10 — Multi-Step Proportion Problem
A recipe for 24 cookies requires $\dfrac{3}{4}$ cup of butter. You want to make 40 cookies. How much butter do you need? If butter costs $\$3.60$ for 1 cup, what is the cost of the butter needed?
Practice Problems
Chapter 3 Practice
Ratios and Rates
- A bag contains 15 red marbles and 25 blue marbles. Write the ratio of red to blue in simplest form.
- A car travels 390 miles in 6 hours. Find the unit rate (speed in mph).
- Brand X: 16 oz for $\$2.40$. Brand Y: 12 oz for $\$1.92$. Which is the better buy?
Proportions
- Solve: $\dfrac{x}{20} = \dfrac{9}{12}$
- Solve: $\dfrac{5}{8} = \dfrac{15}{y}$
- On a blueprint, $\frac{1}{4}$ inch = 1 foot. A room is $3\frac{1}{2}$ inches wide on the blueprint. How wide is the actual room?
Similar Figures
- Two similar rectangles have widths of 4 cm and 10 cm. If the shorter rectangle has length 7 cm, find the length of the larger rectangle.
Direct Variation & Percent Proportions
- $y$ varies directly with $x$. When $x = 5$, $y = 35$. Find $y$ when $x = 9$.
- 36 is what percent of 90?
- 72 is 60% of what number?
Answers: 1) $3:5$ 2) 65 mph 3) Brand X ($0.15$/oz) vs Brand Y ($0.16$/oz); Brand X 4) $x = 15$ 5) $y = 24$ 6) 14 feet 7) 17.5 cm 8) $y = 63$ 9) 40% 10) 120
Chapter Summary
- A ratio compares two quantities; always simplify using the GCF.
- A rate compares quantities with different units; a unit rate has denominator 1.
- A proportion sets two ratios equal: $\dfrac{a}{b} = \dfrac{c}{d}$.
- Cross-multiplication $ad = bc$ lets you solve for any unknown in a proportion.
- Scale drawings use a consistent scale factor to represent real objects.
- Similar figures have equal corresponding angles and proportional corresponding sides.
- Direct variation $y = kx$ produces a line through the origin with slope $k$.
- The percent proportion $\dfrac{\text{Part}}{\text{Whole}} = \dfrac{\text{Percent}}{100}$ solves all three types of percent problems.