Ratios, rates, proportions, direct relationships
Middle School Bridge • 6-8
If a recipe calls for 2 cups of flour for every 3 cups of sugar, and you want to triple the batch, how much of each ingredient do you need? Ratios and proportions let you scale quantities up and down while keeping relationships constant. This kind of reasoning appears in cooking, map reading, science experiments, and countless other situations.
A ratio compares two quantities. It can be written three ways:
All three forms mean the same thing. Ratios can compare a part to a part (2 boys to 3 girls) or a part to a whole (2 boys out of 5 students).
A rate is a ratio that compares two quantities measured in different units, such as miles per hour or dollars per pound. A unit rate has a denominator of 1, making comparisons easy.
A car travels 240 miles on 8 gallons of gas. What is the unit rate?
A proportion is an equation stating that two ratios are equal. If a⁄b = c⁄d, we say the four values are in proportion. To solve a proportion for an unknown, use cross multiplication.
Solve: 3⁄5 = x⁄20
A map uses a scale of 1 inch = 15 miles. Two cities are 4.5 inches apart on the map. What is the actual distance?
When setting up a proportion, make sure the same units are in the same positions. For example, if your first ratio is miles/hours, your second ratio must also be miles/hours -- not hours/miles. Mixing up the order leads to incorrect answers.
1. Write the ratio of 15 apples to 25 oranges in simplest form.
15 : 25. Divide both by 5: 3 : 5.
2. A runner covers 6 miles in 48 minutes. What is the unit rate in minutes per mile?
48 ÷ 6 = 8 minutes per mile.
3. Solve: 4⁄7 = x⁄35
Cross multiply: 4 × 35 = 7x. 140 = 7x. x = 20.
4. If 5 notebooks cost $8.75, how much do 12 notebooks cost?
Unit price: 8.75 ÷ 5 = $1.75 per notebook. Cost for 12: 1.75 × 12 = $21.00.
5. A recipe uses 3 eggs for every 2 cups of milk. If you use 9 eggs, how many cups of milk do you need?
3⁄2 = 9⁄x. Cross multiply: 3x = 18. x = 6 cups of milk.
A ratio compares two quantities; a rate compares quantities in different units. A proportion states that two ratios are equal and can be solved using cross multiplication. Always ensure consistent unit placement when setting up proportions.