In inverse variation, doubling one quantity halves the other. The product stays constant rather than the ratio, and the graph is the hyperbola you have already met.
Today's hook, More workers means less time. More speed means less travel time. Inverse variation is everywhere a fixed total is being shared out, and its graph is the reciprocal curve.
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Recall, your gut answer first
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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Recall, your gut answer first
+5 XP warm-up
If 4 people can paint a fence in 6 hours, how long would 8 people take? What quantity stayed the same while the people and the hours both changed?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Constant product, not constant ratio
Work through the core explanation before applying it.
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Constant product, not constant ratio
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"$y$ varies inversely with $x$" means $y = \dfrac{k}{x}$, equivalently $xy = k$. So the product is constant: 4 people times 6 hours is 24 person-hours, and so is 8 people times 3 hours. The graph is a hyperbola.
For inverse variation $k = xy$. Using $\frac{y}{x}$ by habit is the standard error when switching from direct variation.
The graph is the hyperbola
You met $y = \frac{k}{x}$ as the reciprocal function. Inverse variation is that curve doing a job, with the same asymptotes.
Halving and doubling
If $x$ doubles, $y$ halves. If $x$ triples, $y$ is a third. That is a quick sanity check on any answer.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
Inverse variation means $y = \frac{k}{x}$, equivalently $xy = k$.
$k$ is found by multiplying one known pair, since $k = xy$.
The graph is a hyperbola with asymptotes $x = 0$ and $y = 0$.
Direct variation has a constant ratio; inverse variation has a constant product.
Understand
Concepts
Why a constant product is what inverse variation means.
Why the graph is the reciprocal curve rather than a straight line.
Why doubling the input halves the output exactly.
Can do
Skills
Translate a description of inverse variation into $y = \frac{k}{x}$.
Find $k$ by multiplying a known pair and use the model to find other values.
Decide from a description or data whether a relationship is direct or inverse.
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Key terms
Inverse variationA relationship where one quantity rises as the other falls, with their product fixed. Like this: 4 painters take 6 hours, 8 painters take 3.
Constant productThe test for inverse variation: $xy$ is the same for every pair, and that value is $k$. Like this: $4 \times 6 = 24$ and $8 \times 3 = 24$.
Varies inverselyThe phrase signalling $y = \frac{k}{x}$, sometimes written "is inversely proportional to". Like this: "time varies inversely with speed" means $t = \frac{k}{v}$.
HyperbolaThe two-branch curve that inverse variation produces, the same shape as the reciprocal function. Like this: $t = \frac{24}{n}$ curves down steeply then flattens.
Practical domainThe part of the curve that makes sense in the situation, usually only the positive branch. Like this: a negative number of painters is meaningless, so only $n > 0$ is used.
Direct against inverseTwo different relationships: ratio constant for direct, product constant for inverse. Like this: pay against hours is direct; time against speed is inverse.
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Building the model and finding k
Work through the core explanation before applying it.
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Building the model and finding k
core concept
A description such as "the time taken varies inversely with the number of painters" becomes $t = \dfrac{k}{n}$, or equivalently $tn = k$.
Find $k$ by **multiplying** the known pair. With 4 painters taking 6 hours: $k = 4 \times 6 = 24$. The constant is 24 person-hours, the total work in the job.
The model is $t = \dfrac{24}{n}$. For 8 painters: $t = \dfrac{24}{8} = 3$ hours, which matches the intuition that doubling the workforce halves the time.
Multiply, do not divide. For direct variation $k = \frac{y}{x}$; for inverse variation $k = xy$. Carrying the division habit across is the single most common error in this topic.
Quick check: $y$ varies inversely with $x$, and $y = 5$ when $x = 4$. What is $k$?
"Varies inversely" means $y = \frac{k}{x}$, equivalently $xy = k$. Find $k$ by multiplying a known pair, not dividing. The constant often has a physical meaning, such as the total work in a job.
Pause, copy the translation to $y = \frac{k}{x}$, the painters example giving $k = 24$ person-hours, and the multiply-not-divide warning, into your book.
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Telling direct from inverse
core concept
We just saw how inverse variation is built. That raises a question: given a situation, how do you decide which kind of variation it is before writing anything? This card answers it → ask what happens to one quantity when the other doubles.
If doubling $x$ **doubles** $y$, it is direct variation and the ratio $\frac{y}{x}$ is constant. If doubling $x$ **halves** $y$, it is inverse variation and the product $xy$ is constant.
Pay against hours worked is direct: work twice as long, earn twice as much. Time against speed is inverse: travel twice as fast, arrive in half the time.
With a table, test both: divide each pair, then multiply each pair. Whichever gives a constant identifies the relationship, and that constant is $k$.
Some relationships are neither. If neither the ratio nor the product is constant, the relationship is something else, and forcing it into a variation model will give wrong answers.
Fill the blank: for inverse variation the quantity that stays constant is the of x and y.
Doubling the input doubles the output for direct variation and halves it for inverse. Direct has a constant ratio, inverse a constant product. Test a table both ways, and accept that some relationships are neither.
Pause, copy the doubling test, the two everyday examples, and the note that neither constant means neither model applies, into your book.
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Solving problems, and the practical domain
core concept
We just saw how to identify the type. That raises a question: once the model is built, what does a complete answer look like? This card answers it → find $k$, complete the model, substitute, and respect what the situation allows.
The routine is always the same: translate, find $k$ from the given pair, write the completed model, then substitute for whatever is asked.
Inverse models can be run backwards too. From $t = \frac{24}{n}$, asking how many painters finish in 2 hours gives $2 = \frac{24}{n}$, so $n = 12$.
Only part of the hyperbola is meaningful. A negative number of painters does not exist, so only the branch with $n > 0$ is used. State that restriction when the context calls for it.
Check by multiplying. Every answer pair should give the same product as the original. Here $12 \times 2 = 24$, matching $4 \times 6$. It is a one-line check that catches a divided-instead-of-multiplied error immediately.
Which is NOT true of inverse variation $y = \dfrac{k}{x}$?
Translate, find $k$ by multiplying, complete the model, then substitute. Inverse models run backwards just as easily. Only the physically meaningful branch of the hyperbola is used, and every answer pair should reproduce the same product.
Pause, copy the four-step routine, the backwards example giving 12 painters, and the multiply-to-check habit, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · BUILDING THE MODEL
The time to paint a fence varies inversely with the number of painters. Four painters take 6 hours. How long would 8 painters take?
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$t = \dfrac{k}{n}$, so $k = tn$
Inverse variation means a constant product.
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$k = 6 \times 4 = 24$ person-hours
Multiply the known pair.
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$t = \dfrac{24}{8} = 3$ hours
Doubling the painters halves the time, as expected.
PROBLEM 2 · WORKING BACKWARDS
For the same fence, how many painters are needed to finish in 2 hours?
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$t = \dfrac{24}{n}$, and $t = 2$
Use the completed model.
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$2 = \dfrac{24}{n}$, so $2n = 24$
Multiply both sides by $n$.
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$n = 12$ painters. Check: $12 \times 2 = 24$ ✓
The product matches, confirming the answer.
PROBLEM 3 · IDENTIFYING THE TYPE
Data: $x = 2, y = 18$; $x = 3, y = 12$; $x = 6, y = 6$. Is this direct or inverse variation?
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Ratios: $\frac{18}{2} = 9$, $\frac{12}{3} = 4$, not constant
Inverse variation with $k = 36$, so $y = \dfrac{36}{x}$
A constant product identifies it.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
+10 XP
If $y$ varies inversely with $x$ and $y = 8$ when $x = 3$, find $k$.
Using that model, find $y$ when $x = 6$.
Six machines fill an order in 10 hours. How long would 4 machines take?
Is a constant product the test for direct or inverse variation?
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Revisit the fence
Run the quick drill and copy the summary into your book.
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Revisit the fence
At the start you worked out how long 8 painters would take and named what stayed the same. Give that constant with its units, and explain why halving the time when the painters double is exactly what a constant product predicts.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 43 marks
Q1. The time taken for a journey varies inversely with the average speed. A trip takes 4 hours at 90 km/h. How long does it take at 120 km/h? (3 marks)
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ApplyBand 54 marks
Q2. For a fixed volume of gas, pressure varies inversely with volume. At a volume of 250 mL the pressure is 96 kPa. Find the volume at which the pressure is 60 kPa, and state the constant with its units. (4 marks)
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UnderstandBand 43 marks
Q3. Explain the difference between direct and inverse variation, giving the test for each and one everyday example of each. (3 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: $k = 24$. Practice 2: $y = 4$. Practice 3: $k = 60$ machine-hours, so 4 machines take 15 hours. Practice 4: inverse.
Q1 (3 marks): $t = \frac{k}{v}$, so $k = tv = 4 \times 90 = 360$ km [1]. $t = \frac{360}{120}$ [1]. $t = 3$ hours [1].
Q2 (4 marks): $P = \frac{k}{V}$, so $k = PV$ [1]. $k = 96 \times 250 = 24000$ kPa mL [1]. $60 = \frac{24000}{V}$ [1]. $V = 400$ mL [1].
Q3 (3 marks): Direct variation means $y = kx$, so the ratio $\frac{y}{x}$ is constant and doubling $x$ doubles $y$; for example pay against hours worked [1]. Inverse variation means $y = \frac{k}{x}$, so the product $xy$ is constant and doubling $x$ halves $y$; for example travel time against speed [1]. The tests are therefore constant ratio for direct and constant product for inverse [1].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Boss battle · Variation Sorter
earn bronze · silver · gold
Identify direct against inverse and solve variation problems both ways. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.