Lesson 15Part-Part-Whole Ratios

Learning Goal

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

Learning Targets

I can create tape diagrams to help me reason about problems involving a ratio and a total amount.
I can solve problems when I know a ratio and a total amount.

Lesson Terms

tape diagram

Warm Up: True or False: Multiplying by a Unit Fraction

True or false?

$\frac{1}{5} \cdot 45 = \frac{45}{5}$
$\frac{1}{5} \cdot 20 = \frac{1}{4} \cdot 24$
$42 \cdot \frac{1}{6} = \frac{1}{6} \cdot 42$
$486 \cdot \frac{1}{12} = \frac{480}{12} + \frac{6}{12}$

Activity 1: Cubes of Paint

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

Problem 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.

What amount does each cube represent?
How many milliliters of maroon paint will there be?

Problem 2

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

Problem 3

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

Activity 2: 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.

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?
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?
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?

Are you ready for more?

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?

Activity 3: Invent Your Own Ratio Problem

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.
Create a visual display that includes:
- The new problem that you wrote, without the solution.
- Enough work space for someone to show a solution.
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.
When the solution to the problem you invented is being shared by another group, check their answer for accuracy.

Lesson 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 $4 : 5$ .

A tape diagram with equal parts. For "ducks" there are 4 equal parts. For "swans" there are 5 equal parts.

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.

A tape diagram with equal parts. For "ducks" there are 4 equal parts. The number "2" is labeled in each of the 4 parts. For "swans" there are 5 equal parts. The number "2" is labeled in each of the 5 equal parts. A brace including duck equal parts and swan equal parts is labeled 18.

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 $18 \div 9 = 2$ .

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