PSAT/NMSQT Math: Ratios, Rates, and Proportions
Ratios, rates, and proportions describe how two or more quantities relate to each other, and PSAT/NMSQT questions test this from every angle: simplifying, converting units, calculating rates and averages, and scaling proportional relationships. This free, complete lesson covers simplifying and applying ratios, unit conversion, rates and averages, and multi-part ratios and proportional sampling. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Simplifying and Applying Ratios
A ratio compares two quantities, and simplifying one works exactly like simplifying a fraction, divide both parts by their GCF. To apply a ratio to a real quantity, set up a proportion between the known ratio and the actual numbers involved.
Worked Example: A jar has yellow and blue pencils in a ratio of 5:2. If there are 30 yellow pencils, how many blue pencils are there?
Set up a proportion: 5/2 = 30/b.
5b = 60 → b = 12 blue pencils
1. Simplify the ratio 18:24 to lowest terms.
Explanation: The GCF of 18 and 24 is 6. Dividing both parts by 6: 18/6 : 24/6 = 3:4.
2. A jar has yellow and blue pencils in a ratio of 5:2. If there are 30 yellow pencils, how many blue pencils are there? (see worked example above)
Explanation: Setting up the proportion 5/2 = 30/b and cross-multiplying: 5b = 60 → b = 12.
3. Using known ratio of distance-to-time for sound, a person hears thunder 6 seconds after seeing lightning, at a rate of about 1 mile per 5 seconds. Approximately how far away was the lightning?
Explanation: Setting up the proportion 1 mile/5 sec = d/6 sec: d = 6/5 = 1.2 miles.
2. Unit Conversion
To convert between units, multiply by a conversion factor, a fraction equal to 1 that relates the two units. For a conversion involving two different unit changes at once (like yards per minute to feet per second), apply each conversion factor one at a time.
Worked Example: Convert 45 kilograms to grams (1 kg = 1,000 g).
45 kg × (1,000 g / 1 kg) = 45,000 g
1. Convert 45 kilograms to grams. (see worked example above)
Explanation: 45 × 1,000 = 45,000 grams.
2. Convert 500 yards per minute to feet per second (1 yard = 3 feet, 1 minute = 60 seconds).
Explanation: First convert yards to feet: 500 × 3 = 1,500 feet/minute. Then convert minutes to seconds: 1,500/60 = 25 feet/second.
3. A recipe calls for per-unit ingredient amounts that total 48 ounces. Convert this total to pounds (1 pound = 16 ounces).
Explanation: 48 ÷ 16 = 3 pounds.
3. Rates and Averages
A rate compares two quantities with different units, such as miles per hour or posters per hour. To find a total from a rate, multiply the rate by the amount of time (or other quantity); to find an average rate, divide the total amount by the total time.
Worked Example: A printer produces posters at a rate of 75 per hour. How many posters are produced in 4 hours?
75 × 4 = 300 posters
1. A printer produces posters at a rate of 75 per hour. How many posters are produced in 4 hours? (see worked example above)
Explanation: 75 posters/hour × 4 hours = 300 posters.
2. A turtle travels 1,860 miles over 93 days at a roughly constant pace. Find its average daily distance.
Explanation: Average rate = total distance / total time = 1,860 / 93 = 20 miles/day.
3. A car travels a known distance in a known time. Which operation correctly finds its speed?
Explanation: Speed (a rate) is always distance divided by time.
4. Multi-Part Ratios and Proportional Sampling
A multi-part ratio (such as 6:3:1) compares three or more quantities at once; find the value of "one part" using any known amount, then scale every part by that same factor. Proportional sampling uses a smaller, representative sample to estimate a value for a much larger population.
Worked Example: Markers come in a ratio of 6:3:1 (blue:red:green). If there are 72 blue markers, find the total number of markers.
Each "part" equals 72/6 = 12.
Total parts = 6 + 3 + 1 = 10, so total markers = 10 × 12 = 120
1. Markers come in a ratio of 6:3:1 (blue:red:green). If there are 72 blue markers, find the total number of markers. (see worked example above)
Explanation: One part = 72/6 = 12. Total parts = 6 + 3 + 1 = 10, so total = 10 × 12 = 120.
2. A survey of 200 randomly sampled teachers in a district finds that 150 use an online platform. If the district has 3,000 teachers total, about how many likely use the platform?
Explanation: The sample proportion is 150/200 = 0.75. Applying this rate to the full population: 0.75 × 3,000 = 2,250.
3. Weights on the Moon are about 1/6 of weights on Earth. A 180-pound object on Earth weighs how much on the Moon?
Explanation: Multiply by the fractional relationship: 180 × (1/6) = 30 pounds.
Common Mistakes to Avoid
- Setting up a proportion with mismatched units or categories. Always keep corresponding quantities in the same position across both sides of a proportion.
- Forgetting to convert through an intermediate unit in a two-step unit conversion, applying only one of the two necessary conversion factors.
- Confusing total amount with average rate. Total = rate × time, while average rate = total amount / total time, these are inverse operations for different questions.
- Mismatching which part of a multi-part ratio corresponds to a given quantity. Carefully identify which specific number in the ratio matches the known real-world quantity before solving for "one part."
- Applying a sample proportion incorrectly to a different-sized population, such as forgetting to multiply by the new total rather than just reusing the sample's raw count.
Frequently Asked Questions
What's the difference between a ratio and a rate?
A ratio compares two quantities with the same units (like 5 red marbles to 2 blue marbles), while a rate compares two quantities with different units (like miles per hour). Both are handled with the same proportional-reasoning tools.
How do I know how many "parts" a multi-part ratio has in total?
Add up every number in the ratio. A ratio of 6:3:1 has 6 + 3 + 1 = 10 total parts, even though it describes three separate categories.
Where can I practice more problems like these?
The Ratios, Rates, and Proportions quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.