Unit Rates & Better Buys
Reduce everything to one of something, and comparisons become obvious.
A 12 oz jar costs $3.00. An 18 oz jar costs $4.05.
The big one costs more, and holds more. So which is the better deal?
The prices cannot be compared directly, because they are not buying the same thing.
Nor can the sizes. Every comparison here is between two things that differ in two ways at once.
So make them the same: price for one ounce. 25 cents against 22.5 cents, and the question answers itself. A unit rate is a common footing.
Making it "per one"
To find a unit rate, divide by the second quantity so that it becomes one.
$3.00 for 12 oz is 3.00 ÷ 12 = $0.25 per ounce.
The units tell you which way to divide: "per ounce" means dollars divided by ounces.
Read the word "per"
"Per" means divided by. Dollars per ounce is dollars ÷ ounces, and getting that backwards gives ounces per dollar — a different, though still useful, number.
Comparing buys
Work out the price per unit for each option, then compare those two numbers.
The smaller price per unit is the better buy, whatever the packet size.
$0.25 per ounce against $0.225 per ounce makes the larger jar the winner.
Or go the other way
Ounces per dollar works too — and there the larger number wins. Just be clear which you computed before deciding.
Speed and other rates
Speed is distance per unit time: 210 miles in 3 hours is 70 miles per hour.
Once you have the unit rate, any other amount is one multiplication away.
70 mph for 5 hours covers 350 miles.
Two steps, always
Divide to find the rate for one, then multiply to reach the amount you want. That pattern solves nearly every rate problem.
Converting units
A conversion is a rate: 12 inches per foot, 60 minutes per hour, 100 centimetres per metre.
Multiply or divide depending on which way you are going, and let the size of the answer check you.
Changing metres to centimetres should give a bigger number, since centimetres are smaller.
Smaller units, bigger number
Measuring the same length in smaller units needs more of them. If your answer shrank, you divided when you should have multiplied.
Rates that are not constant
A unit rate assumes the relationship is steady all the way through.
If a shop gives a discount on bulk, the price per ounce changes with the size, and one unit rate will not describe both.
Check whether the situation really is steady before relying on a single rate.
Average is not always constant
A car averaging 60 mph may never have driven at 60. The average rate is still useful; it simply is not the whole story.
Worked examples
Better buy
A 12 oz jar costs $3.00; an 18 oz jar costs $4.05. Which is the better buy?
- Small jar: 3.00 ÷ 12 = $0.25 per ounce. Dollars divided by ounces.
- Large jar: 4.05 ÷ 18 = $0.225 per ounce. Same calculation, same units.
- 0.225 < 0.25, so the large jar is better. Lower price per unit wins.
The 18 oz jar
Speed
A car travels 210 miles in 3 hours. How far in 5 hours at the same speed?
- Unit rate: 210 ÷ 3 = 70 miles per hour. Divide to get "per one hour".
- Multiply by the new time: 70 × 5. The second step of every rate problem.
- = 350 miles. Sensible: five hours is more than three, so further.
350 miles
Converting
How many centimetres are in 2.5 metres?
- There are 100 centimetres per metre. The conversion rate.
- Centimetres are smaller, so the number should grow: multiply. A check before calculating.
- 2.5 × 100 = 250 cm. Bigger number, smaller unit — as expected.
250 cm
Words to know
- Rate
- A comparison of two quantities with different units, like miles per hour.
- Unit rate
- A rate with a denominator of one — per hour, per ounce, per item.
- Better buy
- The option with the lower unit price.
- Constant speed
- A rate that does not change, so distance and time stay proportional.
- Conversion factor
- A rate used to change units, like 12 inches per foot.
Mistakes to watch for
Comparing prices without comparing sizes. A $4.05 jar is dearer than a $3.00 one and still the better buy, because it holds half as much again. Reduce both to a price per unit first.
Dividing the wrong way round. Dollars per ounce is dollars ÷ ounces. Doing it the other way gives ounces per dollar, where the bigger number is the better buy — correct, but the opposite rule.
Multiplying when converting to a larger unit. Metres to centimetres multiplies by 100; centimetres to metres divides. If the number went the wrong way, so did the operation.
Stopping at the unit rate. Finding 70 mph answers only half of "how far in five hours" — the rate then has to be multiplied by the new time.
Assuming a rate stays constant. A bulk discount means the price per ounce changes with size, so one unit rate cannot describe both packets.
Shops in many countries are required by law to print the unit price on the shelf label, precisely because shoppers cannot do this comparison in their heads.
The bigger package is usually but not always cheaper per unit — shops rely on the assumption, and occasionally break it.
Speed, wages, fuel economy and exchange rates are all unit rates: something per one of something else.