6.02 Direct, inverse, and joint variation

A direct variation represents a proportional relationship between two quantities. This means that as one amount increases, the other amount increases at the same rate.

For example, if you earn \$18 per hour, your earnings are directly proportional to the number of hours worked because \text{earnings}=18 \times \text{hours worked}.

The constant of proportionality is the value that relates the two amounts. We can also refer to this as the constant of variation. In the example above, the constant would be 18.

A direct variation can be written in the form: y=kx, where k is the constant of proportionality, or constant of variation (and k \neq 0).

Once we solve the constant of proportionality, we can use it to answer other questions about this relationship.

A very common misconception is that two variables are directly proportional if one increases as the other increases. This is not the case. We can only say that two variables are directly proportional if, and only if, the ratio between the variables stays constant. In other words, both variables increase or decrease at a constant rate.

If we graph direct variation, we will see a linear graph (in other words, a straight line graph) that passes through the origin , (0,\,0).

The diagram shows a linear graph, where A is directly proportional to B.

The next picture explains why A and B are directly proportional. Firstly, we can see that this is a straight line that passes through the origin. Now let's look at the ratios between the A and B coordinates. Look at the point (1,\,1), marked by the black lines. If we write these points as a ratio of A to B, it would be written as 1:1. Now look at the point (2,\,2), marked by the blue lines. If we write these point as a ratio, we would write it as 2:2, which simplifies down to 1:1, which is the same as the first point.

The graph of all points describing a direct variation is a line passing through the origin.

Examples

Example 1

If I pay \$6 for 12 eggs and \$10 for 20 eggs, are these rates directly proportional?

Worked Solution

Create a strategy

Divide each amount paid by the number of eggs.

Apply the idea

\displaystyle \$6 \div 12	\displaystyle =	\displaystyle 50 c / \text{ egg}	Divide \$6 by 12
\displaystyle \$10 \div 20	\displaystyle =	\displaystyle 50 c / \text{ egg}	Divide \$10 by 20

Since both variables show a rate of 50 c per egg, the prices are directly proportional.

Video coming soon

Example 2

Consider the equation P=90t.

a

State the constant of proportionality.

Worked Solution

Create a strategy

Use the equations of the form y=kx, where k is the constant of proportionality.

Apply the idea

The constant of proportionality in P=90t is 90.

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b

Find the value of P when t=2.

Worked Solution

Create a strategy

Substitute the given value into the equation.

Apply the idea

\displaystyle P	\displaystyle =	\displaystyle 90 \times 2	Substitute t=2
	\displaystyle =	\displaystyle 180	Evaluate

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Example 3

Consider the values in each table. Which one of them could represent a directly proportional relationship between x and y?

A

B

C

D

Worked Solution

Create a strategy

Use the equations of the form y=kx, and substitute the value of x and y from the table.

Apply the idea

Use the equations of the form y=kx, where k is some positive constant.

\displaystyle 5 \times 1	\displaystyle =	\displaystyle 5	Mutiply 5 by 1
\displaystyle 5 \times 2	\displaystyle =	\displaystyle 10	Mutiply 5 by 2
\displaystyle 5 \times 3	\displaystyle =	\displaystyle 15	Mutiply 5 by 3
\displaystyle 5 \times 4	\displaystyle =	\displaystyle 20	Mutiply 5 by 4

The correct answer is option C.

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Example 4

Ivan paints 10 plates every 6 hours.

a

Complete the proportion table:

Plates painted	0	10	20		40
Hours worked		6	12	18

Worked Solution

Create a strategy

Remember that for every 6 extra hours there are 10 extra plates will be painted.

Apply the idea

Complete proportion table:

Plates painted	0	10	20	30	40
Hours worked	0	6	12	18	24

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b

Using the data from the table above, plot a graph.

Worked Solution

Create a strategy

Choose two values from the table that you can plot on the graph.

Apply the idea

5

10

15

20

\text{number of plates}

5

10

15

20

\text{hours}

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In Direct Variation as a Linear Function, we learned about direct proportion, where one amount increased at the same constant rate as the other amount increased. Now we are going to look an inverse proportion or inverse variation.

Inverse variation means that as one amount increases the other amount decreases. Mathematically, we write this as {y}\propto \dfrac{1}{x}. For example, speed and travel time are inversely proportional because the faster you go, the shorter your travel time.

We express these kinds of inversely proportional relationships generally in the form y=\dfrac{k}{x} where k is the constant of proportionality (or constant or variation) and x and y are any variables.

Just like identifying directly proportional relationships, when identifying inversely proportional relationships we need to find the constant of proportionality or the constant term that reflects the rate of change. The we can substitute any quantity we like into our equation.

We can see that the graph of a direct variation is linear, while the graph of an inverse variation is nonlinear.

Examples

Example 5

Consider the equation s=\dfrac{375}{t}.

a

State the constant of proportionality.

Worked Solution

Create a strategy

Use the equations of the form y=\dfrac{k}{x}, where k is the constant of proportionality.

Apply the idea

The constant of proportionality in s=\dfrac{375}{t} is 375.

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b

Find the value of s when t=6. Give your answer as an exact value.

Worked Solution

Create a strategy

Substitute the given value into the equation.

Apply the idea

\displaystyle s	\displaystyle =	\displaystyle \dfrac{375}{6}	Substitute t=6
	\displaystyle =	\displaystyle \dfrac{125}{2}	Simplify

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c

Find the value of s when t=12. Give your answer as an exact value.

Worked Solution

Create a strategy

Substitute the given value into the equation.

Apply the idea

\displaystyle s	\displaystyle =	\displaystyle \dfrac{375}{12}	Substitute t=12
	\displaystyle =	\displaystyle \dfrac{125}{4}	Simplify

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Example 6

Consider the values in each table. Which two of them could represent an inversely proportional relationship between x and y?

A

B

C

D

Worked Solution

Create a strategy

Remember that if x and y are inversely proportional, as x increases, y decreases.

Apply the idea

Options A and B have an increasing x and decreasing y.

So, the correct answers are options A and B.

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Example 7

Is the variation relating the distance between two locations on a map and the actual distance between the two locations an example of a direct variation or an inverse variation?

Worked Solution

Create a strategy

Think about whether the distance between two locations on the map increases or decreases when the actual distance between the two locations increases.

Apply the idea

The relationship between the distances on a map and the actual distances is a direct variation.

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So far, the examples discussed explored variation between two variables. It is possible to have situations involving more than two variables where one variable is directly proportional to the product or quotient of the other variables.

Consider the following table:

Does x vary according to the product of y and z? This would imply that x\propto yz or x=kyz for some constant k. Check this with the table below:

Immediately, it becomes clear that x=2yz, so the constant of proportionality is 2 and x does vary as the product of y and z varies.

For joint variation, one variable is directly proportional to the product or quotient of two or more other variables.

Examples