Inverse Variation Revision Notes for HSC SSCE Mathematics Standard

Inverse Variation

What is inverse variation?

Inverse variation (also called inverse proportion) describes a relationship where as one quantity increases, the other decreases proportionally. The two quantities are related but move in opposite ways.

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Real-World Connection

Think about painting a house. The time needed to complete the job depends inversely on the number of painters available. If you have more painters, the job takes less time. If you have fewer painters, the job takes more time.

The inverse variation formula

When two variables have an inverse relationship, we can express this mathematically using the formula:

$y = \frac{k}{x}$

Where:

$y$ and $x$ are the two variables that are inversely related
$k$ is the constant of variation (a fixed number that doesn't change)

We say that " $y$ is inversely proportional to $x$ ".

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Key Feature of Inverse Variation

The product of the two variables always equals the constant: $xy = k$

This means that $k$ stays the same no matter what values $x$ and $y$ take, as long as they follow the inverse relationship.

Solving inverse variation problems

Follow these three steps to solve any inverse variation problem:

Step 1: Write the equation

Set up the inverse variation equation relating the two variables: $y = \frac{k}{x}$

Remember that $k$ is the constant of variation that stays the same throughout the problem.

Step 2: Find the constant of variation

Use the given values to calculate $k$ by substituting the known values of $x$ and $y$ into the equation, then solve for $k$ .

Step 3: Solve the problem

Write the equation using the value of $k$ you found in Step 2, then substitute the new given value to find the unknown quantity.

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Problem-Solving Tip

Always check that your answer makes sense! If one variable increases, the other should decrease. If both go up or both go down, you may have made an error.

Worked examples

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Worked Example: Reception Centre Hiring Costs

Problem: The cost per person ( $c$ ) to hire a reception centre is inversely proportional to the number of people attending ( $n$ ). When there are $50$ people, the cost per person is $36.

a) What is the cost per person when there are 20 people attending?

Solution:

Cost is inversely proportional to the number of people.

Use the formula $y = \frac{k}{x}$ by replacing $y$ with $c$ and $x$ with $n$ :

$c = \frac{k}{n}$

Substitute $36$ for $c$ and $50$ for $n$ :

$36 = \frac{k}{50}$

Evaluate to find $k$ :

$k = 1800$

Write the formula using the constant of variation:

$c = \frac{1800}{n}$

Substitute $20$ for $n$ :

$c = \frac{1800}{20}$

$c = 90$

Answer: The cost is $90 per person.

b) How many people are required for the cost per person to be $25?

Solution:

Write the formula using the constant of variation:

$c = \frac{1800}{n}$

Substitute $25$ for $c$ :

$25 = \frac{1800}{n}$

Solve for $n$ :

$n = \frac{1800}{25}$

$n = 72$

Answer: 72 people are required.

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Worked Example: Ferry Waiting Times

Problem: The number of people ( $n$ ) waiting at Circular Quay for a ferry is inversely proportional to the time ( $t$ , in minutes) until the next ferry arrives. When there are $64$ people waiting, the ferry will arrive in $3\frac{1}{2}$ minutes.

a) Find the constant of variation.

Solution:

The number of people is inversely proportional to the time.

Use the formula $y = \frac{k}{x}$ by replacing $y$ with $n$ and $x$ with $t$ :

$n = \frac{k}{t}$

Substitute $64$ for $n$ and $3.5$ for $t$ :

$64 = \frac{k}{3.5}$

Evaluate:

$k = 224$

Answer: The constant of variation is 224.

b) Write the inverse variation equation.

Solution:

Substitute $224$ for $k$ in the equation:

$n = \frac{224}{t}$

c) How many people are waiting at Circular Quay 4 minutes before the ferry arrives?

Solution:

Write the inverse variation equation:

$n = \frac{224}{t}$

Substitute $4$ for $t$ :

$n = \frac{224}{4}$

$n = 56$

Answer: 56 people are waiting.

d) How long before the ferry arrives if there are 28 people waiting?

Solution:

Write the inverse variation equation:

$n = \frac{224}{t}$

Substitute $28$ for $n$ :

$28 = \frac{224}{t}$

Solve for $t$ :

$t = \frac{224}{28}$

$t = 8$

Answer: The ferry will arrive in 8 minutes.

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Key Points to Remember:

Inverse variation means as one quantity increases, the other decreases proportionally
The formula is $y = \frac{k}{x}$ , where $k$ is the constant of variation
The constant $k$ always equals the product of the two variables: $k = xy$
To solve problems: (1) write the equation, (2) find k using given values, (3) substitute to solve for the unknown
Real-world examples include: more workers means less time, more people sharing costs means lower cost per person, shorter waiting time means more people waiting

Inverse Variation (HSC SSCE Mathematics Standard): Revision Notes