Ratios, rates, proportional relationships, and units
Use ratios, unit rates, proportions, scale and unit conversions to solve problems in context. It is part of Problem-Solving and Data Analysis in the Math section. The book teaches it in Chapter 23, Ratios, Rates, Proportions, Units and Percentages, with 9 practice questions, and tests it 4 more times across the two full-length practice tests.
About Chapter 23
Problem-Solving and Data Analysis makes up about 15% of the Math section, or 5 to 7 scored questions per section. This chapter covers two of its skills: Ratios, rates, proportional relationships, and units, and Percentages. The arithmetic is rarely hard. Points are lost to setup errors: dividing the wrong way, using the wrong base for a percent, or answering a middle step. Two habits prevent most of them: write a unit beside every number, and turn every percent into a multiplier.
What the questions look like
Most questions in these two skills are short word problems. About 30% of all Math questions are set in a real-world context, and most of those use 50 words or fewer. Some are four-option multiple choice; others are student-produced responses (SPR), where you type your own answer. Typical tasks:
Find a unit rate, such as dollars per pound or points per game, and use it to predict a total.
Convert a quantity from one unit to another, sometimes with squared or cubed units.
Solve a proportion or apply a scale from a map, a model or a recipe.
Use density or another rate that links two quantities.
Find a percent of a number, or find the whole when a part and its percent are given.
Find a percent change, combine two percent changes, or work backward to an original value.
How to recognize them
A rate compares two quantities with different units, such as 12 miles per 2 hours. A unit rate gives the amount for exactly one unit, such as 6 miles per hour. A proportion is an equation that sets two ratios equal. The wording usually tells you which tool to use.
"per," "for each," "every": A rate. Divide to find the amount for one unit.
"at this rate," "at the same rate," "the same ratio": A proportion. Scale a known pair of values.
"a scale of 1 centimeter : 2 meters": Multiply lengths by the scale. Square it for areas.
Two different units for the same kind of quantity: A conversion. Chain conversion factors.
"density," "grams per cubic centimeter": Mass = density × volume.
"% of," "is what percent of": A percent of a number, or the whole from a part.
"increased by," "decreased by," "discount," "markup": A percent change. Use a multiplier.
"original," "before the change": A reverse percent. Divide by the multiplier.
Common traps
Dividing the wrong way in a rate: The leftover unit is upside down, such as hours per mile when the question asks for miles per hour.
Using the linear factor for area or volume: The question mentions square or cubic units, so the factor must be squared or cubed.
Inverting a proportion: Your answer moved in the wrong direction: more muffins should need more flour, not less.
Using the new value as the base: Percent change always divides by the original value.
Adding successive percents: Two changes happen one after the other, so multiply the multipliers.
Multiplying when you should divide: The question gives the value after a change and asks for the value before it.
Mixing percentage points and percent: Both values are percents; their difference is in points, not a percent change.
Answering a middle step: The choices include the unit rate, the amount of the increase, or the area in the wrong unit. Reread the last sentence of the question.
Video lessons
Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.
Ratios, rates and proportions (Brain Buffs Tutoring, 3:48). Watch for: Setting up a proportion with matching units.
Ratio and rate essentials (Math Vibes Hub, 12:03). Watch for: Unit conversions done as a chain of fractions.
Practice this skill
The full method, worked examples and 9 practice questions with every answer choice explained are in Chapter 23 of the book and in the study workspace, where an AI coach can talk you through each one.
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