PSAT 8/9 Math: Ratios, Rates, and Proportional Relationships
Ratios and rates describe how quantities relate to each other, and the PSAT 8/9 tests this through scaling, unit conversion, and real-world constraints. This free, complete lesson covers scaling ratios and part-to-part ratios, unit rates, unit conversion, and multi-step conversions and constraints. 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 quiz below.
1. Scaling Ratios and Part-to-Part Ratios
A ratio compares two quantities. To scale a ratio to a new amount, set up a proportion and cross-multiply. A part-to-part ratio (like boys to girls) lets you find one part if you know the other part and the ratio between them.
Worked Example: The ratio of boys to girls in a class is 3:2. If there are 24 boys, how many girls are there?
Set up a proportion: 3/2 = 24/g.
Cross-multiply: 3g = 48.
g = 16 girls
1. The ratio of boys to girls in a class is 3:2. If there are 24 boys, how many girls are there? (see worked example above)
Explanation: Set up the proportion 3/2 = 24/g. Cross-multiplying: 3g = 48, so g = 16.
2. The distance-to-time ratio for sound is about 1 mile to every 4 seconds. If thunder is heard 32 seconds after lightning, about how far away did the lightning strike?
Explanation: Set up the proportion 1/4 = d/32. Cross-multiplying: 4d = 32, so d = 8 miles.
3. A fruit basket has a ratio of 3 apples to 2 oranges. If there are 18 oranges, how many apples are there?
Explanation: Set up the proportion 3/2 = a/18. Cross-multiplying: 2a = 54, so a = 27.
2. Unit Rates
A unit rate compares a quantity to a single unit of another (like miles per day or dollars per pound). To find a unit rate, divide the total amount by the number of units.
Worked Example: A cyclist covers 1,800 miles in 90 days, at a roughly constant pace. Find the average daily distance.
1,800 ÷ 90 = 20 miles per day
1. A cyclist covers 1,800 miles in 90 days, at a roughly constant pace. Find the average daily distance. (see worked example above)
Explanation: Unit rate = total distance / total days = 1,800 / 90 = 20 miles/day.
2. A shopper spends \$48 buying apples at \$4 per pound. How many pounds of apples did they buy?
Explanation: Pounds = total cost / unit price = 48 / 4 = 12 pounds.
3. Find the total cost of 8 pounds of coffee at \$7.50 per pound.
Explanation: Total cost = unit price × quantity = 7.50 × 8 = \$60.00.
3. Unit Conversion
To convert between units, multiply or divide by a conversion factor relating the two units. Dividing by the conversion factor shrinks the number (like inches to feet); multiplying grows it (like kilograms to grams).
Worked Example: Convert an 18 kg dog's mass to grams (1 kg = 1,000 g).
18 × 1,000 = 18,000 grams
1. Convert an 18 kg dog's mass to grams. (see worked example above)
Explanation: 18 × 1,000 = 18,000 grams.
2. Convert 2,160 inches to feet (1 foot = 12 inches).
Explanation: 2,160 ÷ 12 = 180 feet.
3. A factory produces items at a rate of 45 per hour. Find the output over 4 hours.
Explanation: 45 items/hour × 4 hours = 180 items.
4. Multi-Step Conversions and Constraints
Some problems require chaining two conversion factors together, or dividing a total amount by a per-unit constraint (like "at least 6 square feet per person") to find a maximum or minimum.
Worked Example: Convert a car's speed of 72 mph to feet per second (1 mile = 5,280 feet, 1 hour = 3,600 seconds).
First convert miles to feet: 72 × 5,280 = 380,160 feet per hour.
Then convert hours to seconds: 380,160 ÷ 3,600 = 105.6.
72 mph = 105.6 feet per second
1. Convert a car's speed of 72 mph to feet per second (1 mile = 5,280 feet, 1 hour = 3,600 seconds). (see worked example above)
Explanation: First convert miles to feet: 72 × 5,280 = 380,160 feet/hour. Then convert to seconds: 380,160 ÷ 3,600 = 105.6 feet/second.
2. A theater's floor space is 4,800 square feet, and fire code requires at least 6 square feet per person. Find the maximum number of people allowed.
Explanation: Maximum occupancy = total area / minimum space per person = 4,800 / 6 = 800 people.
3. A runner's pace is 8 minutes per mile. Convert this pace to miles per hour.
Explanation: At 8 minutes per mile, the runner covers 60/8 = 7.5 miles in 60 minutes (1 hour), so the pace is 7.5 mph.
Common Mistakes to Avoid
- Setting up a proportion with mismatched categories, always keep the same type of quantity (like boys, or miles) in the same position across both fractions.
- Dividing instead of multiplying (or vice versa) during unit conversion, think about whether the new unit should produce a bigger or smaller number before calculating.
- Forgetting one of the two steps in a multi-step conversion, such as converting miles to feet but forgetting to also convert hours to seconds.
- Confusing a total amount with a unit rate. A unit rate is always a total divided by the number of units, not the total itself.
- Interpreting "at least X per person" backwards when finding a maximum, more space required per person means fewer people fit, not more.
Frequently Asked Questions
What's the difference between a ratio and a rate?
A ratio compares two quantities with the same units (like 3 boys to 2 girls), while a rate compares two quantities with different units (like miles per hour). Both are solved using the same proportional reasoning.
How do I handle a conversion with more than one step?
Break it into smaller, single-unit conversions, and apply them one at a time (or multiply all the conversion factors together). It often helps to write out each unit explicitly to make sure they cancel correctly.
Where can I practice more problems like these?
The Ratios, Rates, and Proportional Relationships quizzes in the PSAT 8/9 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 8/9. You can also browse the full PSAT 8/9 Math Question Bank.