Math · Ratios & measurement
Rates, Unit Rates, and Rate of Change
Estimated time includes reading and one quiz. Take the time you need.
Start here: key ideas
- A rate compares quantities with different units.
- A unit rate has a denominator of one unit.
- Common units are required before rates can be compared.
- Rate of change is vertical change divided by horizontal change.
What you’ll be able to do
- Compute and interpret unit rates with units.
- Compare rates after converting them to common units.
- Find and interpret rate of change from a table or graph.
A rate compares quantities with different units. A unit rate has a denominator of one unit.
Optional review
If you’d like to review the basics, visit Ratios and Proportions. You can start this lesson without completing those first.
Divide to find “per one”
A rate compares different units, such as dollars per pound. A unit rate has 1 in the denominator. If 6 notebooks cost $15, divide both quantities by 6: $15 ÷ 6 = $2.50 per notebook. Keep the units attached; 2.50 alone does not say what was measured.
Worked example 1: A car travels 156 miles in 3 hours. 156 miles ÷ 3 hours = 52 miles per hour. Multiplying 52 by 3 returns 156, which checks the result.
Compare the same unit
Rates are comparable only when their units match. Suppose one bottle is 24 ounces for $3.60 and another is 1.5 pounds for $3.84. Convert 1.5 pounds to 24 ounces. Their unit prices are $3.60/24=$0.15 per ounce and $3.84/24=$0.16 per ounce, so the first costs less per ounce.
Connect tables and graphs
Rate of change is output change divided by input change: slope=(y₂−y₁)/(x₂−x₁). Match subtraction order and label vertical units per horizontal unit.
Scroll the table sideways if needed.
| Time (min) | Volume (mL) |
|---|---|
| 0 | 10 |
| 2 | 18 |
| 5 | 30 |
Worked example: From (2,18) to (5,30), slope=(30−18)/(5−2)=12/3=4 mL/min. The earlier interval is also (18−10)/(2−0)=4 mL/min, so the rate is constant.
A line y=4x+10 has slope 4 and y-intercept 10. It is linear but not a direct proportion because it does not pass through the origin.
Change in y divided by change in x
From (2, 3) to (7, 6), y rises by 6 − 3 = 3 while x rises by 7 − 2 = 5. The slope is 3/5 = 0.6 y-units per x-unit. The small 1 and 2 labels name the two points; they are not extra values to multiply.
Change in y divided by change in x — OpenStax, via Mathematics LibreTexts. CC BY 4.0. Source image reproduced unchanged.
Use direction and context
A positive slope rises left to right; a negative slope falls; zero slope is horizontal. A curved graph has changing slope, so specify the interval when calculating its average rate.
Choose the requested orientation: dollars per pound differs from pounds per dollar. Convert to common units before comparing rates. For an interval, use both endpoints; do not divide by the final x-value unless the interval begins at zero.
For the graph-and-table comparison of direct proportion, a line with a fixed starting amount, and inverse variation, see Reading Data: relationships.
Terms to remember
- rate
- A ratio comparing quantities with different units.
- unit rate
- A rate per one unit.
- rate of change
- The change in one variable divided by the change in another.
- slope
- The rate of change on a coordinate graph.
- rise
- Vertical change between two points.
- run
- Horizontal change between two points.
Your quick summary
- A rate compares quantities with different units.
- A unit rate has a denominator of one unit.
- Common units are required before rates can be compared.
- Rate of change is output change divided by input change, and its units describe how the variables vary together.
Lesson quiz
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