MathematicsUnit Rates: Finding How Much Per One
A bustling outdoor farmers market with colorful produce stalls; Lumi stands at a table holding two bags of apples and a notepad, calculating prices and comparing deals with a focused, curious expression.
- Explain what a unit rate is and why the bottom number (denominator) equals one.
- Calculate a unit rate by placing the desired quantity on top and the number of units on the bottom, then dividing.
- Compare two unit rates to identify the better deal or faster speed.
- Identify the difference between a ratio and a unit rate.
- Apply unit rate reasoning to real-world pricing and distance problems.
Key terms
- Ratio
- A comparison of two quantities, such as 4 apples for $2.00, that does not yet tell you the amount for one unit.
- Unit rate
- A rate that tells you how much of one quantity goes with exactly one unit of another, like dollars per single apple.
- Denominator
- The bottom number of a fraction or rate, which is always 1 once you have found a unit rate.
- Per
- A word meaning "for each one," which almost always signals that you should find a unit rate.
- Better deal
- The option with the lower price per single unit, found by comparing the two unit rates after dividing.
Why divide to get a unit rate
A unit rate is really just a ratio rewritten so the bottom number becomes 1. To shrink the bottom from 4 apples down to 1 apple, you divide both the top and the bottom by 4. The bottom becomes 4 ÷ 4 = 1, and the top becomes $2.00 ÷ 4 = $0.50. That is why the single step of dividing the top by the bottom gives you the price for exactly one apple.
Comparing deals fairly
You cannot compare two deals just by looking at the totals, because the quantities are different sizes. Turning each deal into a unit rate puts both on the same footing — dollars per one apple for each stall. Once both deals are measured "per one," the smaller number is always the better buy. This same fairness trick works for miles per hour, calories per serving, or cost per ounce.
Spotting unit rates in real life
Whenever you read a label or a sign with the word "per" — miles per hour, words per minute, dollars per pound — someone has already done the unit-rate division for you. If you only see a total, like "3 pounds for $7.50," you can find the unit rate yourself by dividing. Grocery shelf tags often print the unit price in tiny letters so shoppers can compare brands quickly.
Worked examples
Find the unit price of strawberries sold at 3 pounds for $7.50.
- Decide what you want per one: dollars per one pound, so put the price $7.50 on top and 3 pounds on the bottom.
- Divide the top by the bottom: $7.50 ÷ 3 = $2.50.
- The bottom is now 1 pound, so the unit rate is $2.50 per pound.
Answer: $2.50 per pound
Which jar of honey is the better deal: 12 ounces for $3.60 or 20 ounces for $5.40?
- Find the unit price of the small jar: $3.60 ÷ 12 ounces = $0.30 per ounce.
- Find the unit price of the large jar: $5.40 ÷ 20 ounces = $0.27 per ounce.
- Compare the per-ounce prices: $0.27 is less than $0.30, so the 20-ounce jar costs less per ounce.
Answer: The 20-ounce jar at $0.27 per ounce is the better deal.
Activity
Match each ratio card to its correct unit rate by dragging it to the right answer.
Practice
A vendor sells 6 ears of corn for $4.20. Find the unit rate in dollars per ear of corn.
Stall A sells 4 peaches for $3.20 and Stall B sells 5 peaches for $3.75. Which stall has the lower price per peach?
Common mistakes to avoid
- The total price is the unit rate.The total covers all the items, so you must divide the total by the number of units to get the price for just one.
- The deal with the smaller total always wins.A smaller total can still cost more per item, so compare the unit rates and not the totals to find the true better deal.
Check your understanding
A store sells 5 notebooks for $6.25. What is the unit rate in dollars per notebook?
Store X sells 3 pens for $2.40. Store Y sells 5 pens for $3.75. Which store has the lower price per pen?
Mia says the unit rate of '12 eggs for $3.00' is $3.00 per egg because $3.00 is already in the ratio. What is wrong with her thinking?
Recap
A unit rate tells you how much goes with exactly one unit, and you find it by putting the amount you want on top, the count on the bottom, and dividing so the bottom becomes 1. Comparing unit rates instantly reveals the better deal.
Reflect
Where in your own week could finding a unit rate help you make a smarter choice?