Ratios and Proportions

Compare quantities and scale with ease

Have you ever doubled a recipe? Split a bill with friends? Figured out how long a road trip would take based on your speed? If so, you have already been working with ratios and proportions without even realizing it.

These concepts might sound intimidating at first, with their formal-sounding names, but here is the truth: ratios and proportions are just ways of comparing things and keeping those comparisons fair when numbers change. You do this naturally all the time. This chapter will give you the vocabulary and techniques to do it with confidence, whether you are in the kitchen, on a road trip, or facing a math test.

Core Concepts

What is a Ratio?

A ratio is simply a comparison between two quantities. That is it. When you say “there are twice as many dogs as cats at the park,” you are describing a ratio.

Think of it this way: ratios answer the question, “How does this quantity relate to that quantity?”

You already use ratios constantly:

“I got 8 out of 10 questions right” (comparing correct answers to total questions)
“The recipe calls for 2 cups of flour for every 1 cup of sugar” (comparing flour to sugar)
“We drove 120 miles in 2 hours” (comparing distance to time)

Equivalent Ratios

Here is something important: the same ratio can be written in many different ways. Just like $\frac{1}{2}$ and $\frac{2}{4}$ represent the same amount, the ratio 1:2 and the ratio 2:4 represent the same relationship.

Equivalent ratios are ratios that express the same comparison. You create them by multiplying or dividing both parts of the ratio by the same number.

For example, these are all equivalent ratios: $$1:2 = 2:4 = 3:6 = 10:20 = 50:100$$

Each one says the same thing: the second quantity is twice the first.

Unit Rates

A unit rate is a special kind of ratio where the second quantity is exactly 1. Unit rates make comparisons easy because they tell you “how much per one.”

You see unit rates everywhere:

$60 per hour (dollars per one hour)
25 miles per gallon (miles per one gallon)
$3.50 per pound (dollars per one pound)

Unit rates are incredibly useful because they let you quickly calculate for any amount. If you know something costs $3.50 per pound, you can instantly figure out the cost of 4 pounds ($14.00) or 0.5 pounds ($1.75).

What is a Proportion?

A proportion is an equation that says two ratios are equal. That is all it is: two ratios with an equals sign between them.

$$\frac{1}{2} = \frac{3}{6}$$

This proportion says “1 is to 2 as 3 is to 6.” Both ratios represent the same relationship (one half).

Proportions become powerful when one of the numbers is unknown. If you know three of the four numbers in a proportion, you can always find the fourth.

Notation and Terminology

Ways to Write Ratios

Ratios can be written in three different ways. All three mean exactly the same thing:

Notation	Example	Read as
Colon	$3:4$	“3 to 4”
Fraction	$\frac{3}{4}$	“3 to 4” or “3 over 4”
Words	3 to 4	“3 to 4”

The fraction form is especially useful when you need to do calculations.

Key Terms

Ratio: A comparison of two quantities
Terms: The numbers in a ratio (in $3:4$, the terms are 3 and 4)
Equivalent ratios: Ratios that represent the same comparison
Unit rate: A ratio with a denominator of 1
Proportion: An equation stating that two ratios are equal
Cross products: The products you get when you multiply diagonally across a proportion

Cross-Multiplication

When you have a proportion like $\frac{a}{b} = \frac{c}{d}$, there is a powerful shortcut called cross-multiplication:

$$a \times d = b \times c$$

The “cross products” are always equal in a true proportion. This gives you a way to:

Check if two ratios are equivalent
Solve for an unknown value

Examples

Example 1: Writing Ratios

A classroom has 12 boys and 18 girls. Write the ratio of boys to girls in three ways, then simplify.

Solution:

The ratio of boys to girls can be written as:

$12:18$ (colon notation)
$\frac{12}{18}$ (fraction notation)
12 to 18 (word notation)

To simplify, find the greatest common factor (GCF) of 12 and 18, which is 6. Divide both terms by 6:

$$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$$

The simplified ratio is $2:3$, which means for every 2 boys, there are 3 girls.

Example 2: Finding a Unit Rate

You drove 240 miles and used 8 gallons of gas. What is your car’s fuel efficiency in miles per gallon?

Solution:

We need to find the unit rate: miles per one gallon.

$$\text{Unit rate} = \frac{240 \text{ miles}}{8 \text{ gallons}} = \frac{240 \div 8}{8 \div 8} = \frac{30 \text{ miles}}{1 \text{ gallon}}$$

Your car gets 30 miles per gallon.

Now you can easily calculate for any amount of gas. With 12 gallons, you could drive $30 \times 12 = 360$ miles.

Example 3: Solving a Proportion

If 3 notebooks cost $7.50, how much would 7 notebooks cost?

Solution:

Set up a proportion. Let $x$ be the cost of 7 notebooks.

$$\frac{3 \text{ notebooks}}{$7.50} = \frac{7 \text{ notebooks}}{x}$$

Cross-multiply: $$3 \times x = 7.50 \times 7$$ $$3x = 52.50$$

Divide both sides by 3: $$x = \frac{52.50}{3} = 17.50$$

7 notebooks would cost $17.50.

You can check this makes sense: 7 notebooks is a bit more than double 3 notebooks, and $17.50 is a bit more than double $7.50.

Example 4: Scaling a Recipe

A cookie recipe makes 24 cookies and calls for:

2 cups flour
1 cup sugar
3 eggs

You want to make 60 cookies for a party. How much of each ingredient do you need?

Solution:

First, find the scale factor by setting up a proportion: $$\frac{24 \text{ cookies}}{60 \text{ cookies}} = \frac{24}{60} = \frac{2}{5}$$

This means the original recipe is $\frac{2}{5}$ of what we need, so we need to multiply everything by $\frac{60}{24} = \frac{5}{2} = 2.5$.

Flour: $$2 \text{ cups} \times 2.5 = 5 \text{ cups}$$

Sugar: $$1 \text{ cup} \times 2.5 = 2.5 \text{ cups}$$

Eggs: $$3 \text{ eggs} \times 2.5 = 7.5 \text{ eggs}$$

For 60 cookies, you need 5 cups flour, 2.5 cups sugar, and 7.5 eggs (round to 7 or 8 eggs in practice).

Example 5: Map Scale Problem

On a map, 2 inches represents 35 miles. Two cities are 5.5 inches apart on the map. What is the actual distance between them? If you drive at 50 mph, how long will the trip take?

Solution:

Part 1: Find the actual distance

Set up a proportion: $$\frac{2 \text{ inches}}{35 \text{ miles}} = \frac{5.5 \text{ inches}}{x \text{ miles}}$$

Cross-multiply: $$2 \times x = 35 \times 5.5$$ $$2x = 192.5$$ $$x = 96.25 \text{ miles}$$

Part 2: Find the travel time

Now use the speed to find time. Since speed = distance/time, we have time = distance/speed:

$$\text{Time} = \frac{96.25 \text{ miles}}{50 \text{ mph}} = 1.925 \text{ hours}$$

Convert to hours and minutes: $0.925 \times 60 = 55.5$ minutes.

The cities are 96.25 miles apart, and the trip would take about 1 hour and 56 minutes.

Key Properties and Rules

Properties of Ratios

Order matters: The ratio $3:4$ is different from $4:3$. Always be clear about which quantity comes first.
Same units: When comparing quantities, make sure they are in the same units first. Do not compare 2 feet to 18 inches without converting.
Simplifying: Ratios can be simplified just like fractions. Divide both terms by their GCF.

Rules for Proportions

Cross-multiplication: If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$
Means and extremes: In the proportion $a:b = c:d$, the product of the means ($b \times c$) equals the product of the extremes ($a \times d$)
Solving proportions: When solving $\frac{a}{b} = \frac{c}{x}$:
- Cross-multiply: $a \cdot x = b \cdot c$
- Divide: $x = \frac{b \cdot c}{a}$

Finding Unit Rates

To convert any ratio to a unit rate, divide both terms by the second term: $$\frac{a}{b} = \frac{a \div b}{b \div b} = \frac{a \div b}{1}$$

Real-World Applications

Cooking and Baking

Every recipe is built on ratios. Professional bakers often work with “baker’s percentages” where all ingredients are expressed as ratios to the flour weight. When you double or halve a recipe, you are working with proportions.

Maps and Scale Models

Maps use ratios like 1:24,000, meaning 1 inch on the map equals 24,000 inches (2,000 feet) in real life. Architects use similar scales for blueprints, and model builders use them for miniatures.

Shopping and Best Deals

Unit prices (cost per ounce, per item, etc.) let you compare deals. A 16 oz jar for $4.00 ($0.25/oz) is a better deal than a 10 oz jar for $3.00 ($0.30/oz).

Speed and Travel

Speed is a unit rate: miles per hour or kilometers per hour. When you calculate “how long until we get there?” you are solving a proportion.

Sports Statistics

Batting averages, shooting percentages, and yards per carry are all ratios. A basketball player shooting 45% from the field has a ratio of made shots to attempted shots of roughly 9:20.

Medicine and Science

Medication dosages are often based on body weight ratios. A prescription might call for 5 mg per kg of body weight. Solution concentrations in chemistry are ratios of solute to solution.

Finance

Interest rates are ratios. A 5% annual interest rate means you earn $5 for every $100 invested per year. Exchange rates between currencies are also ratios.

Self-Test Problems

Problem 1 (Easy)

A bag contains 15 red marbles and 25 blue marbles. Write the ratio of red to blue marbles in simplest form.

Show Answer

$$\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}$$

The ratio is $3:5$ (3 red marbles for every 5 blue marbles).

Problem 2 (Easy)

A car travels 195 miles on 6.5 gallons of gas. What is the fuel efficiency in miles per gallon?

Show Answer

$$\frac{195 \text{ miles}}{6.5 \text{ gallons}} = 30 \text{ miles per gallon}$$

Problem 3 (Medium)

Solve for $x$: $\frac{4}{9} = \frac{x}{27}$

Show Answer

Cross-multiply: $$4 \times 27 = 9 \times x$$ $$108 = 9x$$ $$x = 12$$

Check: $\frac{4}{9} = \frac{12}{27}$ (both simplify to $\frac{4}{9}$)

Problem 4 (Medium)

A pancake recipe calls for 1.5 cups of flour to make 10 pancakes. How much flour do you need to make 35 pancakes?

Show Answer

Set up and solve the proportion: $$\frac{1.5 \text{ cups}}{10 \text{ pancakes}} = \frac{x \text{ cups}}{35 \text{ pancakes}}$$

Cross-multiply: $$1.5 \times 35 = 10 \times x$$ $$52.5 = 10x$$ $$x = 5.25 \text{ cups}$$

You need 5.25 cups (or 5 and 1/4 cups) of flour.

Problem 5 (Hard)

On a blueprint, 0.5 inches represents 4 feet. If a room measures 3.25 inches by 2.75 inches on the blueprint, what is the actual area of the room in square feet?

Show Answer

First, find the scale: $\frac{0.5 \text{ in}}{4 \text{ ft}}$ means 1 inch = 8 feet.

Find actual dimensions:

Length: $3.25 \times 8 = 26$ feet
Width: $2.75 \times 8 = 22$ feet

Calculate area: $$\text{Area} = 26 \times 22 = 572 \text{ square feet}$$

Summary

A ratio compares two quantities. It can be written as $a:b$, $\frac{a}{b}$, or “a to b.”
Equivalent ratios represent the same comparison. Create them by multiplying or dividing both terms by the same number.
A unit rate has a denominator of 1 (like $30 per hour or 25 miles per gallon) and makes calculations easy.
A proportion is an equation showing two ratios are equal: $\frac{a}{b} = \frac{c}{d}$
Cross-multiplication is your go-to tool for solving proportions: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$.
Ratios and proportions appear everywhere: cooking, maps, shopping, travel, sports, medicine, and finance.
To scale something up or down, find your scale factor and multiply all quantities by it.
Always check that your answer makes sense. If you are scaling up, your answer should be larger. If scaling down, it should be smaller.