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Ratios, rates, proportional relationships, and units: SAT Math notes
These questions test whether you can compare amounts with ratios and rates, scale them up or down, and change units without losing track of what each number means.
Study it with worked examples and 8 practice questionsPractise Ratios, rates, proportional relationships, and units questions
What this topic tests
A ratio compares two amounts, like 7 red cards to 28 blue cards. A rate is a ratio with units attached, like 12 centimeters per second or 353 kilograms per cubic meter. When two quantities are proportional, their ratio never changes: if one doubles, the other doubles too.
Almost every question here comes down to one move: multiply or divide by the right number. The hard part is choosing which. The safest habit is to write the units next to every number and check that the units you don't want cancel out, leaving the units the question asks for.
Expect lots of word problems with real units (miles, ounces, furlongs, pesos), some questions that use letters instead of numbers, and a few density problems where you first need the volume of a cube. You have a calculator, so put your effort into setting the problem up correctly.
Key ideas
- A ratio of a to b can be written as the fraction a/b. Two ratios are equal when their fractions are equal.
- Proportional means y=kx for some constant k, so y/x is the same for every pair of values.
- Every rate works the same way: total = rate × amount. Distance = speed × time, and mass = density × volume.
- To convert units, multiply by a fraction that equals 1, such as (60 min)/(1 hour), arranged so the old unit cancels.
- Square and cubic units use the conversion factor squared or cubed. One square foot is 122=144 square inches, not 12.
- If one quantity is multiplied by a number, a proportional quantity is multiplied by the same number. Three times the hours means three times the pay.
- Before you pick an answer, ask whether it should be bigger or smaller than what you started with. Converting to a smaller unit gives a bigger number.
The 4 kinds of Ratios, rates, proportional relationships, and units question
Sorted from every Ratios, rates, proportional relationships, and units question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Ratios and proportions: find the missing value
How to spot it. Two quantities are in a fixed ratio or are proportional. You get one complete pair and have to find a missing value. Look for the words ratio, to, the same ratio, or proportional, or a table where y/x stays the same.
How to answer it:
- Write the known ratio as a fraction, in the same order as the question (the first quantity goes on top).
- Find the scale factor: how many times bigger the new known value is than the matching number in the ratio.
- Multiply the other part of the ratio by that same factor. Or set up a/b=c/d and cross-multiply.
- Check that your two new values still simplify to the original ratio.
The trap. Adding instead of multiplying. In a ratio, both parts grow by the same factor, not by the same amount. Adding 3 more ounces of each paint color changes the mix. Also keep the order straight: if the ratio is m to k and you're given k, it goes on the bottom.
2. Unit conversions
How to spot it. A measurement is given in one unit and you need it in another, usually with a conversion fact in parentheses, like (1 mile = 1,760 yards). Some questions convert a rate, such as kilometers per hour to meters per second, and some convert an area, such as square inches to square feet.
How to answer it:
- Write the starting amount with its units.
- Write each conversion fact as a fraction equal to 1, with the unit you want to get rid of on the opposite side (top or bottom) from where it is now.
- Multiply, cancel the units, and check that only the units the question asks for are left.
- For square units, use the conversion factor squared. For cubic units, cube it.
- Sense check: a smaller unit means a bigger number, and a bigger unit means a smaller number.
The trap. Multiplying when you should divide. Let the units decide: if they don't cancel, flip the fraction. In area questions, the classic mistake is dividing square inches by 12 instead of by 122=144.
3. Rates and density
How to spot it. A rate links two different quantities: miles per day, revolutions per minute, people per square mile, kilograms per cubic meter, dollars per ounce. You're given two of the three pieces (rate, total, amount) and asked for the third, sometimes after an extra step like finding a volume or a land area.
How to answer it:
- Name the rate and its units, for example kilograms per cubic meter.
- Do any setup first, such as finding the volume of a cube with s3 or subtracting water to get land area.
- Use total = rate × amount. To find the rate, divide the total by the amount. To find the amount, divide the total by the rate.
- Check that the units of your answer match what the question asks for.
The trap. Using the wrong amount. People per square mile of land means you must remove the water area first. In density questions, don't use a cube's edge length where you need its volume: cube it first.
4. Scaling with letters instead of numbers
How to spot it. Letters stand in for the numbers, and the answer choices are expressions. For example: r objects have a total mass of t, so what is the mass of 146r objects? Or: a plant grows 87 millimeters every m months, so how much does it grow in k years?
How to answer it:
- Compare the new amount with the old one: how many times bigger is it? (146r is 146 times r.)
- Multiply the matching quantity by that same factor.
- If the units differ, like months and years, convert first, then build a per-unit rate and multiply.
- If you're unsure, plug in easy numbers for the letters and test each choice.
The trap. Choosing an answer that adds, like 146+t. It looks natural, but proportional means multiplying: 146 times as many objects weigh 146 times as much. Also watch for upside-down answers that use the right numbers with the letters flipped.
Practise Ratios, rates, proportional relationships, and units
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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