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\(a : b\)

Ratio
Problem-Solving

Master ratio reasoning to solve multi-step problems involving recipes, mixtures, and scaling with confidence.

Pre-Algebra

What Is a Ratio?

A ratio is a comparison of two quantities that tells you how much of one thing there is relative to another. If a smoothie calls for 2 cups of fruit and 3 cups of yogurt, the ratio of fruit to yogurt is \(2 : 3\).

Fruit Fruit : Yogurt Yogurt Yogurt

Equivalent Ratios

Two ratios are equivalent if you can multiply (or divide) both parts by the same nonzero number. Think of it like simplifying fractions:

\(2 : 3\)
\(\times 2\)
\(4 : 6\)
\(\times 3\)
\(12 : 18\)
 All equivalent!
\(\frac{2}{3}\)
\(\frac{4}{6}\)
\(\frac{12}{18}\)
\(=\)
\(=\)

Why Ratios Matter

Ratios are everywhere: doubling a recipe, mixing paint colors, calculating speed, converting currencies, and reading maps. Mastering ratio reasoning means you can scale any real-world quantity accurately.

Recipes
Scale ingredients up or down while keeping the same taste.
Maps
A scale of \(1 : 50{,}000\) means 1 cm on the map is 50,000 cm in real life.
Mixtures
Get the exact shade of paint by keeping color ratios constant.

The Core Rule: Proportions

A proportion is an equation that says two ratios are equal. If you know three of the four values, you can find the missing one.

\(\frac{\textcolor{#4ade80}{a}}{\textcolor{#fbbf24}{b}} = \frac{\textcolor{#4ade80}{c}}{\textcolor{#fbbf24}{d}}\)
Cross-multiply: \(a \times d = b \times c\)

Proportions and Unit Rates

A unit rate tells you the amount per one unit. It is a special ratio where the second quantity is 1. Finding the unit rate makes scaling effortless.

Original
\(6 \text{ cups} : 3 \text{ batches}\)
\(\div 3\)
 Unit Rate
\(2 \text{ cups} : 1 \text{ batch}\)
\(5 \text{ batches} \rightarrow 2 \times 5 = \textcolor{#4ade80}{10 \text{ cups}}\)

Ratio vs. Fraction vs. Rate

These three concepts are closely related but not identical. Understanding the differences helps you choose the right approach for each problem.

Concept Ratio Fraction Rate
Compares Part to part or part to whole Part to whole only Two different units
Example \(3 : 5\) \(\frac{3}{8}\) \(60 \text{ mi} / 1 \text{ hr}\)
Units Same or different Same Always different

How to Solve Ratio Problems

1
Identify the ratio. Write down the given ratio clearly, labelling what each part represents.
2
Set up a proportion. Write a second ratio with the unknown, making sure matching quantities are in the same position.
3
Cross-multiply. Use \(a \times d = b \times c\) to form an equation.
4
Solve for the unknown. Divide both sides to isolate the variable.
5
Check your answer. Simplify both ratios to confirm they are equivalent.

Example 1: Recipe Scaling

A recipe for 4 servings uses 6 cups of flour. How much flour do you need for 10 servings?

Step 1: Proportion
\(\frac{\textcolor{#4ade80}{6}}{\textcolor{#fbbf24}{4}} = \frac{\textcolor{#4ade80}{x}}{\textcolor{#fbbf24}{10}}\)
 Step 2: Cross-multiply
\(6 \times 10 = 4 \times x\)
 Step 3: Solve
\(60 = 4x \rightarrow x = \textcolor{#4ade80}{15}\)
 Answer: 15 cups
\(\frac{15}{10} = \frac{3}{2} = \frac{6}{4}\) ✓

Example 2: Mixing Paint

To make orange paint, you mix red and yellow in a ratio of \(3 : 5\). If you have 20 cups of yellow, how many cups of red do you need?

Set up proportion
\(\frac{\textcolor{#f87171}{3}}{\textcolor{#fbbf24}{5}} = \frac{\textcolor{#f87171}{x}}{\textcolor{#fbbf24}{20}}\)
 Cross-multiply
\(3 \times 20 = 5 \times x\)
 Solve
\(60 = 5x \rightarrow x = \textcolor{#f87171}{12}\)
 Answer: 12 cups red
\(\frac{12}{20} = \frac{3}{5}\) ✓

Example 3: Multi-Step Scaling

A map has a scale of \(1 : 25{,}000\). Two cities are 7.2 cm apart on the map. What is the actual distance in kilometres?

1
Use the scale ratio: \(\frac{1}{25{,}000} = \frac{7.2}{x}\), so \(x = 7.2 \times 25{,}000 = 180{,}000\) cm.
2
Convert units: \(180{,}000 \text{ cm} \div 100 = 1{,}800 \text{ m}\).
3
Convert again: \(1{,}800 \text{ m} \div 1{,}000 = \) \(1.8\) km.

Part-to-Part vs. Part-to-Whole

A class has boys and girls in the ratio \(3 : 7\). There are 40 students total. How many are boys?

Bar Model (each block = 1 part) Boys: 3 parts Girls: 7 parts Solution
\(\text{Total parts} = 3 + 7 = 10 \rightarrow 1 \text{ part} = 40 \div 10 = \textcolor{#fbbf24}{4} \rightarrow \text{Boys} = 3 \times 4 = \textcolor{#818cf8}{12}\)

Using a Ratio Table

A ratio table organizes equivalent ratios in rows. Each column is found by multiplying by the same scale factor.

Scale Factor \(\times 1\) \(\times 2\) \(\times 3\) \(\times 5\)
Lemons 2 4 6 10
Cups of Water 5 10 15 25
Sugar (tbsp) 3 6 9 15

Common Mistakes

✕ Flipping the Ratio
Writing \(5 : 3\) when the problem says \(3 : 5\). Order matters: the first number matches the first quantity named.
✕ Adding Instead of Multiplying
To scale \(2 : 3\) up by 4, multiply both parts: \(8 : 12\). Adding 4 gives \(6 : 7\), which is a completely different ratio.
✕ Forgetting Units
Getting the right number but in the wrong unit. Always convert at the end if the problem asks for different units than you started with.

Strategy Summary

Problem Type Method Key Step
Find missing value Proportion + cross-multiply \(\frac{a}{b} = \frac{c}{x}\)
Scale a recipe Unit rate or ratio table Find amount per 1 serving
Part-to-whole Add parts, divide total \(\text{Total} \div \text{sum of parts}\)
Mixture problems Ratio table (multiple rows) Same scale factor every row
Map/scale drawing Proportion + unit conversion Don't forget unit change

Question 1 of 5

A recipe calls for 4 cups of flour for every 6 cups of sugar. If you use 10 cups of flour, how many cups of sugar do you need?

Question 2 of 5

The ratio of red marbles to blue marbles in a bag is \(5 : 8\). If there are 40 blue marbles, how many total marbles are in the bag?

Question 3 of 5

On a map with scale \(1 : 50{,}000\), two towns are 3.6 cm apart. What is the actual distance in kilometres?

Question 4 of 5

A smoothie uses fruit and yogurt in the ratio \(2 : 3\). You want to make 20 cups total. How many cups of fruit do you need?

Question 5 of 5

A painter mixes white and blue paint in the ratio \(7 : 2\). She uses 36 litres of paint in total. How many litres of blue paint did she use?