Inversely proportional describes two quantities whose product never changes: y = kx, or xy = k, for a fixed nonzero number k. When one quantity doubles, the other is cut in half. On a 120-mile trip, speed and time are inversely proportional: 60 mph takes 2 hours and 30 mph takes 4 hours.
For every point on an inverse-proportion curve, the rectangle under it has the same area: xy = k. Doubling x halves y.
Formal definition and notation
A variable y is inversely proportional to x (also, y varies inversely with x) if there is a constant k ≠ 0 such that y = kx for all x ≠ 0 in the relationship; equivalently xy = k. The graph for positive x is a curve (a branch of a hyperbola) that never touches either axis. More generally, y varies inversely with the nth power of x when y = kxn.
Symbol
Read it as
Meaning
y ∝ 1x
y is proportional to one over x
shorthand for y = kx
k
k
the constant of variation, equal to the product xy
Key facts
Test with multiplication: xy is the same for every pair.
Multiplying x by a number divides y by that number: triple x and y becomes one third as large.
The graph is a curve, not a line, and it never reaches 0 on either axis.
Not every decreasing relationship is inverse: y = 10 − x goes down, but xy is not constant.
Everyday examples: speed and travel time for a fixed distance, workers and days for a fixed job, the number of people and each share of a fixed prize.
Inverse proportion
Inverse proportion
y = kx, so x1y1 = x2y2
k = xy
the constant, found by multiplying any one pair
x1y1 = x2y2
the equation to solve when one value is missing
To test a table, multiply x by y in every row. Inversely proportional means every product is the same.
Inversely Proportional examples, step by step
Example 1: Find k, then predict
Warm-up
y is inversely proportional to x, and y = 8 when x = 3. Find y when x = 6.
Directly proportional: y = kx, constant ratio, both grow together, line through the origin. Inversely proportional: y = kx, constant product, one grows as the other shrinks, curved graph.
A line with negative slope, like y = 12 − 2x, also decreases, but it changes by subtracting a fixed amount and crosses the axes. An inverse curve never touches the axes.
Quick check: 3 questions
Question 1 of 3
Which table shows y inversely proportional to x?
Show answer and explanation
Answer: B. In B every product is 12: 1 × 12, 2 × 6, 4 × 3. A is direct (constant ratio). C and D decrease, but their products change.
Question 2 of 3
At 40 mph a trip takes 3 hours. How long does the same trip take at 60 mph?
Show answer and explanation
Answer: D. Distance is fixed at 40 × 3 = 120 miles, and 120 ÷ 60 = 2 hours. 4.5 sets up a direct proportion, and 1 and 2.5 subtract instead of dividing.
Question 3 of 3
y varies inversely with x, and y = 5 when x = 9. What is y when x = 15?
Show answer and explanation
Answer: A.k = 9 × 5 = 45, so y = 4515 = 3. 8.3 uses a direct proportion, and 11 and −1 add or subtract the change in x.
How do you know if something is inversely proportional?
Multiply the two quantities in each pair. If the product is always the same number, they are inversely proportional. In words, ask whether doubling one would halve the other. Speed and time for a fixed distance pass this test: 30 mph for 4 hours and 60 mph for 2 hours both make 120 miles.
What does the graph of an inverse proportion look like?
For positive values it is a curve that drops quickly and then flattens out, getting closer and closer to both axes without touching them. It is not a straight line. The curve y = 12x passes through (1, 12), (2, 6), (3, 4) and (6, 2).
Is inversely proportional the same as negatively correlated?
No. Negative correlation only means one variable tends to go down as the other goes up, often loosely. Inversely proportional is an exact rule, xy = k, so doubling one variable exactly halves the other. Many decreasing relationships, such as y = 50 − 3x, are not inverse proportions.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Inversely Proportional.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/inversely-proportional/
Flashcard
TermInversely Proportionaly = kx
DefinitionThe product xy stays the same, so doubling one halves the other. 6 workers take 10 days; 12 workers take 5 days.
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In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7Proportional RelationshipA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio yx is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8, Algebra 1