Topics
3. Ratios & Proportions
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United States of America
Grade 7

3.02 Proportional relationships in tables

Lesson
Practice

Introduction

Recall that a  ratio  uses division to compare two numbers. There are three ways to write a ratio of two numbers, and all three statements have the same meaning. Here are some examples, noting that the denominator in a ratio cannot be zero:

WordsNumbersAlgebra
\text{wins to losses}5 \text{ to } 4a \text{ to } b
\dfrac{\text{wins}}{\text{losses}}\dfrac{5}{4}\dfrac{a}{b}
\text{wins : losses}5:4a:b

In this lesson, we will extend our understanding of ratios by exploring the definition of proportion and how to identify proportional relationships from a table.

Proportions

A proportion is a statement of equality between two ratios. A proportion is true if both sides of the proportion simplify to be equivalent ratios. It can be represented in words or as an equation.

WordsAlgebra
a\text{ is to } b\text{ as } c\text{ is to } d\dfrac{a}{b} = \dfrac{c}{d} \text{ or } a:b = c:d
2\text{ is to } 5\text{ as } 4\text{ is to } 10\dfrac{2}{5} = \dfrac{4}{10} \text{ or } 2:5 = 4:10

Have you ever had to double a recipe to make enough for everyone? If you forget to double one of the ingredients, the recipe doesn't quite turn out. That's because the ingredients weren't in proportion. We use proportions a lot in everyday life.

Examples

Example 1

Is the following proportion true or false? 2:8 = 4:10

Worked Solution
Create a strategy

If the proportion is true, both sides of the proportion should simplify to be equivalent ratios. We can check for equivalence by multiplying or dividing one side to match the other.

Apply the idea

We know that 2 \times 2=4. Therefore, we can multiply both parts of the left hand side by 2.

\displaystyle 2:8\displaystyle =\displaystyle 2 \times 2 : 2 \times 8Multiply both parts of the ratio by 2
\displaystyle =\displaystyle 4 : 16Simplify
\displaystyle 4:10\displaystyle \neq\displaystyle 4:16

The ratios are not equivalent, therefore the proportion is false.

Reflect and check

We could also have done this problem in a similar way if we had converted the ratios to be fractions.

\displaystyle 2:8\displaystyle =\displaystyle \dfrac{2}{8}
\displaystyle 4:10\displaystyle =\displaystyle \dfrac{4}{10}
\displaystyle \dfrac{2}{8}\displaystyle =\displaystyle \dfrac{2\times2}{8\times2}Multiply both parts of the ratio by 2
\displaystyle =\displaystyle \dfrac{4}{16}Simplify
\displaystyle \dfrac{4}{10}\displaystyle \neq\displaystyle \dfrac{4}{16}
Idea summary

A proportion is a statement of equality between two ratios. It can be represented in words or as an equation.

WordsAlgebra
a\text{ is to } b\text{ as } c\text{ is to } d\dfrac{a}{b} = \dfrac{c}{d} \text{ or } a:b = c:d
2\text{ is to } 5\text{ as } 4\text{ is to } 10\dfrac{2}{5} = \dfrac{4}{10} \text{ or } 2:5 = 4:10

Ratio tables

We say that two quantities have a proportional relationship if the values always maintain the same ratio. When two quantities are proportional, we can use a ratio table to represent equivalent ratios, as well as determine unknown values.

For example, if a pie recipe calls for 1 tablespoon of brown sugar per 2 cups of flour, we could write this as a ratio: 2:1.

In a ratio table, we have:

Sugar2468
Flour1234

We can also use a ratio table to help us determine unknown values. For example, if we wanted to find out how much flour is needed when we use 12 tablespoons of brown sugar, we have the following:

Sugar246812
Flour1234⬚

As we know that the ratios in the table are proportional, we can determine the corresponding amount of flour to 12 tablespoons of brown sugar by equating ratios.

\displaystyle 6:3\displaystyle =\displaystyle 12 : ⬚Equivalent ratios
\displaystyle 6:3\displaystyle =\displaystyle 6\times 2 : 3 \times 2Multiply both parts of the ratio by 2
\displaystyle =\displaystyle 12:6

Therefore, for every 12 tablespoons of brown sugar, we can use 6 cups of flour.

Examples

Example 2

State whether or not the following ratio table represents a proportional relationship:

x01234
y0714628
Worked Solution
Create a strategy

If the table represents a proportional relationship, it will be made up entirely of equivalent ratios. We can check to see if each ratio is equal by first writing the ratios in fraction form.

Apply the idea

\dfrac{1}{7} = \dfrac{2}{14} = \dfrac{4}{28} \neq \dfrac{3}{6}

As we can see, all but one of the ratios are equal. \dfrac{3}{6} simplifies to \dfrac{1}{2}, which is not equal to the rest of the ratios which simplify to \dfrac{1}{7}.

Example 3

Complete the pattern of equivalent ratios by filling in the gaps in the following table:

2610a18
cb152127
Worked Solution
Create a strategy

Use equivalent ratios to determine the values of the unknown variables.

Apply the idea
\displaystyle 2:c\displaystyle =\displaystyle 10:15Equate the ratios
\displaystyle 10:15\displaystyle =\displaystyle 10 \div 5 : 15\div 5Divide both parts of the ratio by 5
\displaystyle =\displaystyle 2:3Simplify
\displaystyle c\displaystyle =\displaystyle 3
\displaystyle 6:b\displaystyle =\displaystyle 2:3Equate the ratios
\displaystyle 2:3\displaystyle =\displaystyle 2 \times 3 : 3 \times 3Multiply both parts of the ratio by 3
\displaystyle =\displaystyle 6:9Simplify
\displaystyle b\displaystyle =\displaystyle 9
\displaystyle a:21\displaystyle =\displaystyle 2:3Equate the ratios
\displaystyle 2:3\displaystyle =\displaystyle 2\times7 : 3 \times 7Multiply both parts of the ratio by 7
\displaystyle =\displaystyle 14:21Simplify
\displaystyle a\displaystyle =\displaystyle 14

The equivalent ratios are:

26101418
39152127
Reflect and check

We could have chosen any ratios to work with since they are all equivalent. We chose the ratios that are easiest to manipulate with multiplication and/or division.

How can we check if this table represents a proportional relationship? Since we know that each ratio is equivalent, the table must represent a proportional relationship.

Idea summary

We can use ratio tables to determine unknown values by multiplying or dividing. All of the values in the table will be equivalent ratios.

Outcomes

7.RP.A.2

Recognize and represent proportional relationships between quantities.

7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g. By testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

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