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Reducing a rate to 'per one', so two offers measured differently can be compared.
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In this lesson you turn a rate into a unit rate, such as miles per hour, dollars per pound or cents per ounce, by dividing so that the second quantity is one. You will use unit rates to compare two offers, to predict a total, and to work backward from a total to the time it takes. It is some of the most useful mathematics in everyday life.
You can divide a total into equal shares, and you know that the word each means one share. You can multiply and divide whole numbers, and you can change dollars to cents by multiplying by 100. You have also met ratios such as 3 to 2, which compare two amounts. A unit rate puts all of these together: it is the amount of one thing that goes with exactly one of another thing.
| Term | What it means |
|---|---|
| Rate | A comparison of two quantities measured in different units, such as 195 miles in 3 hours. |
| Unit rate | A rate in which the second quantity is 1, such as 65 miles per 1 hour. |
| Per | For each one. Miles per hour means miles for each single hour. |
| Unit price | The price of one unit of something: one ounce, one pound, one item. |
| Constant speed | Moving the same distance in every hour (or minute), so one unit rate describes the whole trip. |
| Better buy | The choice with the lower unit price, so you pay less for each unit you get. |
A rate compares two quantities that have different units. A car that goes 195 miles in 3 hours has a rate of 195 miles for every 3 hours. A unit rate is the same rate written so that the second quantity is exactly 1: 65 miles for 1 hour, which we say as 65 miles per hour.
You find a unit rate by dividing. Divide the first quantity by the second:
$$195 \div 3 = 65.$$
This works because a rate stays the same when you divide both of its parts by the same number. Dividing 195 miles and 3 hours both by 3 gives 65 miles and 1 hour. That is the same speed, written for one hour.
Unit rates matter because they let you compare. Store A sells 5 apples for 4 dollars and Store B sells 8 apples for 6 dollars. The totals cannot be compared, because they buy different numbers of apples. The unit prices can: Store A charges 80 cents an apple and Store B charges 75 cents. Store B is cheaper for each apple even though its total is bigger.
Unit rates also let you predict. Once you know the amount for one, multiply to find the amount for any number: at 65 miles per hour, 4 hours cover $65 \times 4 = 260$ miles.
Another way: picture
Draw a double number line. The top line is marked 0, 65, 130, 195 miles and the bottom line is marked 0, 1, 2, 3 hours, each mark straight under its partner. The mark above 1 hour is the unit rate. Walking one step to the right always adds 65 miles and 1 hour, so every step is the same size.
Another way: steps
Every rate can be turned into two unit rates, and choosing the right one is the first decision. A printer prints 84 pages in 6 minutes. Dividing pages by minutes gives $84 \div 6 = 14$ pages per minute. Dividing minutes by pages gives $6 \div 84 = \frac{1}{14}$ of a minute per page. Both are true, and they answer different questions.
The word after per tells you which quantity goes on the bottom. Pages per minute has minutes on the bottom; minutes per page has pages on the bottom. If a question does not say, ask yourself which unit you want one of. To compare prices you want the price of one ounce, so ounces go on the bottom and the answer is cents per ounce. To find how long a job takes you may want the time for one page, so pages go on the bottom.
A good habit is to write the units into the fraction, as in $\frac{84 \text{ pages}}{6 \text{ minutes}}$. The units then tell you what the answer means, and a fraction with the wrong unit on the bottom is easy to spot.
Prices usually come in dollars and cents, such as 3.60 dollars. Dividing a decimal is harder than dividing a whole number, so a useful trick is to change dollars to cents first. One dollar is 100 cents, so 3.60 dollars is $3.60 \times 100 = 360$ cents. Now $360 \div 12 = 30$ cents per ounce is a whole-number division.
Some unit rates do not come out as whole numbers, and that is fine. A runner who goes 10 miles in 4 hours runs $10 \div 4 = 2.5$ miles per hour. A bag of 3 pounds of grapes for 5 dollars costs $500 \div 3 \approx 166.7$ cents per pound, or about 1.67 dollars. When you must round, round only at the very end. If you round the unit rate to 1.70 dollars and then buy 9 pounds, you get 15.30 dollars instead of the true 15.00 dollars, and the error grows every time you multiply.
When two unit prices are very close, keep an extra decimal place so that the comparison is fair. 16.6 cents and 16.4 cents are different; 17 and 16 hide how close they are.
Every unit rate problem in this lesson uses the same few moves.
Step 1: Name the quantities. Write down both numbers with their units, such as 84 pages and 6 minutes. A rate always has two different units.
Step 2: Set up the fraction. Put the quantity you want one of on the bottom. This is the quantity named after the word per.
Step 3: Divide. Divide the top by the bottom. This is allowed because dividing both parts of a rate by the same number keeps the rate the same, and dividing the bottom by itself leaves exactly 1.
Step 4: Use the unit rate. To compare, put two unit rates side by side; they are measured the same way now. To predict forward, multiply the unit rate by the new amount. To work backward from a total, divide the total by the unit rate.
Step 5: Check. There are three quick checks.
If a check fails, the usual cause is dividing the wrong way round, so go back to Step 2 and look at which unit is on the bottom.
A unit rate answers two kinds of question. Going forward, you know how many units and want the total: at 14 pages per minute, 15 minutes print $14 \times 15 = 210$ pages. Going backward, you know the total and want the number of units: 350 pages take $350 \div 14 = 25$ minutes.
The easy way to remember which is which is to think of equal groups. Each minute is one group of 14 pages. Forward, you know the number of groups, so multiply. Backward, you are asking how many groups of 14 make 350, which is a division.
You can always check one direction with the other. If 25 minutes print 350 pages, then 25 groups of 14 must be 350, and $25 \times 14 = 350$.
Many grocery stores in the United States print a unit price on the shelf tag, next to the regular price. The regular price is what you pay; the unit price, often in cents per ounce, is what you use to compare.
Suppose a 64-ounce bottle of apple juice costs 3.84 dollars and a 46-ounce bottle costs 2.99 dollars. Change to cents: 384 and 299. The big bottle costs $384 \div 64 = 6$ cents per ounce. The small bottle costs $299 \div 46 = 6.5$ cents per ounce. The big bottle is cheaper per ounce, by half a cent. That sounds tiny, but over 64 ounces it is $0.5 \times 64 = 32$ cents.
Now suppose the small bottle goes on sale for 2.30 dollars. Its unit price becomes $230 \div 46 = 5$ cents per ounce, and the small bottle is now the better buy. The sale changed the answer, which is exactly why shoppers compare unit prices instead of trusting that bigger is cheaper.
Cars are compared by miles per gallon, the number of miles the car goes on one gallon of gas. A car that goes 348 miles on 12 gallons gets $348 \div 12 = 29$ miles per gallon.
That unit rate lets a family plan a trip. A 435-mile drive needs $435 \div 29 = 15$ gallons. If gas costs 3.20 dollars a gallon, the gas for the trip costs $15 \times 3.20 = 48$ dollars. Two unit rates were used in a row: miles per gallon to find the gallons, and dollars per gallon to find the cost.
The same numbers give a third unit rate, the cost for one mile: $48 \div 435 \approx 0.11$ dollars, or about 11 cents per mile.
A nurse does not count your heartbeats for a whole minute. She counts for 15 seconds and multiplies by 4, because a minute is 4 groups of 15 seconds. If she counts 21 beats, your heart rate is $21 \times 4 = 84$ beats per minute. The American Heart Association gives 60 to 100 beats per minute as a normal resting heart rate for adults, and children often have a faster rate. The nurse turned a count over 15 seconds into a unit rate for one minute, so it can be compared with that range.
Comparing totals instead of unit rates. Six dollars is more than four dollars, but if the six dollars buys eight apples and the four dollars buys five, the six dollars is the better deal. Totals can only be compared when they buy the same amount. Divide first, compare second.
Believing the bigger pack is always cheaper. Stores often charge less per unit for a big pack, but not always. A sale on the small size, or a big size with a fancy label, can turn it around. Only the unit price tells you.
Dividing the wrong way round. For 84 pages in 6 minutes, $6 \div 84$ is not wrong arithmetic, but it gives minutes per page. If the question asks for pages per minute, the pages go on top. Writing units in the fraction stops this mistake.
Adding instead of multiplying. At 14 pages per minute, 15 minutes print $14 \times 15 = 210$ pages, not $14 + 15 = 29$. Adding mixes pages with minutes, which never makes sense.
Rounding too early. Keep the full unit rate until the last step, then round the final answer, usually to the nearest cent.
A family drives 195 miles in 3 hours at a steady speed. Find the speed in miles per hour. First, name the two quantities.
$195 \text{ miles}, \qquad 3 \text{ hours}$
A rate compares two quantities with different units, so both units are written down.
Write the rate as a fraction with hours on the bottom.
$\dfrac{195 \text{ miles}}{3 \text{ hours}}$
The question says miles per hour, so hours is the quantity we turn into 1.
Divide the top and the bottom by 3.
$\dfrac{195 \div 3}{3 \div 3} = \dfrac{65}{1}$
Dividing both parts by the same number gives an equal rate, and $3 \div 3 = 1$ is exactly the one hour we want.
Read the unit rate with its units.
$65 \text{ miles per hour}$
With 1 hour on the bottom, the top is the distance for a single hour.
Check by multiplying back.
$65 \times 3 = 195$
Three hours of 65 miles each must make the whole trip, and they do.
A 12-ounce box of cereal costs 3.60 dollars and an 18-ounce box costs 4.86 dollars. Change both prices to cents.
$3.60 \times 100 = 360 \text{ cents}, \qquad 4.86 \times 100 = 486 \text{ cents}$
One dollar is 100 cents, and whole numbers are easier to divide than decimals.
Find the small box's price for one ounce.
$360 \div 12 = 30 \text{ cents per ounce}$
To compare prices we want the price of one ounce, so ounces go on the bottom.
Find the large box's price for one ounce.
$486 \div 18 = 27 \text{ cents per ounce}$
The same division for the other box, so both answers mean the same thing.
Compare the two unit prices.
$27 < 30$
Both numbers are cents for one ounce, so the smaller number is the lower price.
Find how much you save on each ounce.
$30 - 27 = 3 \text{ cents per ounce}$
The difference of the unit prices is the saving on every single ounce.
Check the large box's unit price by multiplying back.
$27 \times 18 = 486 \text{ cents} = 4.86 \text{ dollars}$
The unit price times the ounces gives back the shelf price, so the large box really is the better buy here.
A printer prints 84 pages in 6 minutes. How many pages does it print in 15 minutes, and how long does it take to print 350 pages? Write the rate as a fraction.
$\dfrac{84 \text{ pages}}{6 \text{ minutes}}$
Both questions are about pages and minutes, so one rate serves both.
Divide to find the pages in one minute.
$84 \div 6 = 14 \text{ pages per minute}$
Minutes are on the bottom, so dividing gives the pages for a single minute.
Going forward: multiply the unit rate by 15 minutes.
$14 \times 15 = 210 \text{ pages}$
Fifteen minutes are 15 equal groups of 14 pages.
Check the forward answer another way: 15 minutes is how many times 6 minutes?
$15 \div 6 = 2.5, \qquad 84 \times 2.5 = 210$
Two and a half times the time must print two and a half times the pages, and both methods agree.
Going backward: divide the 350 pages by 14 pages per minute.
$350 \div 14 = 25 \text{ minutes}$
The question is how many groups of 14 pages make 350, and that is a division.
Check the backward answer by multiplying.
$14 \times 25 = 350$
Twenty-five minutes at 14 pages each gives back the 350 pages.
Do it once more with the other unit rate, minutes per page.
$6 \div 84 = \dfrac{1}{14} \text{ minute per page}, \qquad 350 \times \dfrac{1}{14} = 25$
Every rate has two unit rates. Using the time for one page and multiplying gives the same 25 minutes.
Change the price to cents.
$7.50 \times 100 = 750 \text{ cents}$
Cents are whole numbers, which makes the division easy.
Divide to find the price of one pound.
$750 \div 5 = 150 \text{ cents per pound}$
Pounds go on the bottom because we want the price of one pound.
Multiply by 8 pounds.
Check by dividing back.
Store A sells 4 pens for 16 dollars. Store B sells 5 pens for 10 dollars. Which store charges less for one of the pens?
A small jar of peanut butter holds 15 ounces and costs 3.60 dollars. A large jar holds 27 ounces and costs 8.37 dollars. Complete the worked solution to find the better buy.
Change both prices to cents.
$3.60 \text{ dollars} = 360 \text{ cents}, \quad 8.37 \text{ dollars} = 837 \text{ cents}$
Whole numbers of cents are easier to divide than dollars with decimals.
Divide the small jar's cents by its ounces.
$360 \div 15 =$ u cents per ounce
This is the price of one ounce from the small jar.
Divide the large jar's cents by its ounces.
$837 \div 27 =$ v cents per ounce
The same division for the large jar, so both unit prices are in cents for one ounce.
Subtract the smaller unit price from the larger one.
$(\text{large jar}) - (\text{small jar}) =$ d cents per ounce
The difference is how much more every ounce costs in the large jar.
Name the better buy.
$\text{small jar: fewer cents per ounce}$
Here the smaller jar wins, which is why a bigger size must always be checked, never assumed.
A car travels 240 miles in 8 hours at a steady speed. What is its speed in miles per hour?
The speed is answer miles per hour.
A bottling machine fills 31 bottles per minute. How many bottles does it fill in 14 minutes?
The machine fills answer bottles.
A painter covers 90 square feet of wall per hour. At that rate, how many hours does it take to paint 360 square feet, and how many square feet are still bare after the first 2 hours?
The job takes hours hours, and after 2 hours left square feet are still bare.
A 5-pound bag of rice costs 4.10 dollars. At the same price per pound, how many dollars would 8 pounds of the same rice cost?
One pound costs unit cents, and the rice you want costs total dollars.
A biologist films two hummingbirds. She counts 444 wingbeats for the first bird in 6 seconds and 308 wingbeats for the second bird in 4 seconds. Which bird beats its wings faster?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A 5-pound bag of rice costs 2.50 dollars. At the same price per pound, how many dollars would 7 pounds of the same rice cost?
One pound costs unit cents, and the rice you want costs total dollars.
You can find a unit rate, use it to compare two offers, and use it to predict forward and backward. Without looking: which is the better buy, 3 for 4 dollars or 5 for 6 dollars, and what did you divide to decide?
16. Your turn: 5 pounds of apples cost 7.50 dollars. What do 8 pounds cost?, step 3
$150 \times 8 = 1200 \text{ cents} = 12.00 \text{ dollars}$
Eight pounds are 8 groups of 150 cents; 1200 cents is 12 dollars.
16. Your turn: 5 pounds of apples cost 7.50 dollars. What do 8 pounds cost?, step 4
$1200 \div 8 = 150$
The new total shared among 8 pounds gives the same unit price, so the answer fits.