1. Ratios, Rates, and Proportions

A ratio compares two quantities using division. It can be written as a fraction, with a colon, or in words: the ratio of 3 red marbles to 5 blue marbles is 3/5, 3:5, or '3 to 5'. Ratios tell you the relative size of quantities, not necessarily the actual amounts.

A rate is a special ratio that compares quantities with different units. The most useful form is the unit rate -- the amount per one unit of the denominator. If a car travels 240 miles on 8 gallons of gas, the unit rate is 240 / 8 = 30 miles per gallon.

A proportion sets two ratios equal to each other:

• If 3 pounds of apples cost $4.50, how much do 7 pounds cost?
• Set up: 3/4.50 = 7/x. Cross-multiply: 3x = 31.50, so x = $10.50.

Setting up proportions step by step:

• Identify the two quantities being compared (apples and dollars above).
• Write the known ratio on one side and the unknown on the other, keeping units consistent (apples over dollars on both sides).
• Cross-multiply to solve.

Scale problems use proportions to convert between a model and the real world. If a map scale is 1 inch = 50 miles, then 3.5 inches represents 3.5 x 50 = 175 miles.

Worked example: A recipe calls for 2 cups of flour for every 3 cups of oats. If you want to use 9 cups of oats, how many cups of flour do you need?

• Ratio: flour/oats = 2/3.
• Set up: 2/3 = x/9. Cross-multiply: 3x = 18, so x = 6 cups of flour.

• Part-to-whole ratios: If the ratio of boys to girls in a class is 3:5, then boys make up 3/(3+5) = 3/8 of the class.
• Equivalent ratios: Multiply or divide both parts by the same number to scale a ratio up or down (just like equivalent fractions).

⚠️ Common mistake: confusing part-to-part ratios (3:5) with part-to-whole fractions (3/8). A ratio of 3:5 does NOT mean 3 out of 5 -- it means 3 out of every 8 total.

💡 Tip: always label units when setting up a proportion. Keeping track of labels helps you catch errors before they cost you points.