Ratios & Proportional Relationships

Understanding Ratios

Ratios as a relationship between two quantities.

Foundational15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Comparing Two Quantities Together

A recipe calls for 2 cups of flour for every 1 cup of sugar. If you wanted to make a bigger batch — say, using 3 cups of sugar — how much flour would you need? Before reading on, think about what stays the same between the small batch and the big batch, even though both amounts change.

What stays the same is the relationship between the two ingredients: there's always twice as much flour as sugar. That relationship, not the exact amounts, is what a ratio captures. Scale the batch up or down, and as long as that relationship holds, the recipe still tastes the same.

Definition — Ratio

A comparison of two quantities by division, showing how many times one quantity contains — or is contained in — another. The ratio of flour to sugar in the recipe is written 2:1, 2 to 1, or 2/1.

A ratio is different from a plain difference. Saying 'flour is 1 cup more than sugar' describes a fixed gap that wouldn't scale — doubling the recipe would need 2 more cups of flour than sugar, not 1. A ratio instead describes a multiplicative relationship, which is exactly why it scales correctly: 4 cups of flour to 2 cups of sugar is still the same ratio, 2:1, because 4 is still twice 2.

Worked Example — Writing a Ratio from a Real Situation

A classroom has 12 students and 3 tables. What is the ratio of students to tables? Write the two quantities in the order they're compared: 12 to 3, or 12:3. Since both numbers share a common factor of 3, this simplifies to 4:1 — 4 students for every 1 table.

Worked Example — Using a Ratio to Scale a Quantity

A paint mixture uses a ratio of 3:2, blue to white. If 9 cups of blue paint are used, how much white paint is needed to keep the same ratio? Since 9 is 3 times as much as the original 3 cups of blue, scale the white paint by the same factor: 2 × 3 = 6 cups of white paint.

Tip

A ratio can always be scaled up or down by multiplying or dividing both quantities by the same number — the same idea behind simplifying and generating equivalent fractions.

Common Mistakes

Writing a ratio in the wrong order, such as writing 'students to tables' as 3:12 instead of 12:3.
The order in a ratio matches the order the quantities are named — 'students to tables' always lists the student count first.
Treating a ratio like a difference, and adding the same fixed amount to both quantities when scaling instead of multiplying.
Scale a ratio by multiplying (or dividing) both quantities by the same factor — adding the same number to both breaks the relationship the ratio describes.

Key Takeaways

A ratio compares two quantities by division, describing a multiplicative relationship rather than a fixed difference.
A ratio can be scaled by multiplying or dividing both quantities by the same number without changing the relationship.
The order of a ratio matches the order the quantities are described in.

Summary

Ratios describe how two quantities relate to each other, in a way that scales consistently. The next lesson uses that same relationship to answer a very practical question: how much of one quantity corresponds to just a single unit of the other.

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