Ratios compare two quantities and show how much of one amount there is for another amount. They are used in recipes, maps, scale drawings, speed, prices, mixtures, and probability. A proportion says that two ratios are equal, which lets you use a known comparison to find an unknown value.
Learning ratios and proportions builds a foundation for algebra, geometry, science, and everyday problem solving.
A ratio can be written as 2:3, 2 to 3, or 2/3, depending on the situation. Equivalent ratios are made by multiplying or dividing both parts by the same nonzero number, such as 2:3 = 4:6 = 6:9. In a proportion like 2/3 = 8/12, the cross products are equal because 2 × 12 = 3 × 8.
To solve proportion word problems, define the quantities carefully, keep units consistent, set up equal ratios in the same order, and solve for the unknown.
Key Facts
- A ratio compares two quantities: a:b means a compared with b.
- Equivalent ratios are made by multiplying or dividing both parts by the same nonzero number.
- A proportion is an equation showing two equal ratios: a/b = c/d.
- Cross products are equal in a proportion: if a/b = c/d, then ad = bc.
- To solve a proportion, isolate the unknown after cross-multiplying, such as x/5 = 12/15 gives 15x = 60, so x = 4.
- Unit rates compare a quantity to 1 unit, such as 180 miles/3 hours = 60 miles/hour.
Vocabulary
- Ratio
- A ratio is a comparison of two quantities by division.
- Equivalent Ratio
- Equivalent ratios are ratios that have the same value even though their numbers may look different.
- Proportion
- A proportion is an equation stating that two ratios are equal.
- Cross Product
- A cross product is the product found by multiplying the numerator of one ratio by the denominator of the other ratio in a proportion.
- Unit Rate
- A unit rate is a ratio that compares a quantity to exactly one unit of another quantity.
Common Mistakes to Avoid
- Switching the order of one ratio, such as writing apples/oranges = oranges/apples, is wrong because the comparisons no longer match.
- Adding the same number to both parts of a ratio, such as changing 2:3 to 4:5, is wrong because equivalent ratios require multiplying or dividing both parts by the same nonzero number.
- Cross-multiplying incorrectly, such as using a/b = c/d but writing ab = cd, is wrong because the correct cross products are ad and bc.
- Ignoring units in word problems is wrong because ratios must compare matching quantities, such as miles to hours on both sides or dollars to pounds on both sides.
Practice Questions
- 1 A recipe uses 2 cups of rice for every 3 cups of water. How many cups of water are needed for 10 cups of rice?
- 2 Solve the proportion 5/8 = x/40.
- 3 Two maps use scales of 1 inch to 5 miles and 2 inches to 10 miles. Explain whether the scales are proportional and why.