Inverse Variation Revision Notes for VCE SSCE General Mathematics

Inverse Variation

What is inverse variation?

Inverse variation describes a relationship between two variables where one variable increases as the other decreases. This is different from direct variation, where both variables change in the same direction.

For two variables $x$ and $y$ , if $y = \frac{k}{x}$ where $k$ is a positive constant, we say that " $y$ varies inversely as $x$ ". This can also be expressed as " $y$ is inversely proportional to $x$ ".

Notation: We write this relationship symbolically as:

$y \propto \frac{1}{x}$

The positive constant $k$ is called the constant of variation.

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Key characteristic: As $x$ increases, $y$ decreases, and vice versa. This opposite relationship is the defining feature of inverse variation.

For example, if $y = \frac{3}{x}$ , then $y \propto \frac{1}{x}$ and 3 is the constant of variation.

Understanding inverse variation through a real-world example

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Real-World Example: Bricklayers Building a Wall

Consider a builder who employs bricklayers to build a wall. If three bricklayers can complete the wall in 8 hours, what happens when six bricklayers are employed? The wall will be completed in half the time (4 hours). The more bricklayers employed, the shorter the time taken to complete the wall.

In this scenario:

Time taken ( $t$ ) decreases as the number of bricklayers ( $b$ ) increases
We say: $t$ varies inversely as $b$ , or $t$ is inversely proportional to $b$
We write: $t \propto \frac{1}{b}$

The graph of this relationship shows the characteristic curve of inverse variation:

Recognizing inverse variation from tables

When working with data in table form, inverse variation can be identified by checking if the product of corresponding values remains constant.

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Important property: If $y \propto \frac{1}{x}$ , then $xy = k$ for a positive constant $k$ .

This means the product is always constant, which is a useful way to check for inverse variation.

Example:

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In this table, $y \propto \frac{1}{x}$ and the constant of variation is 6. We can verify this:

When $x = \frac{1}{3}$ , $y = 18$ , so $xy = \frac{1}{3} \times 18 = :success[6]$
When $x = \frac{1}{2}$ , $y = 12$ , so $xy = \frac{1}{2} \times 12 = :success[6]$
When $x = 1$ , $y = 6$ , so $xy = 1 \times 6 = :success[6]$
When $x = 2$ , $y = 3$ , so $xy = 2 \times 3 = :success[6]$

Therefore, we can say that $y = \frac{6}{x}$ .

Graphical representation of inverse variation

When we plot the graph of $y$ against $x$ for inverse variation, we obtain a curved shape called a hyperbola. This curve shows a downward trend as $x$ increases.

However, if we plot the graph of $y$ against $\frac{1}{x}$ , we get a straight line. This line does not pass through the origin because $\frac{1}{x}$ is undefined when $x = 0$ .

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Important note: Not all decreasing trends represent inverse variation. To confirm inverse variation, you must verify that the product $xy$ remains constant.

Finding the constant of variation

To work with inverse variation problems, you often need to find the constant of variation $k$ . Here's the systematic approach:

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Method:

Write the variation expression: $y \propto \frac{1}{x}$
Rewrite as an equation: $y = \frac{k}{x}$
Substitute known values of $x$ and $y$
Solve for $k$
Check your answer with other values if available

Worked example: determining the constant from a table

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Worked Example: Determining the Constant from a Table

Question: Use the table of values to find the constant of variation, $k$ , and complete the table.

$y \propto \frac{1}{x}$

$x$	1	2	3	4
$y$	3		1	0.75	0.6

Solution:

Write the variation expression: $y \propto \frac{1}{x}$

Rewrite as an equation with constant of variation: $y = \frac{k}{x}$

Substitute known values. When $x = 1$ , $y = 3$ : $3 = \frac{k}{1}$

Solve for $k$ : $k = 3$

Write the equation: $y = \frac{3}{x}$

Check with another value. When $x = 3$ : $y = \frac{3}{3} = 1 \text{ ✓}$

Find the missing $y$ value. When $x = 2$ : $y = \frac{3}{2} = 1.5$

Find the missing $x$ value. When $y = 0.6$ :

\begin{aligned} 0.6 &= \frac{3}{x}\\ x &= \frac{3}{0.6}\\ x &= 5 \end{aligned}

Complete table:

$x$	1	2	3	4	5
$y$	3	1.5	1	0.75	0.6

Solving practical inverse variation problems

Many real-world situations involve inverse variation. Here's how to approach these problems systematically.

Worked example: cylinder problem

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Worked Example: Cylinder Problem

Question: For a cylinder of fixed volume, the height ( $h$ cm) is inversely proportional to the square of the radius ( $r$ cm).

If a cylinder of height 15 cm has a base radius of 4.2 cm, how high would a cylinder of equivalent volume be if its radius was 3.5 cm?

Solution:

Write the variation expression and equation: $h \propto \frac{1}{r^2}$

$h = \frac{k}{r^2}$

Substitute known values. When $h = 15$ , $r = 4.2$ : $15 = \frac{k}{(4.2)^2}$

$k = 15(4.2)^2$

$k = 264.6$

Write the equation: $h = \frac{264.6}{r^2}$

Find the height when radius is 3.5 cm: $h = \frac{264.6}{(3.5)^2}$

$h = 21.6$

Answer: A cylinder with radius 3.5 cm has a height of 21.6 cm.

Key properties of inverse variation

Understanding these properties will help you work confidently with inverse variation:

Equation form: If $y$ varies inversely as $x$ , then $y = \frac{k}{x}$ for some positive constant $k$
Proportionality notation: We write $y \propto \frac{1}{x}$
Constant product: If $y \propto \frac{1}{x}$ , then $x_1y_1 = x_2y_2 = k$ for any two pairs of values
Graphical representation: The graph of $y$ against $\frac{1}{x}$ is a straight line (not defined at the origin), and the gradient equals the constant of variation
Opposite direction: As one variable increases, the other decreases

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These properties provide multiple ways to identify and work with inverse variation. When solving problems, you can use whichever property is most convenient for the given situation.

Remember!

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Key Points to Remember:

Inverse variation means one variable increases while the other decreases, following the relationship $y = \frac{k}{x}$
The constant of variation ( $k$ ) stays the same throughout the relationship, and can be found by calculating $xy$ for any pair of values
The graph of $y$ against $x$ forms a hyperbola (curved shape), while the graph of $y$ against $\frac{1}{x}$ is a straight line
To solve inverse variation problems: write the equation $y = \frac{k}{x}$ , find $k$ using known values, then use the equation to find unknown values
Check for inverse variation by verifying that the product $xy$ remains constant for all data pairs

Inverse Variation (VCE SSCE General Mathematics): Revision Notes