Arithmetic ReasoningLesson 5 of 12
Unit Rate and Scaling
Finding the cost or quantity of one, then scaling to many. The most common shape on the subtest.
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If you had to pick one question shape to be good at on Arithmetic Reasoning, this is it. Cost per item, miles per gallon, pages per hour, rounds per minute - the subject changes and the arithmetic does not.
Divide to get one, multiply to get many
Twelve cases of MREs cost $186. What do 20 cases cost?
One case: 186 divided by 12 is 15.50.
Twenty cases: 15.50 times 20 is $310.
The only real decision is which quantity goes on the bottom, and the question answers that: you want dollars, so dollars go on top and cases go on the bottom.
| Question wants | Rate to build | Then |
|---|---|---|
| cost of a different number of items | dollars per item | multiply by the new count |
| how many items for a budget | items per dollar, or divide by the unit cost | either works |
| distance from a time | miles per hour | multiply by the new time |
| time from a distance | hours per mile, or divide by speed | either works |
| fuel used on a trip | gallons per mile, or divide miles by miles-per-gallon | either works |
The rows with two options are there to make a point: you do not have to build the reciprocal rate. Dividing by a rate does the same job as multiplying by its reciprocal, and division is usually the shorter path.
Scale by the factor instead
When the numbers cooperate, the unit rate is a detour.
Eight sandbags weigh 320 pounds. What do 24 weigh?
Twenty-four is three times eight, so the weight is three times 320, which is 960 pounds. No division, no per-bag figure.
Look for the factor first. It is there more often than not, because question writers choose numbers that divide cleanly, and spotting it turns a two-step problem into one multiplication.
When the factor is not whole, it can still be easier than the unit rate:
Six crates cost $54. What do 15 cost?
Fifteen is 2.5 times six, so 2.5 times 54 is 135. Or: one crate is $9, and 15 times 9 is 135. Both are one line. Take whichever you see first.
Two rates in one question
The subtest likes to stack a rate on a rate.
A press prints 450 pages in 15 minutes. Paper costs 2 cents a page. What is the paper cost for 4 hours of printing?
Work in one direction and do not skip the unit change.
- Rate: 450 divided by 15 is 30 pages per minute.
- Time: 4 hours is 240 minutes.
- Pages: 30 times 240 is 7,200.
- Cost: 7,200 times 0.02 is $144.
Every one of those four numbers is a plausible-looking answer, and three of them are wrong. The defense is naming the unit you want before you start - dollars - and noticing you are not there yet.
The unit change is the step people drop. The question gave minutes and asked about hours. Converting at the end works too, but convert somewhere, and say out loud which unit your rate is in.
Better buy questions
A common variant asks which of two options is the better value, which is a unit rate comparison and nothing else.
A 24-ounce bottle costs $3.60. A 40-ounce bottle costs $5.60. Which is the better value, and by how much per ounce?
- Small: 3.60 divided by 24 is 15 cents an ounce.
- Large: 5.60 divided by 40 is 14 cents an ounce.
The large bottle is better by 1 cent per ounce.
Put the price on top both times. Comparing 24 ounces per $3.60 against 40 ounces per $5.60 also works, but then the bigger number is the better deal, and flipping the direction of "better" mid-question is how people get these wrong. Pick one orientation and hold it.
What you can skip
Across the 50 questions on this topic:
- Currency exchange never appears. The unit-rate questions are prices, fuel economy, recipes and travel.
Where people lose points
Stopping at the unit rate. The most common error on the shape. In the printing example above, 30 is a clean number and it is an answer choice, and it is not the answer.
Dividing the wrong way. 186 dollars over 12 cases is a cost per case. 12 cases over 186 dollars is cases per dollar, which is 0.0645 and looks obviously wrong - but with friendlier numbers the reciprocal looks perfectly reasonable. Write the units.
Missing a unit change between the rate and the question, as above.
Assuming the rate holds. Almost always it does, and the question says "at the same rate". When a question deliberately changes the rate partway through, it says so, and it is testing whether you noticed.
Rounding a per-unit price mid-problem. $15.50 per case is exact. A rate like $7.333 per case should stay as a fraction until the end, or the final answer drifts.
Work one in under a minute
A vehicle averages 22 miles per gallon. Fuel is $3.45 a gallon. What does a 385-mile trip cost?
Gallons: 385 divided by 22. Twenty-two times 17 is 374, leaving 11, which is half a gallon. So 17.5 gallons.
Cost: 17.5 times 3.45. Take 17.5 times 3 is 52.50, and 17.5 times 0.45 is 7.875. Total $60.38.
Check: about 17 gallons at about $3.50 is about $60. Good.
Where this leads
Rate, time and distance is this lesson with the units fixed, and work rate is this lesson with the rates added together.
Related lessonsReference
- Rate, Time and Distance - unit rate with miles and hours, plus the two-object versions
- Work and Production Rates - what happens when two rates combine
- Unit Conversion - the step between the rate and the question that people drop
- Money, Cost and Profit - unit rates with discounts, markup and profit layered on
Practice this topic
Check that this lesson stuck. Answer questions on unit rate and scaling only, and see the right answer and why after each one.
Practice Unit Rate and Scaling questions