Compare Two Quantities
A ratio shows how much of one quantity there is compared with another quantity. Order matters.
12 students : 3 tables
Free Math Guide
Ratios and Unit Rates Explained
Learn how to write ratios in the correct order, recognize equivalent ratios, calculate per-one rates, compare choices, and label every answer with meaningful units.
A ratio compares two quantities in a specific order. A unit rate is a special ratio whose second quantity is one. Together, these ideas help describe prices, speeds, recipes, work rates, and many other real-world relationships.
What Is the Difference Between a Ratio and a Unit Rate?
Both compare quantities. The difference is that a unit rate rewrites the comparison so the second quantity is exactly one.
Compare to One Unit
Divide both quantities by the second quantity so the comparison describes the amount for one.
4 students per table
The Connection
The ratio 12 students to 3 tables can be divided by 3 to produce the unit rate 4 students per 1 table.
How to Write a Ratio Correctly
Read the comparison from left to right. If the question asks for the ratio of blue tiles to red tiles, the blue quantity must be written first.
12 blue to 8 red = 12:8 = 12/8
Part-to-Part Ratio
Compare one category with another category. If there are 6 dogs and 4 cats, the ratio of dogs to cats is 6:4.
Part-to-Whole Ratio
Compare one category with the total. With 6 dogs and 4 cats, there are 10 animals, so the ratio of dogs to all animals is 6:10.
Reversing the Order Changes the Ratio
The ratio 12 blue to 8 red is 12:8. The ratio of red to blue is 8:12. These describe opposite comparisons.
Equivalent and Simplified Ratios
Equivalent ratios describe the same relationship using different numbers. Multiply or divide both parts by the same nonzero value.
| Ratio | Operation | Equivalent Ratio |
|---|---|---|
| 12:8 | Divide both parts by 2 | 6:4 |
| 6:4 | Divide both parts by 2 | 3:2 |
| 3:2 | Multiply both parts by 4 | 12:8 |
Simplest Form
Divide both parts by their greatest common factor. Because GCF(12, 8) = 4, the ratio 12:8 simplifies to 3:2.
How to Calculate a Unit Rate
Divide the first quantity by the second quantity. The unit labels follow the same order as the numbers.
Cost per item
$18 ÷ 6 notebooks = $3 per notebook
Miles per hour
150 miles ÷ 3 hours = 50 miles per hour
Pages per day
84 pages ÷ 7 days = 12 pages per day
“Per” Tells You the Division Order
Dollars per pound means dollars ÷ pounds. Miles per hour means miles ÷ hours. Keep the requested unit after the word “per” in the denominator.
How to Compare Unit Rates
Convert every option to the same per-one unit. Then decide whether the problem favors the lower or higher rate.
| Comparison | Option A | Option B | Conclusion |
|---|---|---|---|
| Price per ounce | $4.80 ÷ 12 = $0.40 | $5.25 ÷ 15 = $0.35 | Option B costs less per ounce. |
| Miles per hour | 24 ÷ 3 = 8 mph | 35 ÷ 5 = 7 mph | Option A is faster. |
“Better” Depends on the Situation
A lower price per item is usually the better buy, while a higher speed or production rate may be preferred. Read what the question asks you to compare.
Worked Ratio and Unit Rate Examples
Cats to All Animals
A shelter room contains 6 dogs and 4 cats. Write the ratio of cats to all animals.
- There are 4 cats.
- There are 6 + 4 = 10 animals total.
- The requested order is cats to all animals: 4:10.
- Divide both parts by 2 to simplify.
4:10 = 2:5
Cost per Notebook
Six notebooks cost $14.40. Find the price of one notebook.
- The requested unit is dollars per notebook.
- Divide dollars by notebooks.
- Compute 14.40 ÷ 6 = 2.40.
$2.40 per notebook
Better Price per Ounce
Package A costs $9.60 for 12 ounces. Package B costs $13.50 for 18 ounces.
- Package A: 9.60 ÷ 12 = $0.80 per ounce.
- Package B: 13.50 ÷ 18 = $0.75 per ounce.
- The lower price per ounce is the better buy.
Package B is the better buy.
Round a Unit Rate
A vehicle travels 53 miles in 6.5 hours. Find the average speed to the nearest hundredth.
- Divide miles by hours.
- Compute 53 ÷ 6.5 = 8.153846...
- Round only the final answer.
8.15 miles per hour
Common Ratio and Unit Rate Mistakes
Reversing the Ratio
Write the quantities in the same order used in the question. Apples to oranges is not the same as oranges to apples.
Confusing Part-to-Part With Part-to-Whole
If the question asks for one category compared with the total, add all categories before writing the second part.
Changing Only One Part of a Ratio
To create an equivalent ratio, multiply or divide both parts by the same nonzero number.
Dividing in the Wrong Order
Dollars per pound means dollars ÷ pounds—not pounds ÷ dollars.
Comparing Total Amounts
Different package sizes or travel times cannot be compared fairly until both are converted to the same unit rate.
Dropping the Units
The number 3 does not explain whether the result is 3 dollars per item, 3 miles per hour, or another rate.
Rounding Too Early
Keep the calculator value during your work and round only the final answer unless the problem says otherwise.
How to Solve Ratio and Unit Rate Problems
1. Identify the quantities
Write each number with its label or unit.
2. Follow the order
Match the order of the numbers to the order requested in the question.
3. Choose the goal
Decide whether to write a ratio, simplify it, find a per-one rate, or compare rates.
4. Calculate
For a unit rate, divide the first quantity by the second. For an equivalent ratio, change both parts by the same factor.
5. Interpret and check
Attach the correct units and confirm that the answer matches the original question.
A complete unit-rate answer includes both the number and its meaning, such as $2.40 per notebook.
Practice Ratios and Unit Rates
Write ratios, calculate unit rates, and compare real-world choices with randomized problems, immediate feedback, hints, and explanations.
Open the Ratios and Unit Rates ToolRelated Math Resources
Interactive Ratio and Rate Practice
Write ratios and solve randomized unit-rate problems with immediate feedback.
GCF and LCM Word Problems Guide
Learn how to identify, solve, and interpret common factor and multiple situations.
Divisibility Rules Guide
Use number patterns to determine whether one whole number divides another evenly.
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