Home › Math Lessons › Algebra › Ratios and Proportions
AlgebraBeginner

Ratios and Proportions

Ratios compare two quantities. Proportions state that two ratios are equal. Together they are among the most practically useful math concepts — appearing in cooking, maps, finance, medicine, and almost every quantitative field.

In this lesson

  1. What ratios are
  2. Simplifying ratios
  3. What proportions are
  4. Solving proportions with cross-multiplication
  5. Real-world applications

1 What Ratios Are

A ratio is a comparison of two quantities by division. If a recipe uses 2 cups of flour for every 1 cup of sugar, the ratio of flour to sugar is 2:1 (read "2 to 1"). This means for every 2 cups of flour, you use 1 cup of sugar.

Ratios can be written three ways: 2:1, 2/1, or "2 to 1." They all mean the same thing. Order matters — the ratio of flour to sugar (2:1) is different from sugar to flour (1:2).

Ratios vs fractions

A ratio of 3:4 looks like the fraction 3/4 but means something different. 3/4 as a fraction means 3 out of 4 total. 3:4 as a ratio means 3 of one thing for every 4 of another — a total of 7 parts. Context determines which interpretation applies.

2 Simplifying Ratios

Like fractions, ratios can be simplified by dividing both parts by their GCF (greatest common factor). The ratio 12:8 simplifies to 3:2 because GCF(12, 8) = 4. The simplified ratio expresses the same relationship in smaller numbers.

Simplifying a Ratio
A class has 18 boys and 24 girls. Express the ratio in simplest form.
1Ratio: 18:24
2GCF(18, 24) = 6
3Divide both by 6: 18÷6 : 24÷6 = 3:4
4For every 3 boys there are 4 girls
Answer: 3:4

3 What Proportions Are

A proportion states that two ratios are equal: a/b = c/d. If 2 apples cost $1, then 6 apples cost $3 because 2/1 = 6/3. The ratios are equal — the relationship between apples and cost is consistent.

In a proportion a/b = c/d, the cross products are equal: a × d = b × c. This property is used to solve for unknown values.

4 Solving Proportions with Cross-Multiplication

Solving a Proportion
If 5 items cost $12, how much do 8 items cost?
1Set up the proportion: 5/12 = 8/x
2Cross-multiply: 5x = 12 × 8 = 96
3Solve: x = 96/5 = $19.20
Answer: 8 items cost $19.20
Map Scale
A map has scale 1:50,000. A distance of 3.5 cm on the map represents how many km in reality?
11 cm = 50,000 cm = 500 m = 0.5 km in reality
23.5 cm on map = 3.5 × 0.5 km = 1.75 km
3Or use proportion: 1/50000 = 3.5/x → x = 175,000 cm = 1.75 km
Answer: 1.75 km

5 Real-World Applications

Cooking: scaling recipes up or down uses proportions. If a recipe for 4 serves uses 300g of pasta, a recipe for 10 serves needs (10/4) × 300 = 750g. Every ingredient scales by the same ratio.

Medicine: drug dosing is often proportional to body weight. If the dose is 5mg per kg and a patient weighs 68kg, the dose is 5 × 68 = 340mg. Proportion errors in medicine can be dangerous — this is one reason why calculation accuracy is critical in healthcare.

Finance: currency exchange and interest calculations are proportional. If $1 = €0.92, then $350 = 350 × 0.92 = €322. Unit rates (price per item, speed, pay per hour) are all ratios applied practically.

Practice Problems

Simplify the ratio 45:30.
GCF(45,30) = 15. 45÷15 : 30÷15 = 3:2
Solve: 7/4 = x/20
Cross-multiply: 4x = 140. x = 35
A car travels 150 miles in 3 hours. How far does it go in 5 hours at the same speed?
150/3 = x/5. Cross-multiply: 3x = 750. x = 250 miles
A recipe for 6 cookies needs 2 eggs. How many eggs for 21 cookies?
6/2 = 21/x. Cross-multiply: 6x = 42. x = 7 eggs