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
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).
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.
3:43 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
$19.201.75 km5 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
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.