Math — Ratios

Ratios and Proportions

From recipe scaling to map reading, ratios and proportions are everywhere. Learn to write, simplify, and solve them confidently.

Learning Objectives

  • Write a ratio in three equivalent forms: a:b, a/b, and "a to b"
  • Simplify ratios by dividing by the greatest common factor
  • Identify whether two ratios form a proportion
  • Solve proportion equations using cross-multiplication
  • Calculate unit rates and apply them to real-world problems

Prerequisites

Comfortable with fractions and basic multiplication and division. The Khan Academy Ratios and Rates unit pairs well with this lesson.

The Lesson

Step 1 — What is a ratio?

A ratio compares two quantities. If a class has 12 girls and 8 boys, the ratio of girls to boys is 12:8. This can also be written as 12/8 or "12 to 8." Order matters — always match the ratio order to the words in the question.

Three equivalent ways to write the same ratio:
12 to 8  |  12:8  |  12/8
Simplified (divide both by GCF=4): 3:2

Step 2 — Simplifying ratios

Find the Greatest Common Factor (GCF) of both numbers and divide each part by it.

Simplify 30:45:
GCF(30,45) = 15  →  30÷15 : 45÷15 = 2:3

Step 3 — What is a proportion?

A proportion is a statement that two ratios are equal: a/b = c/d. Check by cross-multiplying — if a×d = b×c, the ratios are proportional.

Are 3/4 and 9/12 proportional?
3×12 = 36  &&  4×9 = 36  →  Yes, they are proportional.

Step 4 — Solving a proportion with a missing value

When one value is unknown, set up the proportion, cross-multiply, then divide.

Example: A recipe uses 2 cups of flour for 3 cookies. How many cups for 12 cookies?
2/3 = x/12  →  cross-multiply: 3x = 24  →  x = 8 cups
Example 2: On a map, 1 cm = 50 km. A road measures 7 cm on the map. Real distance?
1/50 = 7/x  →  x = 7×50 = 350 km

Step 5 — Unit rates

A unit rate is a ratio with a denominator of 1, making comparison easy. Divide both quantities by the denominator.

Find unit rate: A car travels 240 miles in 4 hours.
240 ÷ 4 = 60 miles per hour

Best value: Brand A: 360 g for $2.40 → $2.40÷360 = $0.0067/g. Brand B: 500 g for $3.50 → $3.50÷500 = $0.0070/g. Brand A is cheaper per gram.

Step 6 — Part-to-whole ratios

Sometimes a ratio compares a part to the whole (like a fraction). If a bag contains 5 red and 3 blue marbles, the ratio red:total = 5:8, meaning 5/8 of all marbles are red.

Example: A mixture is 2 parts sugar to 5 parts flour (total 7 parts). What fraction is sugar?
Sugar = 2/7 of the mixture. In a 350 g batch, sugar = (2/7) × 350 = 100 g

Practice Problems

  1. Q: Simplify the ratio 56:84.
    Solution: GCF=28 → 56÷28 : 84÷28 = 2:3
  2. Q: Solve the proportion: 5/8 = x/40.
    Solution: 8x = 200 → x = 25
  3. Q: A runner completes 15 km in 75 minutes. What is her speed in km per minute?
    Solution: 15÷75 = 0.2 km/min
  4. Q: A paint mix requires 3 parts blue to 7 parts white. How many litres of blue are needed for 30 litres total?
    Solution: Blue = 3/10 × 30 = 9 litres
  5. Q: Are 7:4 and 21:11 proportional?
    Solution: 7×11=77; 4×21=84. 77 ≠ 84 → Not proportional.

Common Mistakes

Mistake 1 — Reversing the ratio order. "Girls to boys is 3:2" is different from "boys to girls is 3:2." Always match the order of the words in the problem.
Mistake 2 — Forgetting to simplify before checking proportionality. Work with the simplest form to reduce arithmetic errors.
Mistake 3 — Cross-multiplying when the problem only asks for simplification. Not every ratio problem needs cross-multiplication — save it for equations with unknowns.
Mistake 4 — Comparing unit rates in different units. Always convert to the same units before comparing — kg vs g, hours vs minutes, etc.
Mistake 5 — Treating a part:part ratio as a fraction of the whole. If the ratio is 3:5, the fraction of the whole is 3/(3+5) = 3/8, not 3/5.

Further Practice Resources

Khan Academy — Ratios, Rates, and Proportions Math Is Fun — Ratio and Proportion explained Britannica — Ratio in mathematics Wikipedia — Ratio (mathematical definition and history) MIT OCW — Mathematics foundations

Frequently Asked Questions

What is a ratio?

A ratio compares two quantities by division, written as a:b. For example, 3:5 means for every 3 of one thing there are 5 of another.

What is a proportion?

A proportion is an equation stating that two ratios are equal: a/b = c/d.

How do you solve a proportion?

Use cross-multiplication: if a/b = c/d, then a×d = b×c. Solve for the unknown.

What is a unit rate?

A unit rate expresses a ratio with a denominator of 1, e.g. 60 miles per hour. Divide both parts by the denominator to find it.

How are ratios used in real life?

Ratios appear in cooking (recipe scaling), maps (scale 1:50,000), speed, finance, and medicine (dosage per kg).