Lesson 11

Part-Part-Whole Ratios

Let’s look at situations where you can add the quantities in a ratio together.

11.1: True or False: Multiplying by a Unit Fraction

True or false?

$$\frac15 \boldcdot 45 = \frac{45}{5}$$

$$\frac15 \boldcdot 20 = \frac14 \boldcdot 24$$

$$42 \boldcdot \frac16 = \frac16 \boldcdot 42$$

$$486 \boldcdot \frac{1}{12} = \frac{480}{12}+\frac{6}{12}$$

11.2: Cubes of Paint

A recipe for maroon paint says, “Mix 5 ml of red paint with 3 ml of blue paint.”

1. Use snap cubes to represent the amounts of red and blue paint in the recipe. Then, draw a sketch of your snap-cube representation of the maroon paint.

1. What amount does each cube represent?
2. How many milliliters of maroon paint will there be?
1. Suppose each cube represents 2 ml. How much of each color paint is there?

Red: _______ ml

Blue: _______ ml

Maroon: _______ ml

2. Suppose each cube represents 5 ml. How much of each color paint is there?

Red: _______ ml

Blue: _______ ml

Maroon: _______ ml

1. Suppose you need 80 ml of maroon paint. How much red and blue paint would you mix? Be prepared to explain your reasoning.

Red: _______ ml

Blue: _______ ml

Maroon: 80 ml

2. If the original recipe is for one batch of maroon paint, how many batches are in 80 ml of maroon paint?

11.3: Sneakers, Chicken, and Fruit Juice

Solve each of the following problems and show your thinking. If you get stuck, consider drawing a tape diagram to represent the situation.

1. The ratio of students wearing sneakers to those wearing boots is 5 to 6. If there are 33 students in the class, and all of them are wearing either sneakers or boots, how many of them are wearing sneakers?
2. A recipe for chicken marinade says, “Mix 3 parts oil with 2 parts soy sauce and 1 part orange juice.” If you need 42 cups of marinade in all, how much of each ingredient should you use?
3. Elena makes fruit punch by mixing 4 parts cranberry juice to 3 parts apple juice to 2 parts grape juice. If one batch of fruit punch includes 30 cups of apple juice, how large is this batch of fruit punch?

Using the recipe from earlier, how much fruit punch can you make if you have 50 cups of cranberry juice, 40 cups of apple juice, and 30 cups of grape juice?

11.4: Invent Your Own Ratio Problem

1. Invent another ratio problem that can be solved with a tape diagram and solve it. If you get stuck, consider looking back at the problems you solved in the earlier activity.
2. Create a visual display that includes:

• The new problem that you wrote, without the solution.
• Enough work space for someone to show a solution.
3. Trade your display with another group, and solve each other’s problem. Include a tape diagram as part of your solution. Be prepared to share the solution with the class.

4. When the solution to the problem you invented is being shared by another group, check their answer for accuracy.

Summary

A tape diagram is another way to represent a ratio. All the parts of the diagram that are the same size have the same value.

For example, this tape diagram represents the ratio of ducks to swans in a pond, which is $$4:5$$.

The first tape represents the number of ducks. It has 4 parts.

The second tape represents the number of swans. It has 5 parts.

There are 9 parts in all, because $$4+5=9$$.

Suppose we know there are 18 of these birds in the pond, and we want to know how many are ducks.

The 9 equal parts on the diagram need to represent 18 birds in all. This means that each part of the tape diagram represents 2 birds, because $$18\div9 = 2$$.

There are 4 parts of the tape representing ducks, and $$4 \boldcdot 2=8$$, so there are 8 ducks in the pond.

Glossary Entries

• 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.

• 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 $$4 \boldcdot 5 = 20$$.

• same rate

We use the words same rate to describe two situations that have equivalent ratios.

For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.

• table

A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.

For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.

pet tail length (inches)
dog 22
cat 12
mouse 2
• tape diagram

A tape diagram is a group of rectangles put together to represent a relationship between quantities.

For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.

If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.