Ratios and proportions show up all over the SAT — in word problems, geometry, data analysis, and even coordinate geometry. Fortunately, the underlying math is straightforward.

Ratios: The Basics

A ratio compares two quantities. The ratio of to can be written as or .

Example 1: In a class of 30 students, the ratio of boys to girls is . How many boys are there?

Total parts:
Each part: students
Boys:

Proportions and Cross-Multiplication

A proportion says two ratios are equal:

Solve by cross-multiplying:

Example 2: If , find .



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Setting Up Proportions from Word Problems

Example 3: A recipe calls for 3 cups of flour to make 24 cookies. How much flour for 40 cookies?


Pro tip: Make sure the units match — flour with flour, cookies with cookies.

Part-to-Part vs. Part-to-Whole

Watch out for this SAT trap:

Example 4: The ratio of cats to dogs at a shelter is . What fraction of the animals are cats?

Scale Factor Problems

Example 5: On a map, 1 inch represents 25 miles. If two cities are 3.5 inches apart on the map, how far apart are they?

Direct and Inverse Proportion

Direct proportion: As one quantity doubles, the other doubles.

Inverse proportion: As one quantity doubles, the other halves.

Example 6: If 4 workers can paint a house in 12 hours, how long would 6 workers take?

This is inverse proportion. Total work = worker-hours.


Practice Problems

Problem 1: The ratio of red to blue marbles is . If there are 40 marbles total, how many are red?

Solution

Total parts = 8. Each part = 5. Red = marbles.

Problem 2: If a car travels 180 miles on 6 gallons of gas, how far can it go on 10 gallons?

Solution

miles.

Problem 3: A mixture is juice and the rest is water. If there are 15 liters total, how many liters of water?

Solution

Water fraction = . Water = liters.

Key Takeaways

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