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§ Ratios & Proportions

Ratios & Proportions

§ Ratios & Proportions

Ratios & Proportions

CCSS.6.RPCCSS.7.RP3 min read

A ratio expresses the relationship between two quantities by showing how many times one contains the other, written as a:b or a/b. Proportions occur when two ratios are equal, such as 3:4 = 6:8, forming the foundation for solving problems involving scaling, rates, and comparisons. The cross-multiplication method (if a/b = c/d, then a×d = b×c) provides a systematic way to find unknown values in proportional relationships.

§ 01

Why it matters

Ratios and proportions appear throughout daily life and advanced mathematics. Architects use proportions when scaling blueprints where 1 inch represents 10 feet of actual building dimensions. Pharmacists calculate medication dosages using ratios — if 5 mg works for a 100-pound patient, a 150-pound patient needs 7.5 mg. Recipe scaling relies on proportional reasoning: doubling a recipe that serves 4 people to serve 8 requires maintaining the same ingredient ratios. In finance, gear ratios in cars typically range from 3:1 to 5:1, affecting fuel efficiency and performance. These concepts extend into similar triangles in geometry, probability calculations, and unit conversions in physics and chemistry, making ratio reasoning essential for STEM fields.

§ 02

How to solve ratios & proportions

Ratios & Proportions

  • A ratio compares two quantities (a:b or a/b).
  • To solve a proportion a/b = c/d: cross-multiply (a×d = b×c).
  • Simplify ratios by dividing both by their GCF.

Example: 23 = x/12 → 2×12 = 3x → x = 8.

§ 03

Worked examples

Beginner§ 01

Express as the simplest ratio: 11 boys and 12 girls.

Answer: 11:12

  1. Write the ratio → 11:12 — Boys to girls.
  2. Find GCF of 11 and 12 → GCF = 1 — Divide both by the GCF.
  3. Simplify → 11÷1 : 12÷1 = 11:12 — Simplified ratio.
Easy§ 02

A recipe for 6 people uses 24 grams of butter. How much do you need for 10 people?

Answer: 40 grams

  1. Find amount per person → 24 ÷ 6 = 4 per person — Divide by original servings.
  2. Multiply for new servings → 4 × 10 = 40 — Per-person amount times new number of people.
Medium§ 03

A car drove 208 km in 4 hours. At the same speed, how far will it drive in 5 hours?

Answer: 260 km

  1. Find the speed → 208 ÷ 4 = 52 km/h — Distance divided by time.
  2. Multiply by new time → 52 × 5 = 260 km — Speed times new duration.
§ 04

Common mistakes

  • Setting up proportions backwards, such as writing 3 apples/2 dollars = x apples/8 dollars as 3/2 = 8/x instead of 3/2 = x/8, leading to x = 12 apples instead of the correct 12 apples
  • Adding instead of multiplying when cross-multiplying, solving 4/5 = x/10 as 4 + 10 = 5 + x, giving x = 9 instead of the correct x = 8
  • Forgetting to simplify ratios by dividing by the greatest common factor, leaving 12:18 as the final answer instead of reducing it to 2:3
§ 05

Frequently asked questions

What is the difference between a ratio and a proportion?
A ratio compares two quantities (like 3:4), while a proportion states that two ratios are equal (like 3:4 = 6:8). Think of a ratio as a single comparison and a proportion as an equation showing two ratios have the same relationship.
How do you check if your proportion answer is correct?
Substitute your answer back into the original proportion and cross-multiply. For example, if solving 2/3 = x/12 gives x = 8, check by verifying 2 × 12 = 3 × 8, which equals 24 = 24, confirming the answer is correct.
When should you use unit rates versus proportions?
Use unit rates when finding cost per item or speed per hour (like $3.50 per pound). Use proportions when you know corresponding values and need to find a missing quantity (like if 3 pounds costs $10.50, what does 5 pounds cost).
What does it mean to simplify a ratio?
Simplifying a ratio means dividing both numbers by their greatest common factor to get the smallest possible whole numbers. For example, 8:12 simplifies to 2:3 by dividing both sides by 4, making the ratio easier to understand and work with.
How do you solve word problems involving ratios?
Identify what quantities are being compared, write the ratio using the same units, set up a proportion if needed, and cross-multiply to solve. Always label your units and check that your answer makes sense in the context of the problem.
§ 06

See also

RatioRangeProbabilityProportionFactorEquationMeanSet
§ 06

Where to next?

Prerequisites
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