Math · Ratios & measurement
Rates, Unit Rates, and Rate of Change
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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.
Complete ratios and proportions 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.
| 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.
OpenStax, via Mathematics LibreTexts · CC BY 4.0 · https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/03%3A_Graphs_and_Functions/3.03%3A_Slope_of_a_Line · https://creativecommons.org/licenses/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.
Key terms
- 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.
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.
Read a rate like a fraction: miles over hours means miles per hour.
Sources
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