Learning Goals:
- Understand and demonstrate how to find unit rates and equivalent rates
- Be able to compare rates
- Solve problems involving rates
Have you ever had to find an equivalent rate? Take a look at this soccer dilemma.
Myra’s team scored 10 goals in the last 3 games. At this rate, how many goals will Myra’s team score in 6 games?
In this Concept, you will learn how to figure out an equivalent rate for this situation.
Guidance
A unit rate is a special kind of ratio, where the second number, or the denominator, is equal to one.
With a unit rate, you are comparing a quantity to one. Some common unit rates are miles per gallon, price per pound, and pay rate per hour.
To find a unit rate, simplify the ratio so that you have a 1 in the denominator. You can simply divide the first number in the ratio by the second. Make sure you keep track of the units.
Write this information on unit rates down in your notebook. Then continue with the following situation.
Kayla bought 5.5 pounds of apples. She paid a total of $7.15. What was the unit rate of the apples per pound?
A key word is here “per pound”. When you see the word “per”, you should know that you are working with unit rates.
You need to find the price per pound. We can write the ratio of price to pounds using the information in the problem. That is what we are comparing, so that is how we write the ratio. Then we can fill in the given information.
\begin{align*}\frac{price}{pounds} = \frac{\$ 7.15}{5.5 \ pounds}\end{align*}
Now divide to find the price per pound. We divide the price by the number of pounds that she bought.
\begin{align*}{5.5 \overline{){7.15\; }}}\end{align*}
\begin{align*}\frac{\$7.15}{5.5 \ pounds} = \frac{\$ 1.30}{1 \ pound}\end{align*}
The unit rate is $1.30 per pound.
Here is another one.
Brian worked for 8 hours yesterday and made a total of $86. What is his pay rate?
Write a ratio comparing pay rate to hours worked. The rate that we are looking for is what Brian made “per” hour. Even though the problem doesn’t use the word “per” a pay rate is per hour.
\begin{align*}\frac{\text{pay rate}}{\text{hours worked}} = \frac{\$ 86}{8 \ hours}\end{align*}
Now divide to find the unit rate.
\begin{align*}\frac{\$86}{8 \ hours} = \frac{\$ 10.75}{1 \ hour}\end{align*}
Brian’s pay rate is $10.75 per hour.
You may also see rates that are equal. Let’s look at equivalent rates.
Find an equivalent rate for this comparison.
\begin{align*}\frac{\$2.00}{1} = \frac{?}{8}\end{align*}
Notice that we have a unit rate here. We know that the unit rate is two dollars for every one thing. We want eight. We can figure out how to work on this problem by thinking mathematically.
\begin{align*}1 \times 8 = 8\end{align*}
Just like when we worked with fractions, whichever operation we perform with the denominator must be performed with the numerator too. We multiplied by eight, so we need to do that with the numerator too.
\begin{align*}\frac{\$2.00}{1} = \frac{\$ 16.00}{8}\end{align*}
These two rates are equivalent.
As long as you multiply the numerator and the denominator by the same value, you will always create an equivalent rate!
Write find the unit rate for each ratio.
Example A
\begin{align*}\frac{14}{7}\end{align*}
Solution: \begin{align*}\frac{2}{1}\end{align*}
Example B
\begin{align*}\frac{36}{12}\end{align*}
Solution: \begin{align*}\frac{3}{1}\end{align*}
Example C
\begin{align*}\frac{48}{8}\end{align*}
Solution: \begin{align*}\frac{6}{1}\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
First, write the ratio to show the team’s scoring rate.
\begin{align*}\frac{goals}{games} = \frac{10 \ goals}{3 \ games}\end{align*}
You need to know the equivalent rate for 6 games. Notice that the second ratio has the games in the same spot of the denominator. Be sure that you write the ratios so that the same quantities are being compared. If you mix them up, you get a different result.
\begin{align*}\frac{10 \ goals}{3 \ games} = \frac{? \ goals}{6 \ games}\end{align*}
Look at the two fractions. The denominator is doubled in the second fraction. So, multiply the first fraction by an equivalent of 1 in order to get the second fraction.
\begin{align*}\frac{10 \ goals}{3 \ games} \left(\frac{2}{2}\right) = \frac{20 \ goals}{6 \ games}\end{align*}
An equivalent rate is 20 goals in 6 games. So at this rate, Myra’s team will score 20 goals.
Word Wall
- Ratio
- a way of comparing two numbers or quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.
- Unit Rate
- a ratio that is comparing a quantity to one. The word “per” is a key word with unit rates.
- Equivalent Rate
- two rates that are equal although different values are being used to represent the same quantities.
Guided Practice
Here is one for you to try on your own.
A store sells salmon for $6.99 per pound. What is the rate for 6 pounds of salmon?
Solution
First, think about what you know. You know the rate per pound. You need to find the rate for 6 pounds. So you can multiply the unit rate by 6 to find the equivalent ratio.
\begin{align*}\frac{\$6.99}{1 \ pound} \left(\frac{6}{6} \right) = \frac{\$41.94}{6 \ pounds}\end{align*}
The rate for 6 pounds of salmon is $41.94.
Video Review
Khan Academy Finding Unit Rates
Practice
Directions: Use what you have learned about ratios to solve each problem.
In Kyle’s drawer, there are 14 pairs of white socks and 8 pairs of black socks
- Write the ratio of black socks to white socks.
- Write the ratio of black socks to total socks.
- Write the ratio of white socks to total socks.
- Simplify your answer for number 1.
- Simplify your answer for number 2.
- Simplify your answer for number 3.
There are 150 apartments in the Gray building. Of these, 60 are rented and the rest are owned. There are 65 apartments in the Black building. Of these, 45 are rented and the rest are owned. Simplify each answer.
- What is the ratio of rented to owned in the Gray building?
- What is the ratio of rented to owned in the Black building?
- Write the ratio of rented to total apartments in the Gray building.
- Write the ratio of rented to total apartments in the Black building.
- Write the ratio of owned to total apartments in the Gray building.
- Write the ratio of owned to total apartments in the Black building.
- Holly works at a library re-shelving books. She re-shelved 960 books in 4 hours. What is Holly’s rate of reshelving in books per hour?
- Sam bought 9.5 pounds of peaches to make a pie. The peaches cost $15.39. What was the unit rate of the peaches?
- Don can wrap 8 presents in an hour. What is Don’s rate for 12 presents?