Before you begin
The last quiz used a rate in nearly every question and never once examined it. Meters per minute, all the way through, and the word per was doing quiet work in the middle of it. This quiz is about that word, which turns out to be the most useful preposition in arithmetic.
If ratios are new to you, start with Ratio, Proportion, and Percent, which introduces them from the beginning. Parts 2 to 5 of that page are about ratios and rates.
A ratio compares two amounts
If a crew has 12 deckhands and 18 tons of gear, the ratio of hands to tons is 12 to 18. It can be written three ways, all meaning the same thing: 12 to 18, 12 : 18, or 12/18. And like any fraction it can be simplified, so 12 : 18 is also 2 : 3, which says the same relationship in the smallest whole numbers that will carry it.
A rate compares two different kinds of thing
Miles and hours. Dollars and pounds. Meters and minutes. When the two quantities are different in kind, the comparison is called a rate, and the word that joins them is per.
A unit rate, and how to get one
A unit rate is a rate whose second number is 1. It answers how much for one? And you get it by dividing, because dividing is exactly the operation that shares an amount out evenly.
Three pounds of coffee cost $13.50. To find the price for one pound, split the money across the three pounds: 13.50 ÷ 3 = $4.50 per pound.
Which number goes on top
That same pair of facts will give you a second unit rate if you divide the other way: 3 ÷ 13.50 is about 0.22, so you get roughly a fifth of a pound per dollar. Both are true. They answer different questions, and the question decides which one you want.
The rule is simple: whatever comes after per goes on the bottom. Dollars per pound means dollars ÷ pounds. Miles per gallon means miles ÷ gallons. Read the phrase and it tells you the division.
Why unit rates exist at all
They exist so that things can be compared. A 12-ounce jar at $3.60 and a 20-ounce jar at $5.40 cannot be compared as they stand, because neither the prices nor the sizes match. Bring both to per one ounce and they line up: 30 cents an ounce against 27 cents an ounce. The bigger jar is the better buy, by three cents on every ounce.
That is the whole business of the shelf tag in a supermarket, where the small print under the price is a unit rate somebody has already worked out. It is worth knowing how it was arrived at, since it is the one number on the shelf that lets two different packages be judged against each other.
The trap: a ratio is not always a fraction of the whole
Here is where nearly every lost mark in this topic comes from. Suppose a workplace has workers and supervisors in the ratio 5 : 1. It is very tempting to say that supervisors are one fifth of the staff. They are not.
The same addition solves the other common question. If a room holds 40 people in the ratio 5 adults to 3 children, there are 8 parts altogether, so each part is 40 ÷ 8 = 5 people. Then the adults are 5 × 5 = 25 and the children are 3 × 5 = 15, and those add back to 40, which is the check.
The guard
Two habits. Say the unit rate out loud with the words in it: four dollars fifty per pound, not just four fifty, because the words carry the division and the bare number does not. And before turning any ratio into a fraction, add the parts and ask whether the bottom you are about to write is really the total.
Three to study before you start
A car goes 234 miles on 9 gallons. What is the mileage? The phrase wanted is miles per gallon, and whatever follows per goes on the bottom, so divide miles by gallons: 234 ÷ 9 = 26 miles per gallon. Check it by going backwards, which is always available with a rate: 26 miles for each gallon, nine gallons, and 26 × 9 is 234. Read the answer aloud with its words in it, because "26" alone could be anything and "26 miles per gallon" cannot.
A crew has deckhands and officers in the ratio 5 : 1. What fraction are officers? Add the parts first: 5 + 1 = 6, so the crew comes in six equal shares. Five of them are deckhands and one is an officer, so the officers are 1/6 of the crew and the deckhands are 5/6. Answering 1/5 would be saying that the crew has five shares in it, and it has six. Check the two fractions by adding: 5/6 + 1/6 = 1, the whole crew, which is what a check should look like.
A 12-ounce jar costs $3.60 and a 20-ounce jar costs $5.40. Which is the better buy? As they stand, nothing can be said: the bigger jar costs more and holds more, and both facts are useless on their own. Bring each to a price per ounce. The small jar: 3.60 ÷ 12 = $0.30 an ounce. The large jar: 5.40 ÷ 20 = $0.27 an ounce. The large one is cheaper by three cents on every ounce. That is what a unit rate is for: two things that could not be compared are now standing on the same ground.
Now you
Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Write every rate with its words attached, not as a bare number.
- Write the ratio 12 to 18 in simplest form.
- Three pounds of coffee cost $13.50. What is the price per pound?
- A van travels 234 miles on 9 gallons. What is the mileage, in miles per gallon?
- A 12-ounce jar costs $3.60. A 20-ounce jar of the same thing costs $5.40. Which is the better buy?
- A) The 12-ounce jar, because it costs less money.
- B) The 20-ounce jar, because it works out at 27 cents an ounce against 30 cents.
- C) They are the same value, since both cost 30 cents an ounce.
- D) There is no way to tell without knowing the brand.
- Yuliana is paid $522 for 36 hours of work. What is her hourly rate?
- A room holds 40 people, with adults and children in the ratio 5 : 3. How many children are in the room?
ReadingA reading rest stop, for the stubborn ones.
Turn the pages of the book the whale at the top of this page came out of. This is the picture atlas of d'Orbigny's Dictionnaire universel d'histoire naturelle, scanned whole and free to anybody: 252 pages of hand-colored plates, and more than twelve hundred creatures across the set.
The words in it are French and that does not matter, because you are there for the pictures. Use the arrows to turn the pages. Look for the whales among the mammifères and the ants among the insectes.
Published in Paris from 1841, the atlas volumes reprinted through the 1860s. It is long out of copyright, which is why that whale is at the head of this quiz and not somebody else's photograph.
- A workplace has workers and supervisors in the ratio 5 : 1. Rosa says that supervisors are 1/5 of the staff. What went wrong?
- A) Nothing; 1/5 is correct.
- B) She used one part of the ratio as though it were the whole. Adding the parts gives 6, so supervisors are 1/6 of the staff and workers are 5/6.
- C) She should have written the ratio the other way round, as 1 : 5.
- D) Ratios cannot be turned into fractions at all.
- Which is faster: 240 miles in 4 hours, or 330 miles in 6 hours? Show the unit rate for each.
- Two jars of the same food are different sizes and different prices. Explain why the prices alone cannot tell you which is the better buy, and what has to be worked out first.
- An ant travels about 3 inches per second. (a) How many inches per minute is that? (b) How many feet per minute?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1 | Writing and simplifying a ratio | A ratio simplifies like a fraction. Divide both parts by the same number. |
| 2, 3, 5 | Finding a unit rate | Whatever follows per goes on the bottom. Then divide, and say the answer with its words in it. |
| 4, 8, 9 | Comparing with unit rates | Two things cannot be compared until they stand on the same ground. Bring both to per one. |
| 6, 7 | Part to part, part to whole | The signature error of the topic. Add the parts of the ratio before writing any fraction. |
| 10 | Changing the units of a rate | Ask which unit is getting bigger. A bigger unit of time makes the number grow; a bigger unit of length makes it shrink. |
Eight or more right: per is yours, and Quiz 24 is next. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Take one habit shopping with you for a week: at the shelf, cover the big price and read only the small print underneath, which is the unit rate. Then work out for yourself where that small number came from. Almost everything in this quiz is that one division, done in public, several thousand times a day. Then come back to these same ten.
Next in the series: Quiz 24 — Setting Up and Solving Proportions. You can now find one rate. The next quiz sets two of them equal to each other and finds the number that makes them match, which is how a recipe gets doubled, a map gets read, and a dose gets calculated. It is also where cross-multiplying arrives, and it will be explained rather than announced.
A new plate, and a claim that does not add up
The men in those boats were doing arithmetic. Whaling ran on rates and everybody in it knew them: barrels of oil per whale, whales per voyage, and at the end, dollars per barrel divided among a crew in fixed shares called lays. A hand might sail for two years for a two-hundredth part of what the ship brought home. Every one of those is a ratio, and the ones who could not work them out were paid by the ones who could.
Now a claim worth checking
Here is a sentence about ants and whales, of the kind that circulates:
It reads well. It has precise-looking numbers in it. And it contains an error that this quiz has just taught you to find, because the sentence checks itself: 107,800 × 125 = 13,475,000 tons, which is 13.475 million. The sentence says 13,475 million. Its two halves disagree by a factor of a thousand.
Which half is wrong? The whale arithmetic is sound, so the ant figure is the one that slipped: it should read 13,475 thousand tons. And there is a second check that settles it from outside. All the ants on earth, every species everywhere, are estimated to weigh in the low tens of millions of tons. Thirteen billion tons of fire ants in the United States alone would outweigh every ant on the planet several hundred times over. The smaller figure is the possible one.
None of that required looking anything up. It required multiplying two numbers that were sitting in the sentence, and then asking whether the answer could be true. That is worth more than the fact was, and it is why this series keeps checking things in front of you rather than handing them over finished. A number in print is a claim, not a fact, and the arithmetic you have been doing for twenty-three quizzes is the tool for telling the difference.