17.5: Solving Rate Problems

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Lesson

Let's use unit rates like a pro.

Exercise $\PageIndex{1}$: Grid of 100

How much is shaded in each one?

Figure $\PageIndex{1}$: Three 10 unit by 10 unit square grids labeled A, B, and C. Grid A has the first 5 units in each column shaded in columns 1 through 5. Grid B has 10 units shaded in column 1. Grid C has 10 units shaded in columns 1 through 5 and the last 5 units in each column shaded in columns 6 through 10.

Exercise $\PageIndex{2}$: Card Sort: Is it a Deal?

Your teacher will give you a set of cards showing different offers.

Find card A and work with your partner to decide whether the offer on card A is a good deal. Explain or show your reasoning.
Next, split cards B–E so you and your partner each have two.
1. Decide individually if your two cards are good deals. Explain your reasoning.
2. For each of your cards, explain to your partner if you think it is a good deal and why. Listen to your partner’s explanations for their cards. If you disagree, explain your thinking.
3. Revise any decisions about your cards based on the feedback from your partner.
When you and your partner are in agreement about cards B–E, place all the cards you think are a good deal in one stack and all the cards you think are a bad deal in another stack. Be prepared to explain your reasoning.

Are you ready for more?

Time to make your own deal! Read the information on card F and then decide what you would charge if you were the clerk. When your teacher signals, trade cards with another group and decide whether or not you would take the other group’s offer.

Keep in mind that you may offer a fair deal or an unfair deal, but the goal is to set a price close enough to the value it should be that the other group cannot immediately tell if the deal you offer is a good one.

Exercise $\PageIndex{3}$: The Fastest of All

Wild animals from around the world wanted to hold an athletic competition, but no one would let them on an airplane. They decided to just measure how far each animal could sprint in one minute and send the results to you to decide the winner.

You look up the following information about converting units of length:

1 inch = 2.54 centimeters

Table $\PageIndex{1}$
animal	sprint distance
cougar	1,408 yards
antelope	1 mile
hare	49,632 inches
kangaroo	1,073 meters
ostrich	1.15 kilometers
coyote	3,773 feet

Which animal sprinted the farthest?
What are the place rankings for all of the animals?

Summary

Sometimes we can find and use more than one unit rate to solve a problem.

Suppose a grocery store is having a sale on shredded cheese. A small bag that holds 8 ounces is sold for $2. A large bag that holds 2 kilograms is sold for $16. How do you know which is a better deal?

Here are two different ways to solve this problem:

Compare dollars per kilogram.

The large bag costs $8 per kilogram, because $16\div 2=8$.
The small bag holds $\frac{1}{2}$ pound of cheese, because there are 16 ounces in 1 pound, and $8\div 16=\frac{1}{2}$.
The small bag costs $4 per pound, because $2\div\frac{1}{2}=4$. This is about $8.80 per kilogram, because there are about 2.2 pounds in 1 kilogram, and $4.00\cdot 2.2=8.80$.

The large bag is a better deal, because it costs less money for the same amount of cheese.

Compare ounces per dollar.

With the small bag, we get 4 ounces per dollar, because $8\div 2=4$.
The large bag holds 2,000 grams of cheese. There are 1,000 grams in 1 kilogram, and $2\cdot 1,000=2,000$. This means 125 grams per dollar, because $2,000\div 16=125$.
There are about 28.35 grams in 1 ounce, and $125\div 28.35\approx 4.4$, so this is about 4.4 ounces per dollar.

The large bag is a better deal, because you get more cheese for the same amount of money.

Another way to solve the problem would be to compare the unit prices of each bag in dollars per ounce. Try it!

Glossary Entries

Definition: Pace

Pace is one way to describe how fast something is moving. Pace tells how much time it takes the object to travel a certain distance.

For example, Diego walks at a pace of 10 minutes per mile. Elena walks at a pace of 11 minutes per mile. Elena walks slower than Diego, because it takes her more time to travel the same distance.

Definition: Speed

Speed is one way to describe how fast something is moving. Speed tells how much distance the object travels in a certain amount of time.

For example, Tyler walks at a speed of 4 miles per hour. Priya walks at a speed of 5 miles per hour. Priya walks faster than Tyler, because she travels more distance in the same amount of time.

Definition: Unit Price

The unit price is the cost for one item or for one unit of measure. For example, if 10 feet of chain link fencing cost $150, then the unit price is $150\div 10$, or $15 per foot.

Definition: Unit Rate

A unit rate is a rate per 1.

For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because $12\div 2=6$. The other unit rate is $\frac{1}{6}$ of a pie per person, because $2\div 12=\frac{1}{6}$.

Practice

Exercise $\PageIndex{4}$

This package of sliced cheese costs $2.97.

How much would a package with 18 slices cost at the same price per slice? Explain or show your reasoning.

Exercise $\PageIndex{5}$

A copy machine can print 480 copies every 4 minutes. For each question, explain or show your reasoning.

How many copies can it print in 10 minutes?
A teacher printed 720 copies. How long did it take to print?

Exercise $\PageIndex{6}$

Order these objects from heaviest to lightest.

(Note: 1 pound = 16 ounces, 1 kilogram $\approx$ 2.2 pounds, and 1 ton = 2,000 pounds)

Table $\PageIndex{2}$
item	weight
school bus	$9$ tons
horse	$1,100$ pounds
elephant	$5,500$ kilograms
grand piano	$15,840$ ounces

Exercise $\PageIndex{7}$

Andre sometimes mows lawns on the weekend to make extra money. Two weeks ago, he mowed a neighbor’s lawn for $\frac{1}{2}$ hour and earned $10. Last week, he mowed his uncle’s lawn for $\frac{3}{2}$ hours and earned $30. This week, he mowed the lawn of a community center for 2 hours and earned $30.

Which jobs paid better than others? Explain your reasoning.

(From Unit 3.3.1)

Exercise $\PageIndex{8}$

Calculate and express your answer in decimal form.

$\frac{1}{2}\cdot 17$
$\frac{3}{4}\cdot 200$
$(0.2)\cdot 40$
$(0.35)\cdot 60$

(From Unit 3.1.1)

Exercise $\PageIndex{9}$

Here is a polygon.

Figure $\PageIndex{3}$: A heart-shaped polygon. Arrows along the bottom indicate the width of each half of the polygon is 6. Arrows along the top indicate that each of the six sides at the top span 2 units. Arrows along the right side indicate that the top portion of the polygon is 2 units, the middle portion is 3 units, and the bottom portion is 6 units.

Decompose this polygon so that its area can be calculated. All measurements are in centimeters.
Calculate its area. Organize your work so that it can be followed by others.

(From Unit 1.5.1)

item	weight
school bus	\(9\) tons
horse	\(1,100\) pounds
elephant	\(5,500\) kilograms
grand piano	\(15,840\) ounces