Have you ever tried to figure out a better buy?

Kyle went to the bookstore. There was a big sale going on. In one corner of the store, you could buy four books for twelve dollars. In another part of the bookstore, you could buy three books for nine dollars. Kyle thinks that this is the same deal.

Is he correct?

**To answer this question, you will need to understand equivalent rates. This is the information that will be presented in this Concept.**

### Guidance

**What is a rate?**

**A** *rate***is a special kind of ratio. It compares two different types of units, such as dollars and pounds.**

Suppose you are buying turkey at a supermarket, and you pay $12 for 2 pounds of turkey. That is an example of a rate. The turkey you bought cost $6 ** per** pound for the turkey. That is another example of a rate.

*Notice the word “per”. This word signals us that we are talking about a rate.*

We use rates all the time. We use them when shopping for example, per pound. We use them with gasoline, think per mile. We use them with speed, per gallon or with pricing, $4.00 per yard of material.

**Sometimes, two rates are equivalent or equal.**

**How can we tell if two rates are equivalent?**

**Two rates are equivalent if they show the same relationship between two units of measure. We can use the same strategies to find equivalent rates that we use to find equivalent ratio.**

Determine if these two rates are equivalent 40 miles in 2 hours, and 80 miles in 4 hours.

**You can think of this as the distance and time of two different cars. Did they both travel at an equal rate?**

**First, express each rate as a fraction. Be sure to keep the terms consistent. That is, if the first ratio compares miles to hours, the second ratio should also compare miles to hours.**

\begin{align*}40 \ miles \ \text{in}\ 2 \ hours &= \frac{40 mi}{2h} = \frac{40}{2}\\ 80 \ miles \ \text{in} \ 4 \ hours &= \frac{80 mi}{4h} = \frac{80}{4}\end{align*}

Change the ratio, \begin{align*}\frac{40}{2}\end{align*}, to a ratio with 4 as the second term, or denominator.

Since \begin{align*}2 \times 2 = 4\end{align*}, multiply both terms of the ratio \begin{align*}\frac{40}{2}\end{align*} by 2.

\begin{align*}\frac{40}{2} = \frac{40 \times 2}{2 \times 2} = \frac{80}{4}\end{align*}

**This shows that the ratio \begin{align*}\frac{80}{4}\end{align*} is equivalent to the ratio \begin{align*}\frac{40}{2}\end{align*}.**

**This means that the rate 80 miles in 4 hours is equivalent to the rate 40 miles in 2 hours.**

**The two cars traveled at the same rate.**

**You can also cross multiply to determine if two rates are equivalent. Let’s look at an example where this strategy is applied.**

Determine if these two rates are equivalent 5 meters every 3 seconds and 20 meters every 18 seconds.

**You can think of this in terms of speed. The machine wound 5 meters of wire in three seconds. A second machine wound 20 meters of wire in 18 seconds. Did they wind the wire at the same rate?**

**First, cross multiply to determine if the rates are equivalent or not.**

\begin{align*}\frac{5m}{3 \sec} &\overset{?}{=} \frac{20 m}{18 \sec}\\ \frac{5}{3} &\overset{?}{=} \frac{20}{18}\\ 3 \times 20 &\overset{?}{=} 5 \times 18\\ 60 &\overset{?}{=}90\\ 60 &\neq 90\end{align*}

**Since the cross products are not equal, the rates are not equivalent.**

Determine if each rate is equivalent to the other rate.

#### Example A

3 feet in 9 seconds and 6 feet in 18 seconds

**Solution: Equal**

#### Example B

5 miles in 30 minutes and 6 miles in 42 minutes

**Solution: Not Equal**

#### Example C

5 pounds for $20.00 and 8 pounds for $32.00

**Solution: Equal**

Now back to the books.

Kyle went to the bookstore. There was a big sale going on. In one corner of the store, you could buy four books for twelve dollars. In another part of the bookstore, you could buy three books for nine dollars dollars. Kyle thinks that this is the same deal.

Is he correct?

To figure this out, we can write two ratios and compare them. Let's use each deal as our ratios.

\begin{align*}\frac{4}{12}\end{align*} and \begin{align*}\frac{3}{9}\end{align*}

Simplifying these two ratios will show us that they both are equal to \begin{align*}\frac{1}{3}\end{align*}.

**These two ratios are equivalent.**

### Vocabulary

- Rate
- a special kind of ratio that compares two different quantities.

### Guided Practice

Here is one for you to try on your own.

Write an equivalent rate for 3 out of 10.

**Answer**

We can do this by creating any equal ratio. We do this by multiplying both values in the ratio by the same number.

Here are some possible answer.

6 to 20

9 to 30

12 to 40

15 to 50

**There are many possible answers.**

### Video Review

- This is a James Sousa video on rates and unit rates.

### Practice

Directions: Write an equivalent rate for each rate.

1. 2 for $10.00

2. 3 for $15.00

3. 5 gallons for $12.50

4. 16 pounds for $40.00

5. 18 inches for $2.00

6. 5 pounds of blueberries for $20.00

7. 40 miles in 80 minutes

8. 20 miles in 4 hours

9. 10 feet in 2 minutes

10. 12 pounds in 6 weeks

Directions: Simplify each rate.

11. 2 for $10.00

12. 3 for $15.00

13. 5 gallons for $25.00

14. 40 pounds for $40.00

15. 18 inches for $2.00