Mastering Ratios, Rates, and Proportions Questions on the SAT
Understanding ratios, rates, and proportions is crucial for solving many real-world problems on the SAT math section.
Ratios, rates, and proportions are fundamental concepts that frequently appear on the SAT math section.
A ratio compares two quantities (a:b or a/b). A proportion states two ratios are equal (a/b = c/d). A rate is a ratio with different units (miles per hour, price per item).
Identifying and Expressing Ratios
Ratios can be part-to-part (lemon juice to water = 2:3) or part-to-whole (lemon juice to total = 2:5).
Converting: For part-to-part ratio 3:2, the whole is 5. Part-to-whole: 3/5 and 2/5. Conversely, if part-to-whole is 3/5, other part is 5-3=2, giving 3:2.
Examples
10 boys and 15 girls: part-to-part = 10:15 = 2:3. Part-to-whole: 2/5 boys, 3/5 girls.
4 red and 6 blue marbles: part-to-part = 4:6 = 2:3. Part-to-whole: 2/5 red, 3/5 blue.
Using Proportions to Solve Problems
Set two ratios equal and cross-multiply to solve for the unknown.
Example: Sugar:flour = 1:2. Need 4 cups flour. Set up: 1/2 = x/4. Cross-multiply: 4 = 2x, x = 2 cups sugar.
More Examples
3 eggs per 2 cups milk, have 6 cups: 3/2 = x/6. Cross-multiply: 18 = 2x, x = 9 eggs.
Map scale 1 inch = 5 miles, distance 3 inches: 1/5 = 3/x. x = 15 miles.
Understanding and Using Rates
A rate is a ratio with different units. To find a rate, divide the two quantities.
Example: 100 miles in 2 hours = 50 mph. A train at 60 mph travels 60 x 3 = 180 miles in 3 hours.
More Examples
Marathon (26.2 miles) in 4 hours: 26.2/4 = 6.55 mph.
A ratio compares two quantities (like 2:3). A proportion is an equation stating two ratios are equal (like 2/3 = 4/6). Proportions are solved by cross-multiplication.
For ratio a:b, the whole is a+b. Part-to-whole ratios are a/(a+b) and b/(a+b). For 3:2, whole is 5, giving 3/5 and 2/5.
Set up equivalent fractions and cross-multiply. For 3 eggs per 2 cups milk with 6 cups: 3/2 = x/6. Cross-multiply: 18 = 2x, x = 9 eggs.