A rate is a ratio that compares two quantities measured in different units, such as 150 miles in 3 hours or 12 dollars for 4 pounds. The word “per” signals a rate. Dividing gives the unit rate, the amount for 1 unit: 150 miles in 3 hours is 50 miles per hour.
In Spanish
tasa
Also called
rate of speed, per, ratio of two quantities with different units
A rate pairs two different units. Splitting 150 miles in 3 hours into equal parts shows the unit rate: 50 miles for every 1 hour.
Formal definition and notation
A rate is a ratio a : b of two quantities with different units, often written as the fraction a unitsb units. When b ≠ 0, the associated unit rate is ab units of the first quantity per 1 unit of the second. Equivalent rates have the same unit rate, so they form a proportion.
Symbol
Read it as
Meaning
150 mi3 h
150 miles per 3 hours
a rate written as a fraction with units
50 mi/h
50 miles per hour
a unit rate; the slash means per
r
r
a common letter for a rate, as in d = rt
Key facts
A rate always has two different units, and its name puts them in order: milesper hour is miles divided by hours.
A unit rate has 1 in the denominator. Divide the top number by the bottom: 210 miles on 7 gallons is 210 ÷ 7 = 30 miles per gallon.
A unit price is a unit rate with money on top, such as 2.50 dollars per pound. Use it to find the better buy.
Equal rates make a proportion: 1503 = 2505, because both are 50 miles per hour.
On a graph of a constant rate, the line goes through the origin and its slope is the unit rate.
Rate formulas
Rate formulas
rate = amounttimed = r × t
d
distance (or any total amount)
r
the rate, such as miles per hour
t
the time
Rearrange as needed: r = dt and t = dr. Keep the units matched: a rate in miles per hour needs time in hours.
Try it: build equal rates
Interactive
Change the two quantities and watch the table of equal rates grow. The row with 1 hour shows the unit rate.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Rate examples, step by step
Example 1: Find a unit rate
Warm-up
A printer prints 84 pages in 6 minutes. What is its rate in pages per minute?
Write the rate with units: 84 pages6 minutes.
Divide to get 1 minute on the bottom: 84 ÷ 6 = 14.
Attach the units: 14 pages per minute.
Answer: The printer prints 14 pages per minute.
Example 2: Use a rate to predict
Core
A car uses 9 gallons of gas to go 252 miles. At the same rate, how far can it go on 14 gallons?
Unit rate: 252 ÷ 9 = 28 miles per gallon.
Multiply by the new amount: 28 × 14.
28 × 14 = 392.
Answer: The car can go 392 miles on 14 gallons.
Example 3: Compare rates with a units trap
Test-style
Ana types 1,320 words in 30 minutes. Ben types 50 words per minute. Who types faster, and by how many words per minute?
Ana’s unit rate: 1320 ÷ 30 = 44 words per minute.
Both rates are now in words per minute, so they can be compared.
Ben’s rate is higher: 50 − 44 = 6.
Answer: Ben types faster, by 6 words per minute.
Common mistakes
✗ MistakeFor 180 miles in 4 hours: 4 ÷ 180 ≈ 0.02 miles per hour.
✓ InsteadMiles per hour is miles divided by hours: 180 ÷ 4 = 45 miles per hour.
The unit named first goes on top. 4 ÷ 180 gives hours per mile, a different rate.
✗ MistakeComparing 3 dollars per pound with 20 cents per ounce by the numbers alone and calling 20 the bigger price.
✓ InsteadConvert first: 20 cents per ounce is 20 × 16 = 320 cents, or 3.20 dollars, per pound, which is more than 3 dollars per pound.
Rates can only be compared when both use the same units.
✗ MistakeAveraging two speeds: 30 miles per hour out and 60 back gives 45 miles per hour for the round trip.
✓ InsteadFor 60 miles each way: 2 + 1 = 3 hours for 120 miles, so the average rate is 120 ÷ 3 = 40 miles per hour.
An average rate is total amount divided by total time. You spend more time at the slower speed, so the average is not the middle of the two speeds.
Every rate is a ratio, but a ratio can compare quantities with the same unit (3 boys to 5 girls). A rate compares different units (3 dollars per 5 pounds).
A rate compares two amounts at once. A rate of change compares how much one quantity changes as another changes, such as a balance that drops 20 dollars each week.
Quick check: 3 questions
Question 1 of 3
Which of these is a rate?
Show answer and explanation
Answer: B. Dollars and hours are different units, so 64 dollars for 8 hours is a rate (8 dollars per hour). The other choices compare quantities with the same unit: people to people, cups to cups, points to points.
Question 2 of 3
A pump moves 375 gallons in 15 minutes. What is its unit rate?
Show answer and explanation
Answer: D. Gallons per minute: 375 ÷ 15 = 25. 0.04 divides the wrong way (minutes per gallon). 360 and 390 subtract and add instead of dividing.
Question 3 of 3
A cyclist rides at 18 miles per hour. How long does it take to ride 45 miles?
Show answer and explanation
Answer: A.t = dr = 4518 = 2.5 hours. 810 multiplies, 0.4 divides the wrong way and 27 subtracts.
Rate: frequently asked questions
What is the difference between a rate and a unit rate?
A rate compares two quantities with different units, with any amount in each: 150 miles in 3 hours. A unit rate is the same rate scaled so the second quantity is 1: 50 miles per 1 hour, usually said as 50 miles per hour. To turn any rate into a unit rate, divide the first quantity by the second.
What does "per" mean in a rate?
“Per” means “for each” and tells you to divide. Miles per gallon is miles divided by gallons, and dollars per hour is dollars divided by hours. The quantity before “per” goes on top of the fraction, and the one after “per” goes on the bottom.
What are some real-world examples of rates?
Speed (miles per hour), fuel use (miles per gallon), pay (dollars per hour), prices (dollars per pound), heart rate (beats per minute), typing speed (words per minute), populationdensity (people per square mile) and data speed (megabits per second). Each one compares two different kinds of units.
RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Unit PriceA unit price is the cost of one unit of something, found by dividing the total price by the number of units. For example, 18 dollars for 6 pounds is 18 ÷ 6 = 3 dollars per pound. It is the fairest way to compare products sold in different package sizes: the lower unit price costs less per unit.Ratios and Proportions · Grade 6, Grade 7, Grade 8OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7Rate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2PopulationA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics, Algebra 1DensityDensity is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = mV. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.Problem Solving · Grade 8, Algebra 1, GeometrySquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6Constant of ProportionalityThe constant of proportionality is the constant ratio between two proportional quantities, k = yx. It is the k in y = kx and stays the same for every pair of values, which makes the graph a straight line through the origin. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.Ratios and Proportions · Grade 7, Grade 8, Algebra 1PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Number System · Grade 5, Grade 6, Grade 7Unit ConversionUnit conversion is rewriting a measurement in a different unit without changing the amount, such as writing 3 feet as 36 inches. You multiply by a conversion factor, a fraction equal to 1 such as 12 in1 ft. Converting to a smaller unit gives a larger number; converting to a larger unit gives a smaller number.Measurement · Grade 4, Grade 5, Grade 6