Inverse Variation
Inverse Proportion
Inversely Proportional
A relationship between two variables in
which the product is a constant.
When one variable increases the other decreases in proportion so
that the product is unchanged.
If b is inversely proportional to a, the equation
is of the form b = k/a (where k is a
constant).
Equation: y = 60/x
y is inversely proportional to x.
Doubling x causes y to halve. The product
of x and y is always 60.
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See
also
Direct variation, joint
variation, gravity
Worked Example
Problem: Suppose y varies inversely with x, and y = 8 when x = 5. Find y when x = 10.
Step 1: Write the inverse variation equation.
Step 2: Find the constant of variation k by substituting the known values x = 5 and y = 8.
k=x⋅y=5⋅8=40 Step 3: Write the specific equation using k = 40.
y=x40 Step 4: Substitute x = 10 to find the new value of y.
y=1040=4 Answer: When x = 10, y = 4. Notice that doubling x from 5 to 10 caused y to halve from 8 to 4, which is the hallmark of inverse variation.
Another Example
This example applies inverse variation to a real-world scenario (work problems) rather than abstract variables, and shows how to interpret the constant k in context.
Problem: A team of workers can finish a job in a certain number of days. With 4 workers, the job takes 15 days. How many days will it take with 12 workers, assuming each worker contributes equally?
Step 1: Recognize that the number of days (d) varies inversely with the number of workers (w). More workers means fewer days.
Step 2: Find k using the given information: 4 workers take 15 days.
k=w⋅d=4⋅15=60 Step 3: Write the equation and substitute w = 12.
d=1260=5 Answer: With 12 workers, the job takes 5 days. The constant k = 60 represents the total worker-days needed to complete the job.
Frequently Asked Questions
What is the difference between inverse variation and direct variation?
In direct variation, y = kx, so both variables increase or decrease together and their ratio y/x stays constant. In inverse variation, y = k/x, so one variable increases while the other decreases and their product xy stays constant. The graph of direct variation is a straight line through the origin, while inverse variation produces a hyperbola.
How do you tell if a table of values shows inverse variation?
Multiply each x-value by its corresponding y-value. If every product xy gives the same constant, the data represents inverse variation. For example, if your pairs are (2, 30), (5, 12), and (10, 6), every product equals 60, confirming inverse variation with k = 60.
What does the graph of inverse variation look like?
The graph is a hyperbola with two branches. When k is positive, the branches lie in the first and third quadrants. The curve approaches but never touches either axis — these axes are asymptotes. The graph never passes through the origin because x = 0 is undefined in y = k/x.
Inverse Variation vs. Direct Variation
| Inverse Variation | Direct Variation |
|---|
| Equation | y = k/x | y = kx |
| Constant relationship | Product xy = k is constant | Ratio y/x = k is constant |
| Behavior | As x increases, y decreases | As x increases, y increases |
| Graph shape | Hyperbola (two curved branches) | Straight line through the origin |
| Real-world example | Speed and travel time for a fixed distance | Distance and time at a fixed speed |
Why It Matters
Inverse variation appears throughout algebra, physics, and everyday reasoning. In physics, the gravitational force between objects varies inversely with the square of their distance, and pressure varies inversely with volume (Boyle's Law). You will also encounter inverse variation in work-rate problems and electrical circuits (Ohm's Law rearranged), making it a pattern worth recognizing quickly.
Common Mistakes
Mistake: Writing the equation as y = k − x or y = x/k instead of y = k/x.
Correction: Inverse variation means the product xy is constant, not that one variable is subtracted from or divided into the constant. Always place k in the numerator and the other variable in the denominator: y = k/x.
Mistake: Confusing inverse variation with a negative linear relationship (like y = −3x).
Correction: A negative slope means y decreases as x increases, but that is still direct (linear) variation with a negative constant. Inverse variation follows the equation y = k/x, which produces a curve, not a straight line. Check by computing xy: if the product is constant, it is inverse variation.
