14.2: How Much for One?

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Lesson

Let's use ratios to describe how much things cost.

Exercise $\PageIndex{1}$: Number Talk: Remainders in Division

Find the quotient mentally.

$246\div 12$

Exercise $\PageIndex{2}$: Grocery Shopping

Answer each question and explain or show your reasoning.

Eight avocados cost $4.
1. How much do 16 avocados cost?
2. How much do 20 avocados cost?
3. How much do 9 avocados cost?
Twelve large bottles of water cost $9.
1. How many bottles can you buy for $3?
2. What is the cost per bottle of water?
3. How much would 7 bottles of water cost?
A 10-pound sack of flour costs $8.
1. How much does 40 pounds of flour cost?
2. What is the cost per pound of flour?

Here is an applet you can use if you choose to.

Are you ready for more?

It is commonly thought that buying larger packages or containers, sometimes called buying in bulk, is a great way to save money. For example, a 6-pack of soda might cost $3 while a 12-pack of the same brand costs $5.

Find 3 different cases where it is not true that buying in bulk saves money. You may use the internet or go to a local grocery store and take photographs of the cases you find. Make sure the products are the same brand. For each example that you find, give the quantity or size of each, and describe how you know that the larger size is not a better deal.

Exercise $\PageIndex{3}$: More Shopping

Four bags of chips cost $6.
1. What is the cost per bag?
2. At this rate, how much will 7 bags of chips cost?
At a used book sale, 5 books cost $15.
1. What is the cost per book?
2. At this rate, how many books can you buy for $21?
Neon bracelets cost $1 for 4.
1. What is the cost per bracelet?
2. At this rate, how much will 11 neon bracelets cost?
  Pause here so your teacher can review your work.
Your teacher will assign you one of the problems. Create a visual display that shows your solution to the problem. Be prepared to share your solution with the class.

Here is an applet you may use if you choose to.

Summary

The unit price is the price of 1 thing—for example, the price of 1 ticket, 1 slice of pizza, or 1 kilogram of peaches.

If 4 movie tickets cost $28, then the unit price would be the cost per ticket. We can create a double number line to find the unit price.

Figure $\PageIndex{1}$: Double number line, 6 evenly spaced tick marks. Top line, cost in dollars. Beginning with the first tick mark, labels: 0, 7, 14, 21, 28, 35. Bottom line, number of tickets. Beginning with the first tick mark, labels: 0, 1, 2, 3, 4, 5. There is a circle around 7 and 1 and a second circle around 28 and 4.

This double number line shows that the cost for 1 ticket is $7. We can also find the unit price by dividing, $28\div 4=7$, or by multiplying, $28\cdot\frac{1}{4}=7$.

Glossary Entries

Definition: Double Number Line Diagram

A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.

Definition: Per

The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay $5 for each ticket. Buying 4 tickets would cost $20, because $4\cdot 5=20$

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.

Practice

Exercise $\PageIndex{4}$

In 2016, the cost of 2 ounces of pure gold was $2,640. Complete the double number line to show the cost for 1, 3, and 4 ounces of gold.

Figure $\PageIndex{3}$: A double number line with 6 evenly spaced tick marks. The top number line is labeled cost in dollars; the first tick mark is labeled 0, the third tick mark is labeled 2,640 and the sixth tick mark is labeled 6600. The bottom number line is labeled ounces of gold and starting with the first tick mark 0, 1, 2, 3, 4 and 5 are labeled.

Exercise $\PageIndex{5}$

The double number line shows that 4 pounds of tomatoes cost $14. Draw tick marks and write labels to show the prices of 1, 2, and 3 pounds of tomatoes.

Figure $\PageIndex{4}$: A double number line with one tick mark at the beginning of the line and one at the end. The top number line is labeled pounds of tomatoes with the first tick mark labeled 0 and the last tick mark labeled 4. The bottom number line is labeled cost in dollars with the first tick mark labeled 0 and the last tick mark labeled 14.

Exercise $\PageIndex{6}$

4 movie tickets cost $48. At this rate, what is the cost of:

5 movie tickets?
11 movie tickets?

Exercise $\PageIndex{7}$

Priya bought these items at the grocery store. Find each unit price.

12 eggs for $3. How much is the cost per egg?
3 pounds of peanuts for $7.50. How much is the cost per pound?
4 rolls of toilet paper for $2. How much is the cost per roll?
10 apples for $3.50. How much is the cost per apple?

Exercise $\PageIndex{8}$

Clare made a smoothie with 1 cup of yogurt, 3 tablespoons of peanut butter, 2 teaspoons of chocolate syrup, and 2 cups of crushed ice.

Kiran tried to double this recipe. He used 2 cups of yogurt, 6 tablespoons of peanut butter, 5 teaspoons of chocolate syrup, and 4 cups of crushed ice. He didn’t think it tasted right. Describe how the flavor of Kiran's recipe compares to Clare's recipe.
How should Kiran change the quantities that he used so that his smoothie tastes just like Clare's?

(From Unit 2.2.1)

Exercise $\PageIndex{9}$

A drama club is building a wooden stage in the shape of a trapezoidal prism. The height of the stage is 2 feet. Some measurements of the stage are shown here.

Figure $\PageIndex{5}$: Trapezoidal prism is shown on left. Trapezoid is shown on right. Trapezoidal prism has bottom base of 20 feet, top base of 10 feet, legs of 13 feet, and depth 2 feet. The trapezoid has top base 10 feet, legs 13 feet, and bottom base 20 feet. Two vertical dashed lines are drawn from the top two vertices to the bottom base of the trapezoid creating two right triangles and a rectangle. The rectangle has top and bottom base of 10 feet and vertical sides 12 feet. In the two triangles, horizontal legs 5 feet, vertical legs 12 feet, and hypotenuse 13 feet.

What is the area of all the faces of the stage, excluding the bottom? Show your reasoning. If you get stuck, consider drawing a net of the prism.

(From Unit 1.5.4)