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Direct and Inverse Variation

Introduction

Direct and Inverse Variation helps you describe how two quantities change together. In Grades 9–10 algebra, these relationships connect ratios, rates, linear functions, rational functions, tables, graphs, equations, and real-world modeling.

A direct variation has the form $y = k x$ . The number $k$ is the constant of variation or constant of proportionality. For nonzero $x$ , the ratio $y / x$ is constant. In a positive real-world context, if $x$ doubles, $y$ doubles.

An inverse variation has the form $y = k / x$ , where $x \neq 0$ and $k \neq 0$ . Equivalently, $x y = k$ . The product of corresponding values is constant. In a positive real-world context, if $x$ doubles, $y$ is halved.

These two patterns are related but visually different: direct variation produces a straight line through the origin, while inverse variation produces a rectangular hyperbola.

The image compares graphs of direct and inverse proportionality. As you work through this course, focus on three questions: What stays constant? What equation fits the relationship? What does the graph show?

Learning Goals

By the end of this aiMOOC, you should be able to:

Direct variation: Identify, represent, and solve relationships of the form $y = k x$ .
Inverse variation: Identify, represent, and solve relationships of the form $y = k / x$ .
Constant of proportionality: Find and interpret $k$ from equations, tables, graphs, and contexts.
Mathematical modeling: Translate realistic situations into variation equations and check whether the model is reasonable.
Graph interpretation: Compare the shapes, intercepts, asymptotic behavior, and rates of change of direct and inverse variation graphs.
Problem solving: Use units, estimates, and substitutions to justify answers.

Direct Variation

Definition and Equation

Two variables vary directly when one is a constant multiple of the other. The standard equation is $y = k x$ . If you know one nonzero ordered pair $(x, y)$ , you can find the constant with $k = y / x$ .

For example, suppose a student earns 14 dollars per hour. If $h$ is the number of hours and $p$ is the pay in dollars, then $p = 14 h$ . The constant of variation is 14 dollars per hour.

A direct variation has these key features:

Origin: Its graph passes through $(0, 0)$ .
Slope: The slope is the constant of variation $k$ .
Constant ratio: For every nonzero input, $y / x = k$ .
Scaling: Multiplying $x$ by a factor multiplies $y$ by the same factor.

The graph above shows several direct-proportion lines with different constants. A larger positive constant produces a steeper line.

Direct Variation from a Table

Consider these ordered pairs:

x	y	y divided by x
2	10	5
3	15	5
5	25	5

Because the ratio is always 5, the relationship is direct variation with $k = 5$ , so $y = 5 x$ .

A useful test is to calculate $y / x$ for several rows. If the ratio is constant and the relationship is defined appropriately, you have evidence for direct variation.

Direct Variation from a Graph

A graph represents direct variation when it is a straight line through the origin. A line such as $y = 3 x + 2$ is linear, but it is not direct variation because its vertical intercept is 2 rather than 0.

If a direct-variation graph contains the point $(4, 12)$ , then $k = 12 / 4 = 3$ . The entire relationship is $y = 3 x$ . You do not need every point on the graph to determine the model; one nonzero point is enough when direct variation is already established.

Solving a Direct Variation Problem

Suppose $y$ varies directly with $x$ and $y = 28$ when $x = 4$ .

First find the constant: $28 = 4 k$ , so $k = 7$ .

Then write the model: $y = 7 x$ .

If $x = 9$ , substitute: $y = 7 \cdot 9 = 63$ .

Always check the meaning of the constant. Here, every increase of 1 unit in $x$ corresponds to an increase of 7 units in $y$ .

Inverse Variation

Definition and Equation

Two variables vary inversely when their product remains constant. The standard equations are $y = k / x$ and $x y = k$ . Since division by zero is undefined, $x = 0$ is not in the domain.

For example, imagine a fixed 240-kilometer trip. If $v$ is constant speed in kilometers per hour and $t$ is travel time in hours, then $t = 240 / v$ . At 60 kilometers per hour the trip takes 4 hours; at 80 kilometers per hour it takes 3 hours. In both cases, $v t = 240$ .

An inverse variation has these key features:

Reciprocal: One variable is proportional to the reciprocal of the other.
Constant product: Corresponding values satisfy $x y = k$ .
Hyperbola: Its graph is a rectangular hyperbola.
Axis restriction: When $k \neq 0$ , the graph does not cross either coordinate axis.
Scaling: In a positive context, multiplying one variable by a factor divides the other by that same factor.

This graph of $1 / x$ is a basic inverse-variation model. The axes act as asymptotes: the curve can get arbitrarily close to an axis without crossing it.

Inverse Variation from a Table

Consider these ordered pairs:

x	y	x times y
2	30	60
3	20	60
5	12	60

Because the product is always 60, the relationship is inverse variation with $k = 60$ , so $y = 60 / x$ .

When testing a table for inverse variation, multiply corresponding values. Do not rely only on the fact that one variable decreases while the other increases; many decreasing relationships are not inverse variations.

Inverse Variation from a Graph

For positive $k$ , the graph of $y = k / x$ has branches in Quadrants I and III. For negative $k$ , the branches lie in Quadrants II and IV.

The graph is not a straight line. Its rate of change is not constant: changes are steep near zero and become flatter farther from the axes. This is one reason you should not confuse inverse variation with a decreasing linear function.

Solving an Inverse Variation Problem

Suppose $y$ varies inversely with $x$ and $y = 18$ when $x = 4$ .

Find the constant: $k = x y = 4 \cdot 18 = 72$ .

Write the model: $y = 72 / x$ .

If $x = 12$ , then: $y = 72 / 12 = 6$ .

A quick reasonableness check works here: $x$ became three times as large, so $y$ should become one third as large. Eighteen divided by three is six.

Comparing Direct and Inverse Variation

Direct and inverse variation can be compared by the quantity that stays constant.

Feature	Direct variation	Inverse variation
Standard equation	$y = k x$	$y = k / x$
Constant relationship	$y / x = k$	$x y = k$
Typical graph	Straight line through the origin	Rectangular hyperbola
Positive scaling rule	Double x and y doubles	Double x and y halves
Zero input	Usually gives $y = 0$	Not allowed when written as $k / x$

A strong strategy is to ask whether a ratio stays constant or a product stays constant.

Recognizing Variation in Real Situations

Direct Variation Examples

Many familiar rates create direct variation when the rate is fixed. If apples cost 3 dollars per kilogram, cost varies directly with mass. If a car moves at a constant 80 kilometers per hour, distance traveled varies directly with time. If a circle's diameter changes, circumference varies directly with diameter because $C = π d$ .

The model is only as good as its assumptions. A taxi fare that includes a starting fee is not a direct variation between distance and total fare, because the graph does not pass through the origin.

Inverse Variation Examples

Inverse variation often appears when a fixed total is shared across changing groups or when one factor must compensate for another.

For a fixed distance, ideal travel time varies inversely with constant speed. For a fixed amount of work, completion time may be modeled as inversely proportional to the number of equally productive workers, provided coordination losses and other practical limits are ignored. For a rectangle with fixed area $A$ , its height satisfies $h = A / w$ , so height varies inversely with width.

Do not assume that every situation described with words such as "more" and "less" is inverse variation. Test the model mathematically.

Modeling with Units and Assumptions

The constant of variation has meaning and units.

In $p = 14 h$ , the constant 14 has units of dollars per hour. In $v t = 240$ , the inverse-variation constant has units of kilometers because kilometers per hour multiplied by hours gives kilometers.

A model should also state its assumptions. The inverse model $t = d / v$ assumes constant speed and ignores stopping time. A worker-time inverse model assumes equal productivity and perfect divisibility of work. These assumptions help you decide where the model is useful and where it breaks down.

A Reliable Problem-Solving Method

Use this process for variation problems:

Identify variables: Decide what quantities are changing and define symbols with units.
Choose a model: Test whether a constant ratio suggests direct variation or a constant product suggests inverse variation.
Find the constant: Use a known pair of values to calculate $k$ .
Write the equation: Use $y = k x$ or $y = k / x$ .
Substitute and solve: Insert the requested input and calculate the output.
Check reasonableness: Use scaling, units, and the context to see whether your answer makes sense.

Common Misconceptions

Misconception 1: Every straight line is a direct variation. A direct-variation line must pass through the origin. A nonzero vertical intercept means the relationship is linear but not direct variation.

Misconception 2: Every decreasing relationship is inverse variation. Inverse variation requires a constant product. A decreasing line, exponential decay, or another curve may decrease without satisfying $x y = k$ .

Misconception 3: Inverse variation means subtracting. It does not. Inverse variation uses reciprocals and multiplication. If one positive variable is multiplied by 4, the other is divided by 4.

Misconception 4: Zero works in every variation equation. Direct variation can include the origin, but $y = k / x$ is undefined at $x = 0$ .

Misconception 5: A mathematical model is automatically exact in real life. Models depend on assumptions. Real systems can include fixed costs, delays, friction, unequal productivity, measurement error, and other effects.

Worked Examples

Example: Direct Variation with a Unit Rate

A printer produces 36 pages in 3 minutes at a constant rate. Let $p$ be pages and $t$ be time in minutes.

The constant is $k = 36 / 3 = 12$ pages per minute. The model is $p = 12 t$ . In 8 minutes, the printer produces $p = 12 \cdot 8 = 96$ pages.

The result is reasonable because increasing the time from 3 minutes to 8 minutes should increase the number of pages in the same ratio when the printing rate stays constant.

Example: Inverse Variation with Fixed Work

Suppose 6 identical pumps can empty a tank in 10 hours under ideal conditions. If time $t$ varies inversely with the number of pumps $n$ , then $n t = k$ .

The constant is $k = 6 \cdot 10 = 60$ pump-hours. With 12 pumps, $t = 60 / 12 = 5$ hours.

The answer follows the inverse scaling rule: doubling the number of pumps halves the ideal completion time.

Example: Decide Whether Data Show Variation

Table A contains $(1, 4)$ , $(2, 8)$ , and $(5, 20)$ . The ratios are 4, 4, and 4, so this is direct variation with $k = 4$ .

Table B contains $(2, 24)$ , $(3, 16)$ , and $(6, 8)$ . The products are 48, 48, and 48, so this is inverse variation with $k = 48$ .

Table C contains $(1, 5)$ , $(2, 8)$ , and $(3, 11)$ . Neither the ratios nor the products are constant, so the table does not represent direct or inverse variation.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement identifies a direct variation? (The ratio of output to input stays constant) (!The product of input and output stays constant) (!The output always decreases) (!The graph must be curved)

Which statement identifies an inverse variation? (The product of corresponding values stays constant) (!The difference between corresponding values stays constant) (!The graph must pass through the origin) (!The output always equals the input)

What is true about the graph of a direct variation? (It is a straight line through the origin) (!It is always a horizontal line) (!It is a hyperbola) (!It can have any vertical intercept)

What happens in a positive direct variation when the input doubles? (The output doubles) (!The output halves) (!The output stays unchanged) (!The output becomes zero)

What happens in a positive inverse variation when the input doubles? (The output halves) (!The output doubles) (!The output increases by two) (!The output becomes zero)

Which calculation finds the constant in a direct variation from one data pair? (Divide the output by the input) (!Multiply the output by the input) (!Add the output and input) (!Subtract the input from the output)

Which calculation finds the constant in an inverse variation from one data pair? (Multiply the output by the input) (!Divide the output by the input) (!Add the output and input) (!Subtract the output from the input)

Why is a line with a nonzero vertical intercept not a direct variation? (It does not pass through the origin) (!It has a constant slope) (!It contains ordered pairs) (!It can be drawn on coordinates)

Which condition rules out zero as an input for an inverse variation? (Division by zero is undefined) (!Multiplication by zero is too large) (!Every inverse graph is linear) (!The constant must be zero)

Which check is most useful after solving a variation problem? (Compare the answer with units and scaling) (!Ignore the context) (!Replace every value with zero) (!Assume every relationship is exact)

Memory Game

Direct variation	Relationship with a constant output-to-input ratio
Inverse variation	Relationship with a constant product of paired values
Constant	Fixed value that connects the two variables
Origin	Point where both coordinate values are zero
Hyperbola	Curve associated with an inverse variation graph
Slope	Constant rate of change of a direct variation line

Drag and Drop

Match the correct terms.	Topic
Constant ratio	Direct variation test
Constant product	Inverse variation test
Line through the origin	Direct variation graph
Rectangular hyperbola	Inverse variation graph
Check units and assumptions	Model validation

...

Crossword Puzzle

Proportion	What idea describes two quantities linked by a constant multiplicative relationship?
Constant	What fixed value is usually represented by k in a variation equation?
Hyperbola	What curve is produced by a basic inverse variation?
Reciprocal	What number is obtained by reversing numerator and denominator?
Origin	What point must every direct variation line pass through?
Scaling	What idea describes multiplying one quantity by a factor and tracking the corresponding change?

LearningApps

Cloze Text

In a direct variation, the equation has the form

. The ratio of output to input is the constant of

. A direct variation graph passes through the

. In an inverse variation, corresponding values have a constant

. The equation can be written as y equals k divided by

. A basic inverse variation graph is a rectangular

. When the input doubles in a positive direct variation, the output

. When the input doubles in a positive inverse variation, the output is

. A strong model includes correct units and states important

. Testing ratios and products helps you distinguish the two types of

.

Open-Ended Tasks

Easy

Variation Photo Hunt: Find or create four images showing situations that might involve direct or inverse variation, label the variables, and explain your classification in one sentence per image.
Table Detective: Create two six-row tables, one showing direct variation and one showing inverse variation, then exchange them with a partner who must determine each constant.
Graph Sketch: Draw one direct-variation graph and one inverse-variation graph by hand, label key features, and write a short comparison of their shapes.
Variation Vocabulary Video: Record a one-minute explanation using the terms constant, ratio, product, direct variation, and inverse variation accurately.

Standard

Unit Rate Investigation: Collect prices for different quantities of one product, decide whether cost is approximately a direct variation, and explain any fixed fees or discounts that break the model.
Travel Model: Choose a fixed journey distance, calculate travel time for at least six constant speeds, graph the data, and explain why the model is inverse variation under ideal assumptions.
Interview about Proportional Reasoning: Interview a craftsperson, technician, cook, or other professional about a task involving scaling, then translate one example into a mathematical model and evaluate whether it is direct or inverse.
Variation Comparison Poster: Design a poster that compares equations, tables, graphs, scaling rules, units, and real-world examples for direct and inverse variation.

Advanced

Model Breakdown Experiment: Test a real situation that is often modeled proportionally, gather data, identify where the direct or inverse model stops fitting well, and explain why.
Spreadsheet Variation Lab: Build a spreadsheet that generates direct and inverse tables for adjustable constants, graph both relationships, and write observations about how changing the constant changes each graph.
Inverse Work Model Critique: Investigate whether adding workers to a real or simulated task reduces completion time inversely, collect timing data, and discuss coordination effects and limitations.
Teach the Concept: Produce a three-to-five-minute tutorial video that teaches learners how to distinguish direct, inverse, and neither relationships using one table, one graph, and one word problem.

Learning Assessment

Model Selection: Given several real contexts, choose direct variation, inverse variation, or neither and justify each choice using a constant ratio, constant product, or evidence that neither condition holds.
Data Interpretation: Analyze an unfamiliar table, determine whether a variation model fits, calculate the constant if appropriate, and explain what that constant means in context with units.
Graph Reasoning: Compare a line through the origin, a line with a nonzero intercept, and a hyperbola, then explain which can represent direct or inverse variation and why.
Transfer Problem: Create and solve a new problem in which a fixed total produces an inverse relationship, then state the assumptions that make the model reasonable.
Error Analysis: Correct a solution in which a learner tests a decreasing table by ratio instead of product, and explain a reliable method that prevents the same mistake.
Model Evaluation: Use real or simulated measurements to decide whether a proportional model is exact, approximate, or unsuitable, and support the conclusion with numerical evidence.

Evidence of Learning

Strong evidence of learning includes:

Knowledge evidence: You can state and interpret the equations for direct and inverse variation and explain what remains constant in each relationship.
Representation evidence: You can move accurately among words, tables, graphs, equations, and scaling statements.
Calculation evidence: You can find the constant of variation, solve for unknown values, and check calculations using ratio or product structure.
Reasoning evidence: You can distinguish direct variation, inverse variation, linear but non-proportional relationships, and other nonlinear patterns.
Communication evidence: You can explain the meaning of constants and variables using correct units and clear mathematical language.
Product evidence: You can create a graph, table, poster, spreadsheet, report, or video that demonstrates the concepts accurately.
Transfer evidence: You can apply variation ideas to a new context and state where real-world assumptions limit the mathematical model.

OERs on the Topic

The English Wikipedia article on proportionality in mathematics provides additional background on direct and inverse proportionality.

For further review, the embedded Khan Academy videos in this aiMOOC explain direct and inverse variation and how to recognize them from examples. The Wikimedia Commons graphs provide reusable visual representations for discussing shape, scaling, and proportionality.

Linked Learning Areas

Direct and Inverse Variation

Direct and inverse variation connects algebra with measurement, science, economics, technology, and data reasoning. Understanding these relationships prepares you for work with functions, rational expressions, similarity, rates, scaling, and mathematical models.