Have you ever argued with a friend about which snack is a better deal, or wondered how fast a car is really going on the highway? Those everyday questions are secretly about unit rates. When you say something like “10 miles per hour” or “2 dollars for each slice of pizza,” you are using unit rates without even thinking about it!
Before we talk about unit rates, we need to be clear on what a ratio is. A ratio compares two quantities. These quantities must be measured in the same unit or in clearly related units.
Some ways to write a ratio of quantity \(a\) to quantity \(b\) are:
For example, if there are 5 red marbles and 3 blue marbles in a bag, the ratio of red to blue marbles can be written as:
These all represent the same relationship: for every 5 red marbles, there are 3 blue marbles.
Another example: A soccer team scores 4 goals in 2 games. The ratio of goals to games is:
This ratio compares two different kinds of quantities: goals and games. When a ratio compares two different units (like dollars and hours, miles and hours, pages and minutes), we often call it a rate.
A rate is a special type of ratio that compares two quantities with different units. For example:
We often want to know how much of one quantity there is for one unit of the other quantity. That is a unit rate.
A unit rate is a rate where the second quantity is exactly 1. In math form, if we have a ratio \(a:b\) (with \(b \neq 0\)), the unit rate is:
\[ \textrm{unit rate} = \frac{a}{b} \textrm{ per } 1 \textrm{ of the second quantity} \]
For example, suppose a recipe has a ratio of 3 cups of flour to 4 cups of sugar. As described earlier, the ratio of flour to sugar is 3 to 4, or \(3:4\), or \(\dfrac{3}{4}\). As shown in [Figure 1], we can think about how much flour there is for each single cup of sugar.
To find the unit rate of flour per 1 cup of sugar, we divide:
\[ \frac{3 \textrm{ cups of flour}}{4 \textrm{ cups of sugar}} = \frac{3}{4} \textrm{ cup of flour per 1 cup of sugar} \]
This means there is \(\dfrac{3}{4}\) cup of flour for each 1 cup of sugar.
Another example: A group of friends buys 15 hamburgers for 75 dollars. The rate is “75 dollars for 15 hamburgers.” To find the unit rate (the cost for 1 hamburger), we divide the first quantity by the second quantity:
\[ \frac{75 \textrm{ dollars}}{15 \textrm{ hamburgers}} = 5 \textrm{ dollars per hamburger} \]
So the unit rate is 5 dollars per hamburger.
Notice how the fraction \(\dfrac{a}{b}\) helps us find “how much of the first quantity for 1 of the second.” This is the key idea of unit rate.
When we talk about unit rates, we often use words like “per” or “for each” or “for every”. These words help turn the numbers into clear sentences.
For the recipe example, we found a unit rate of \(\dfrac{3}{4}\) cup of flour per 1 cup of sugar. We can say this in different ways:
For the hamburger example, we found a unit rate of 5 dollars per hamburger:
We can do the same thing with other unit rates:
This kind of clear language makes it easier to explain and compare rates in real-world situations.
Stores often sell the same product in different sizes and prices. To decide which is a better deal, we can use a unit rate called the unit price. The unit price tells us the cost for 1 unit of something, such as 1 ounce, 1 liter, or 1 item.
We usually calculate unit price by dividing the total cost by the number of units:
\[ \textrm{unit price} = \frac{\textrm{total cost}}{\textrm{number of units}} \]
Imagine you are at a grocery store comparing two brands of cereal, and you want to know which one is cheaper per ounce. As shown in [Figure 2], you can set up a small comparison table.
Suppose:
To find the unit price for Brand A, calculate:
\[ \frac{3.60}{12} = 0.30 \textrm{ dollars per ounce} \]
To find the unit price for Brand B, calculate:
\[ \frac{5.04}{18} = 0.28 \textrm{ dollars per ounce} \]
Brand A costs 0.30 dollars per ounce, and Brand B costs 0.28 dollars per ounce. Brand B is the better deal because it has a lower unit price.
Unit pricing is helpful for many items: snacks, drinks, cleaning supplies, pet food, and more. When you use unit price, you can make smart choices with your money.
Another common type of unit rate is speed. Speed tells us how far something travels in a certain amount of time, like miles in hours or meters in seconds.
For example, if a car travels 180 miles in 3 hours at a constant speed, we can find the speed as a unit rate:
\[ \textrm{speed} = \frac{\textrm{distance}}{\textrm{time}} = \frac{180 \textrm{ miles}}{3 \textrm{ hours}} = 60 \textrm{ miles per hour} \]
We can say:
Once we know the unit rate of speed, we can answer more questions. For example, if the same car continues at 60 miles per hour, how far will it go in 5 hours?
We can multiply the unit rate by the number of hours:
\[ 60 \textrm{ miles per hour} \times 5 \textrm{ hours} = 300 \textrm{ miles} \]
So the car will travel 300 miles in 5 hours at that speed.
Speed can also be in other units, such as:
No matter the units, it is still a unit rate because it tells us how much distance is traveled in 1 unit of time.
Unit rates appear in many areas of life, not just shopping and driving. Here are some more examples that connect to things you experience every day.
In each case, we are dividing the first quantity by the second to find “how much for 1.” This is always a unit rate.
Now let’s walk through some complete examples step by step, to see how to find and use unit rates in different situations.
Example 1: Recipe Ratio
A smoothie recipe uses 6 cups of strawberries for 4 cups of yogurt. What is the unit rate of cups of strawberries per 1 cup of yogurt?
Step 1: Identify the ratio and what it compares.
The recipe compares 6 cups of strawberries to 4 cups of yogurt. So the ratio of strawberries to yogurt is \(6:4\) or \(\dfrac{6}{4}\).
Step 2: Write the rate as a fraction with strawberries on top and yogurt on the bottom.
\[ \frac{6 \textrm{ cups of strawberries}}{4 \textrm{ cups of yogurt}} \]
Step 3: Divide to find the unit rate (per 1 cup of yogurt).
\[ \frac{6}{4} = 1.5 \]
This means the unit rate is 1.5 cups of strawberries per 1 cup of yogurt.
Step 4: Say it in words.
“There are 1.5 cups of strawberries for each cup of yogurt,” or “1 and a half cups of strawberries per cup of yogurt.”
Example 2: Shopping and Unit Price
A pack of 8 markers costs 6.40 dollars. What is the unit price per marker?
Step 1: Identify the total cost and the number of units.
Total cost: 6.40 dollars
Number of markers: 8
Step 2: Use the unit price formula.
\[ \textrm{unit price} = \frac{\textrm{total cost}}{\textrm{number of units}} = \frac{6.40}{8} \]
Step 3: Divide.
\[ \frac{6.40}{8} = 0.80 \]
This means the markers cost 0.80 dollars per marker (80 cents per marker).
Step 4: Say it clearly.
“The unit price is 80 cents per marker,” or “Each marker costs 80 cents.”
If you remember the cereal example and the comparison table in [Figure 2], the same strategy is being used here: divide cost by number of items to get the price for one item.
Example 3: Constant Speed
A bicyclist travels 27 miles in 3 hours at a constant speed. What is the unit rate speed in miles per hour, and how far will the bicyclist travel in 5 hours at this speed?
Step 1: Find the speed as a unit rate.
The bicyclist travels 27 miles in 3 hours, so:
\[ \textrm{speed} = \frac{\textrm{distance}}{\textrm{time}} = \frac{27 \textrm{ miles}}{3 \textrm{ hours}} \]
Divide:
\[ \frac{27}{3} = 9 \]
The unit rate speed is 9 miles per hour.
Step 2: Use the unit rate to predict distance in 5 hours.
Speed is 9 miles per hour, so in 5 hours the distance is:
\[ 9 \textrm{ miles per hour} \times 5 \textrm{ hours} = 45 \textrm{ miles} \]
Step 3: State your answers.
Just like the flour-and-sugar situation in [Figure 1] shows “how much for each 1,” the speed tells us how many miles for each 1 hour.
Example 4: Mixed Context – Pages per Hour
A student reads 90 pages in 4.5 hours.
Step 1: Write the rate.
\[ \frac{90 \textrm{ pages}}{4.5 \textrm{ hours}} \]
Step 2: Divide to find the unit rate.
\[ \frac{90}{4.5} = 20 \]
The student reads 20 pages per hour.
Step 3: Explain in words.
“The student reads 20 pages for every hour of reading.”
