We say that two quantities have a proportional relationship if the ratio between quantities is constant. This ratio, called the constant of proportionality, can be any non-zero number. When two quantities are proportional, we can use a ratio table to represent equivalent ratios, as well as determine unknown values.

If a car can drive 51 miles on 3 gallons of gas, we could write this as a ratio: 51:3.

In a ratio table, we have:

A table showing gas (gallon), 1,3,4,5 for the first row and distance (mi),17,51,68,85 for the second row. An arrow labeled 'x17' is on 1 pointing to 17 and on 5 pointing to 85. 3 and 57 are highlighted.

This relationship is proportional because the ratio of miles driven to gas is constant:

\dfrac{17}{1} = \dfrac{51}{3} = \dfrac{68}{4} = \dfrac{85}{5}

Since our first ratio has a denominator of 1, we know 17 is the unit rate, or rate of change. We can also use a ratio table or unit rate to help us determine unknown values. For example, if we wanted to find out how far we could drive with 11 gallons of gas, we have the following:

Gas (gallon)	1	3	4	5	11
Distance (mi)	17	51	68	85	⬚

As we know that the ratios in the table are proportional, we can determine the corresponding distance to 11 gallons of gas by equating ratios.

\displaystyle 17:1	\displaystyle =	\displaystyle ⬚ : 11	Equivalent ratios
\displaystyle 17:1	\displaystyle =	\displaystyle 17 \cdot 11 : 1 \cdot 11	Multiply both parts of the ratio by 11
	\displaystyle =	\displaystyle 187:11

Therefore, for every 11 gallons of gas, we can drive 187 miles.

The constant of proportionality, also called the rate of change, can also be negative:

x	1	2	3	4	10
y	-5	-10	-15	-20	-50

\dfrac{-5}{1} = \dfrac{-10}{2} = \dfrac{-15}{3} = \dfrac{-20}{4}

The unit rate is -5 which means as x increases by 1, we know y decreases by 5.

Examples

Example 1

Determine whether each table of values represents a proportional relationship, and explain your reasoning.

x	2	6	10	14	18
y	1	3	5	7	9

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, \dfrac{y}{x}.

Apply the idea

\displaystyle \frac{1}{2}	\displaystyle =	\displaystyle \frac{1}{2}
\displaystyle \frac{3}{6}	\displaystyle =	\displaystyle \frac{1}{2}
\displaystyle \frac{5}{10}	\displaystyle =	\displaystyle \frac{1}{2}
\displaystyle \frac{7}{14}	\displaystyle =	\displaystyle \frac{1}{2}
\displaystyle \frac{9}{18}	\displaystyle =	\displaystyle \frac{1}{2}

As we can see, all of the ratios are equal.

The table of values represent a proportional relationship because the ratio \dfrac{1}{2} is constant.

Reflect and check

\dfrac{1}{2} is called the constant of proportionality.

x	0	8	16	27	32
y	0	1	2	3	4

Worked Solution

Create a strategy

Apply the idea

\displaystyle \frac{1}{8}	\displaystyle =	\displaystyle \frac{1}{8}
\displaystyle \frac{2}{16}	\displaystyle =	\displaystyle \frac{1}{8}
\displaystyle \frac{3}{27}	\displaystyle =	\displaystyle \frac{1}{9}
\displaystyle \frac{4}{32}	\displaystyle =	\displaystyle \frac{1}{8}

The table of values does not represent a proportional relationship because one of the the ratios is \dfrac{1}{9}, while the rest are \dfrac{1}{8}.

x	1	4	6	8	12
y	3.2	12.8	19.2	25.6	38.4

Worked Solution

Create a strategy

Apply the idea

\displaystyle \frac{3.2}{1}	\displaystyle =	\displaystyle 3.2
\displaystyle \frac{12.8}{4}	\displaystyle =	\displaystyle 3.2
\displaystyle \frac{19.2}{6}	\displaystyle =	\displaystyle 3.2
\displaystyle \frac{25.6}{8}	\displaystyle =	\displaystyle 3.2
\displaystyle \frac{38.4}{12}	\displaystyle =	\displaystyle 3.2

The table of values represent a proportional relationship because the ratio is constant 3.2.

Example 2

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

4	6
8
	18
16	24
20

Worked Solution

Create a strategy

Since we are told the ratios of each row are equivalent, each value in a row must be multiplied by the same number to get values for a new row.

Apply the idea

Since we need to multiply 4 by 2 to get 8, we can find the first missing value by creating an equivalent ratio:

\displaystyle 4:6	\displaystyle =	\displaystyle 8:⬚	Equate the ratios
\displaystyle 4:6	\displaystyle =	\displaystyle 4 \cdot 2 : 6 \cdot 2	Multiply both parts of the ratio by 2
	\displaystyle =	\displaystyle 8:12

To create a ratio equivalent to 4:6 in the form x:18, we must multiply by 3 since 6 \cdot 3 = 18.

\displaystyle 4:6	\displaystyle =	\displaystyle ⬚:18	Equate the ratios
\displaystyle 4:6	\displaystyle =	\displaystyle 4 \cdot 3 : 6 \cdot 3	Multiply both parts of the ratio by 3
	\displaystyle =	\displaystyle 12:18

To create a ratio equivalent to 4:6 in the form 20:x, we must multiply by 5 since 4 \cdot 5 = 20.

\displaystyle 4:6	\displaystyle =	\displaystyle 20:⬚	Equate the ratios
\displaystyle 4:6	\displaystyle =	\displaystyle 2\cdot 5 : 6 \cdot 5	Multiply both parts of the ratio by 5
	\displaystyle =	\displaystyle 20:30

The equivalent ratios are:

4	6
8	12
12	18
16	24
20	30

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.

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.

-7	-2
	-4
-28	-8
	-16
-112	-32

Worked Solution

Create a strategy

Since we are told the ratios of each row are equivalent, each value in a row must be multiplied by the same number to get values for a new row.

Apply the idea

In order to create a ratio equivalent to -7:-2 in the form ⬚:-4, we must multiply by 2 since -2 \cdot 2 = -4.

\displaystyle -7:-2	\displaystyle =	\displaystyle ⬚:-4	Equate the ratios
\displaystyle -7:-2	\displaystyle =	\displaystyle -7 \cdot 2 : -2 \cdot 2	Multiply both parts of the ratio by 2
	\displaystyle =	\displaystyle -14:-4

To create an equivalent ratio in the form ⬚:-16, we must multiply by 8 since -2 \cdot 8 = -16.

\displaystyle -7:-2	\displaystyle =	\displaystyle ⬚:-16	Equate the ratios
\displaystyle -7:-2	\displaystyle =	\displaystyle -7 \cdot 8 : -2 \cdot 8	Multiply both parts of the ratio by 8
	\displaystyle =	\displaystyle -56:-16

The equivalent ratios are:

-7	-2
-14	-4
-28	-8
-56	-16
-112	-32

-4
	3
-36
108	27
-324	-81

Worked Solution

Create a strategy

Since we are told the ratios of each row are equivalent, each value in a row must be multiplied by the same number to get values for a new row.

Apply the idea

In this table, we don't know the values in the first row so we will need to start with a ratio that has both values completed.

Let's use -324:-81. In order to create a ratio equivalent to -324:-81 in the form -4:⬚, we must divide by 81 since -324 \div -4 = 81.

\displaystyle -324:-81	\displaystyle =	\displaystyle -4:⬚	Equate the ratios
\displaystyle -324:-81	\displaystyle =	\displaystyle -324 \div 81 : -81 \div 81	Divide both parts of the ratio by 81
	\displaystyle =	\displaystyle -4:-1

To create a ratio equivalent to 108:27 in the form ⬚:3, we must divide by 9 since 27 \div 3 = 9.

\displaystyle 108:27	\displaystyle =	\displaystyle ⬚:3	Equate the ratios
\displaystyle 108:27	\displaystyle =	\displaystyle 108 \div 9 : 27 \div 9	Divide both parts of the ratio by 9
	\displaystyle =	\displaystyle 12:3

To create a ratio equivalent to -324:-81 in the form -36:⬚, we must divide by 9 since -324 \div 9 = -36.

\displaystyle -324:-81	\displaystyle =	\displaystyle -36:⬚	Equate the ratios
\displaystyle -324:-81	\displaystyle =	\displaystyle -324 \div 9 : -81 \div 9	Divide both parts of the ratio by 9
	\displaystyle =	\displaystyle -36:-9

The equivalent ratios are:

-4	-1
12	3
-36	-9
108	27
-324	-81

Example 3

A potter requires 4 kilograms of clay to make 18 tea cups. How many kilograms of clay are needed to make 54 tea cups?

Worked Solution

Create a strategy

Use equivalent ratios to determine how many kilograms of clay are needed to make 54 tea cups.

Apply the idea

We can set up a ratio table to better understand the relationship between the quantities. Let's use c to represent the weight of the clay needed for 54 tea cups.

Clay	4	c
Tea cup	18	54

We can use equivalent ratios to determine c. In order to create a ratio equivalent to 4:18 in the form c:54, we must multiply both parts by 3 since 18 \cdot 3 = 54.

\displaystyle 4:18	\displaystyle =	\displaystyle c:54	Equate the ratios
\displaystyle 4:18	\displaystyle =	\displaystyle 4 \cdot 3 : 18 \cdot 3	Multiply both parts of the ratio by 3
	\displaystyle =	\displaystyle 12:54

The potter needs 12 kilograms of clay to make 54 tea cups.

Idea summary

A proportional relationship is a collection of equivalent ratios.

We can use ratio tables or equivalent ratios to determine unknown values. If a relationship is proportional, all of the values in the table will represent equivalent ratios.

The constant of proportionality can be used to solve for missing values in a proportional relationship as well. It tells us the rate necessary to go from one quantity to another.

Outcomes

7.CE.2

The student will solve problems, including those in context, involving proportional relationships.

7.CE.2a

Given a proportional relationship between two quantities, create and use a ratio table to determine missing values.

7.CE.2c

Apply proportional reasoning to solve problems in context, including converting units of measurement, when given the conversion factor.

2.01 Proportional relationships and ratio tables

Ideas

Proportional relationships and ratio tables

Examples

Example 1

Example 2

Example 3

Outcomes

7.CE.2

7.CE.2a

7.CE.2c

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