Once you know a unit rate, many problems become one multiplication or one division. Unit rates also make fair comparisons: when both offers are written "per one", you are comparing like with like.
Key idea
Find the rate for one unit. Then: total = rate $\times$ number of units, and number of units = total $\div$ rate.
Why it works
A unit rate is the cost, distance or amount for exactly one unit. Seven notebooks cost 7 times the price of one. A trip at 50 miles per hour takes one hour for every 50 miles, so the number of hours is how many 50s fit in the distance.
Worked example: The better buy
Cereal comes in a 12-ounce box for \$3.00 or a 20-ounce box for \$4.40. Which is the better buy?
- Small box: $\$3.00 \div 12 = \$0.25$ per ounce.
- Big box: $\$4.40 \div 20 = \$0.22$ per ounce.
- 22 cents is less than 25 cents per ounce.
Answer: The 20-ounce box is the better buy.
Worked example: Constant speed
A cyclist rides 36 miles in 3 hours at a steady speed. How long does it take to ride 60 miles?
- Unit rate: $36 \div 3 = 12$ miles per hour.
- Each hour covers 12 miles, so count the 12s in 60: $60 \div 12 = 5$.
Answer: 5 hours.
Worked example: How many can you buy?
6 bagels cost \$4.50. How many bagels can you buy with \$12?
- Price of one bagel: $\$4.50 \div 6 = \$0.75$.
- Number of bagels: $\$12 \div \$0.75 = 16$.
Answer: 16 bagels.
Common mistake
Comparing the totals instead of the unit prices. The 20-ounce box costs more in total (\$4.40 against \$3.00), so it can look like the worse deal, but you also get more cereal. Quick check: before comparing, make sure both numbers are "per one" of the same unit.