Ratios, rates and proportions — Digital SAT Math practice
Set up and solve proportional relationships.
Key terms
- Ratio: A comparison of two quantities by division, written as a:b or a/b.
- Rate: A ratio comparing two different units, such as miles per hour or dollars per item.
- Proportion: An equation stating that two ratios are equal; solved by cross-multiplication.
- Unit Rate: A rate with a denominator of 1, used to compare quantities directly.
- Work Rate: The fraction of a job completed per unit of time by one worker or machine.
Common traps
- Averaging speeds directly gives wrong results; use total distance divided by total time.
- Mixing up the order of a ratio; keep the same units in the numerator and denominator on both sides.
- Adding the times instead of the rates when two workers share a job.
Key ideas to know
Solving Proportions with Cross-Multiplication
Solve for x in the proportion 3/4 = x/20. Show your work step by step.
Cross-multiply to get 4x = 60, then divide by 4 to find x = 15. Always multiply diagonally across the equals sign.
Watch out: Forgetting to multiply both sides by the denominators or mixing up which numbers multiply together.
Matching Quantity Order in Ratios
A car travels 150 miles in 3 hours. Which proportion correctly finds the time t to travel 250 miles? A) 150/3 = 250/t B) 3/150 = t/250 C) 150/t = 3/250
Option A is correct. You must keep the same order of quantities in both ratios (miles over hours equals miles over hours). Option B flips the order, and C mixes units incorrectly.
Watch out: Flipping one ratio so miles are over hours on one side but hours are over miles on the other, which breaks the proportion.
Calculating True Average Speed
A train goes 60 mph to a station and returns at 40 mph over the same distance. What is its average speed for the whole trip? A) 50 mph B) 48 mph C) 45 mph
The correct answer is 48 mph. Average speed always equals total distance divided by total time, not the arithmetic mean of the two speeds.
Watch out: Averaging the two speeds directly by adding them and dividing by 2, which ignores how much time was spent at each speed.
Using Ratio Parts to Divide a Total
The ratio of boys to girls in a class is 2 : 3. If there are 40 students total, how many are girls? Show the setup.
Add the ratio parts (2 + 3 = 5 parts). Each part represents 40 / 5 = 8 students. Multiply the girls part by 8 to get 3 × 8 = 24 girls.
Watch out: Assuming the first number in the ratio directly equals the quantity without dividing the total by the sum of the parts first.
Sample questions
Question 1
At a location, the ratio of cats to dogs is 5 to 3. If there are 75 cats, how many dogs are there?
- 50
- 45
- 75
- 120
Show answer and explanation
Answer: 45
75 ÷ 5 = 15 groups; dogs: 3 × 15 = 45.
Question 2
A recipe uses 3 cups of flour for every 12 cookies. How many cups of flour are needed to make 20 cookies?
- 4
- 5
- 6
- 8
Show answer and explanation
Answer: 5
3/12 = 0.25 cup per cookie; 0.25 x 20 = 5 cups.