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

3.04 Constant of proportionality

Lesson
Practice

Introduction

In the past lessons, we have identified that two quantities are proportional when all of the ratios relating to the quantities are equivalent. They vary in such a way that there is a constant multiplier between the quantities. In other words, they always vary by the same constant.

Constant of proportionality

You may have noticed that when we look at proportional relationships represented in a table or in graphs, that a pattern emerges. Here is an example of a proportional relationship where the pattern between the top row and bottom row is \div 2.

A table of values consisting of x, 2, 4, 6, 8 in the top row, and y, 1, 2, 3, 4 in the bottom row. There is an arrow pointing from the top row to the bottom row that says divide by 2.

This pattern between variables is a constant multiplicative factor, also known as the constant of proportionality. The constant of proportionality is a constant, positive multiplier between two variables which is often represented with the variable k.

We know that dividing by 2 is the same as multiplying by \dfrac{1}{2}. Therefore, in the table above, we can say that the constant of proportionality, or k, is \dfrac{1}{2}. We can check our value for k using the equation and the given values in the table, for example, 3 = \dfrac{1}{2} \times 6. If the equation is true, we know our value for k is correct.

Determining k is very helpful as we can now determine the corresponding values for any given variable. If we have 20 in the top row of the table, we know that the corresponding value in the bottom row will be 10 because that is half of 20.

A relationship is proportional if there is a constant multiplier between the two variables, so a proportional relationship between x and y will look like y = kx, where k is a constant.

Examples

Example 1

For each table below, determine whether they show a proportional relationship:

a
x12345
y246810
Worked Solution
Create a strategy

Check that the constant of proportionality is the same between each pair of numbers.

Apply the idea

As you can see, we can multiply each of the numbers in the top row by 2 in order to get the corresponding number in the bottom row. This means the constant of proportionality is k=2.

Since the constant of proportionality is the same for each set of numbers, this table represents a proportional relationship.

b
xy
00
17
214
36
428
Worked Solution
Create a strategy

Check that the constant of proportionality is the same between each pair of numbers.

Apply the idea

As you can see, we can multiply each of the numbers in the left-hand column by 7 to get the corresponding number in the right-hand column, except for the row with 3 and 6. As you can see, we can multiply each of the numbers in the left hand column by 7 in order to get the corresponding number in the right hand column except for the row that has 3 and 6.

3\times7=21 \neq 6

Since the constant of proportionality is not the same for each set of numbers, this table does not represent a proportional relationship.

Example 2

A physiotherapist charges the same rate per patient at their clinic. Their income per number of patients seen in a week is shown in the table below:

a

Complete the table:

No. of patients seen in the week1225324251
Weekly income (dollars)13752310
Worked Solution
Create a strategy

Since the physiotherapist charges the same rate per patient, we know that this table represents a proportional relationship. Therfore, we can find the value of k and then use it to find the missing values in the table.

Apply the idea

To find k, we can find the relationship between pairs of numbers in the table.

\displaystyle 25 \times ⬚\displaystyle =\displaystyle 1375
\displaystyle 25 \times ⬚ \div 25\displaystyle =\displaystyle 1375 \div 25Divide both sides by 25
\displaystyle ⬚\displaystyle =\displaystyle 1375 \div 25Simplify
\displaystyle ⬚\displaystyle =\displaystyle 55Simplify

This value is our constant of proportionality, which means that k=55. We can use this to find the missing values in the table.

\displaystyle 12 \times 55\displaystyle =\displaystyle 660Multiply 12 by 55
\displaystyle 32 \times 55\displaystyle =\displaystyle 1760Multiply 32 by 55
\displaystyle 51 \times 55\displaystyle =\displaystyle 2805Multiply 51 by 55
No. of patients seen in the week1225324251
Weekly income (dollars)6601375176023102805
b

How much would be earned in a week where they treated 0 patients?

Worked Solution
Create a strategy

Multiply the number of patients by the constant of proportionality found in the above part.

Apply the idea
\displaystyle \text{Weekly income}\displaystyle =\displaystyle 0 \times 55Multiply 0 by 55
\displaystyle =\displaystyle \$0 Evaluate
c

How much does the physiotherapist charge per customer?

Worked Solution
Create a strategy

The bottom row of the table represents income. We can consider how income is being calculated.

Apply the idea

We know that income is calculated by multiplying the constant of proportionality, k=55, by the amount of patients seen per week. This means that the amount being charged per customer is \$55.

Idea summary

A relationship is proportional if there is a constant multiplier, called the constant of proportionality, between the two variables, so a proportional relationship between x and y will look like y = kx, where k is a constant.

Outcomes

7.RP.A.2

Recognize and represent proportional relationships between quantities.

7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

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