Comparing Quantities: Ratios, Rates & Proportions
Ratios, rates, and proportions describe how two quantities relate to each other β whether comparing ingredients in a recipe, calculating speed, or scaling a map to real-world distances. These concepts bridge arithmetic and algebra by introducing the idea that relationships between numbers can be expressed, simplified, and solved as equations.
Components of Ratios, Rates & Proportions
This section breaks down the three related concepts:
- Ratios: A comparison of two quantities written as a:b, a/b, or "a to b," showing relative size without units.
- Rates: A special ratio that compares two quantities with different units, such as miles per hour or dollars per pound.
- Unit Rates: A rate simplified so the second quantity is 1, making comparisons easy (e.g., $3.50 per gallon).
- Proportions: An equation stating that two ratios are equal, solved by cross-multiplying to find an unknown value.
Examples of Ratios, Rates & Proportions
Ratio Examples
- A class has 12 boys and 18 girls. The ratio of boys to girls is 12:18, which simplifies to 2:3.
- A recipe uses 3 cups of flour for every 2 cups of sugar, giving a flour-to-sugar ratio of 3:2.
- A bag contains 5 red marbles and 20 total marbles. The ratio of red to total is 5:20, which simplifies to 1:4.
Rate & Unit Rate Examples
- A car travels 180 miles in 3 hours. The rate is 180 miles / 3 hours = 60 miles per hour.
- A 12-pack of juice costs $6.00. The unit rate is $6.00 Γ· 12 = $0.50 per bottle.
- A factory produces 250 widgets in 5 hours, giving a rate of 250 Γ· 5 = 50 widgets per hour.
Proportion Examples
- If 2/3 = x/12, cross-multiply: 2 Γ 12 = 3 Γ x, so 24 = 3x, and x = 8.
- A map scale says 1 inch = 25 miles. Two cities are 3.5 inches apart, so the real distance is 3.5 Γ 25 = 87.5 miles.
- A recipe serves 4 people with 6 cups of rice. To serve 10 people, set up 6/4 = x/10, cross-multiply to get 60 = 4x, so x = 15 cups.