Ratios and proportions are among the most practical math skills tested on the Digital SAT. A ratio compares two quantities, and a proportion states that two ratios are equal. These concepts appear in recipes, maps, scale models, unit conversions, and many other real-world contexts.
Core Concepts
Ratios
A ratio compares two quantities: or .
A ratio of means for every 3 of one thing, there are 5 of the other.
Proportions
A proportion is an equation stating two ratios are equal:
Solve by cross-multiplying: .
Example:
→ →
Part-to-Part vs. Part-to-Whole
In a ratio of boys to girls :
- Part-to-part: boys : girls
- Part-to-whole: boys : total
Scaling
If a recipe for 4 servings uses 6 cups of flour, how much flour for 10 servings?
→ cups
Equivalent Ratios
. Multiply or divide both parts by the same number.
Strategy Tips
Tip 1: Cross-Multiply to Solve
This is the fastest method for solving proportions.
Tip 2: Part-to-Whole Ratio
If a ratio is , the total parts are . Each part is or of the total.
Tip 3: Set Up With Matching Units
When writing a proportion, keep the same quantity on top (or bottom) on both sides.
Tip 4: Scale Factor for Similar Figures
Corresponding sides of similar figures are in proportion.
Tip 5: Check Reasonableness
If doubling the recipe, you should need roughly double the ingredients.
Worked Example: Example 1
In a class, the ratio of boys to girls is . If there are 36 students total, how many boys are there?
Total parts = . Boys = .
Worked Example: Example 2
A map has scale . Two cities are 8 cm apart on the map. How far apart are they in reality?
cm km.
Worked Example: SAT-Style
If 3 workers can paint 5 rooms in 2 days, how many rooms can 9 workers paint in 2 days (at the same rate)?
→ → rooms.
Worked Example: Example 4
Concrete is mixed in the ratio cement:sand:gravel = 1:2:3. How much sand is needed for 180 kg of concrete?
Total parts = . Sand = kg.
Practice Problems
Problem 1
The ratio of cats to dogs in a shelter is 4:9. If there are 52 animals total, how many cats?
Problem 2
Solve: .
Problem 3
A recipe uses 2 cups of sugar for 5 cups of flour. How much sugar for 12.5 cups of flour?
Problem 4
A photo is 4 inches by 6 inches. It is enlarged so the shorter side is 10 inches. What is the longer side?
Problem 5
In a school, the student-to-teacher ratio is 18:1. If there are 54 teachers, how many students?
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Common Mistakes
- Setting up the proportion with mismatched units. Keep the same quantity in corresponding positions.
- Confusing part-to-part with part-to-whole. A ratio gives 8 total parts, not 5.
- Forgetting to simplify ratios. . Always reduce.
- Cross-multiplying incorrectly. gives .
Key Takeaways
Ratio: comparison of two quantities. Proportion: two ratios set equal.
Cross-multiply to solve proportions.
Part-to-whole: if ratio is , total parts = .
Set up proportions with matching units on each side.
Ratios appear in scaling, maps, recipes, mixing, and similar figures.