The concept of variation

Variation explores the way two or more variables interact with each other.

Exploration

To start, consider the equation xy=36, with x and y as positive variables.

Since 36 is a constant, any increase in x will cause a decrease in y, so that their product remains at 36. Some possible solutions for this equation are x=18, y=2 and x=2, y=18, but the concept of variation is not just about finding the value of y for a given value of x, or vice-versa. Variation is about understanding the nature of the change in y with a change in x.

There are different types of variation. In this lesson, direct and inverse variation will be explored.

Variation

Variation deals with the way two or more variables interact with each other and describes how a change in one variable results in a change in the other variable(s).

Direct variation

Direct variation is when a change in one variable leads to a directly proportional change in the other variable.

Let's say there are two variables, x and y. If x is directly proportional to y, then an increase in x, will lead to a proportional increase in y. In a similar way, a decrease in x, will lead to a proportional decrease in y. This direct variation relationship can be written as:

{y}\propto{x}

where the symbol \propto means 'is directly proportional to'.

As another example, the statement:

"Earnings, E, are directly proportional to the number of hours, H, worked."

could be written as:

{E}\propto{H}

For the purposes of calculation, a proportionality statement can be turned into an equation using a constant of proportionality (or the constant of variation).

Direct variation

If {y}\propto{x}, then

y = kx

where k is the constant of proportionality.

To solve a direct variation problem, the key step is to find the constant of proportionality, k.

Identifying direct variation

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.

Worked example

Example 1

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

Think: \$6 / 12 eggs = 50c / egg

\$10 / 20 eggs = 50c / egg

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

Reflect: When paying for items that have a set cost, will the relationship between the amount of that item and total payment always be a direct variation?

Graphs that show direct variation

A graph representing direct variation, will always be a straight line that passes through the origin \left(0, 0\right). In other words, its vertical and horizontal intercepts will always be zero. The gradient of the line will be equal to the constant of variation.

The diagram below show a linear graph, where variable 2 is directly proportional to variable 1.

Practice question

question 1

Remember

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

Worked examples

Example 2

If the amount of petrol a car uses is directly proportional to the distance it travels, and it uses 12 L/100 km, how much petrol does the car use when it travels 350 km?

Think: 350 km is 3.5 times bigger than 100, so we also need to multiply 12 by 3.5 to keep the variables in direct proportion.

Do: 3.5 \times 12 = 42

The car uses 42 L of petrol over 350 km.

Reflect: Would this problem have been easier if we had written an equation to find the constant of proportionality first?

Example 3

The cost of repairing a bicycle is directly proportional to the amount of time spent working on it. If it takes 3 hours to complete a repair job that costs \$255, calculate:

The cost of a repair that takes 2.5 hours.
The length of time it takes to complete a repair job that costs \$357.

Think: We will use C for the cost of repairs and T for the time taken. Then we can find the constant of proportionality.

Do:

Since C is directly proportional to T, we can write,

C \propto T

Convert to an equation, using a constant of proportionality:

C = k T

To find the constant of proportionality k, substitute known values for C and T.

C = k T

255 = k \times 3
(Substitute C = 255 and T = 3)

255 = 3 k

\frac{255}{3} = \frac{3 k}{3}
(Divide both sides by 3)

85 = k

k = 85

So, the constant of proportionality is 85. The equation for this problem can now be written as: C = 85 T
If the repair job takes 2.5 hours,

C = 85 T

C = 85 \times 2.5
(Substituting T = 2.5)

C = \$212.50
If a repair job costs \$357.

C = 85 T

357 = 85 T
(Substituting C = 357)

\frac{357}{85} = \frac{85 T}{85}

4.2 = T

T = 4.2 hours

Direct variation doesn't need to be linear. Another example of direct variation is the relation given by y=3x^2 for x\ge 0. Note that y varies directly with the square of x, so that when x^2 increases (or decreases) then so does y. In this example, think of the graph of y=3x^2 for x\ge 0 as one half of a parabola with the vertex at \left(0,0\right). To show the direct variation of y with x^2, an alternative approach is to graph y against x^2. The "x" axis becomes the "x^2" axis, and the graph becomes a straight line with gradient 3 passing through the origin. That is, the constant of proportionality is 3.

Practice questions

Question 2

Question 3

Question 4

question 5

Question 6

Question 7

Question 8

Question 9

9.03 Direct variation

Outcomes

U2.AoS3.1

U2.AoS3.4

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