When One Quantity Goes Up and the Other Goes Down
Imagine a trip of exactly 120 miles. If you travel at an average speed of 30 miles per hour, the trip takes 4 hours. At 40 miles per hour, it takes 3 hours. At 60 miles per hour, it takes 2 hours.
\[ 30(4)=120, \qquad 40(3)=120, \qquad 60(2)=120. \]The speed and travel time are changing in opposite directions, but something stays the same: their product is always 120.
This is an example of inverse proportion, also called inverse variation.
What Is Inverse Proportion?
Two quantities are inversely proportional when their product remains constant.
If \(x\) and \(y\) vary inversely, then
\[ xy=k, \]where \(k\) is a nonzero constant.
Solving for \(y\) gives the equivalent form
\[ y=\frac{k}{x}, \qquad x\neq0. \]Unlike direct proportion, where the ratio \(\frac{y}{x}\) remains constant, inverse proportion is identified by the constant product
\[ xy=k. \]Inverse Proportion and Inverse Variation
The terms inverse proportion and inverse variation describe the same basic mathematical structure.
You may encounter statements such as:
- \(y\) is inversely proportional to \(x\);
- \(y\) varies inversely with \(x\);
- \(y\) is an inverse variation of \(x\); or
- \(y\propto\frac{1}{x}\).
The statement
\[ y\propto\frac{1}{x} \]means that there is some constant \(k\) such that
\[ y=\frac{k}{x}. \]The Product Stays Constant
Consider
\[ y=\frac{24}{x}. \]Here,
\[ k=24. \]Several corresponding values are:
| \(x\) | \(y\) | \(xy\) |
|---|---|---|
| 1 | 24 | 24 |
| 2 | 12 | 24 |
| 3 | 8 | 24 |
| 4 | 6 | 24 |
| 6 | 4 | 24 |
| 8 | 3 | 24 |
The individual values change, but
\[ xy=24 \]in every row.
The Inverse Scaling Property
In inverse proportion, multiplying one quantity by a factor divides the other quantity by that same factor.
Suppose
\[ xy=24. \]One possible pair is
\[ x=4, \qquad y=6. \]If \(x\) doubles from 4 to 8, \(y\) is divided by 2:
\[ 6\longrightarrow3. \]The product remains unchanged:
\[ 4(6)=24 \]and
\[ 8(3)=24. \]If \(x\) is multiplied by 3, \(y\) must be divided by 3. If \(x\) is divided by 5, \(y\) must be multiplied by 5.
This opposite scaling behavior is one of the clearest ways to recognize inverse proportion.
Direct and Inverse Proportion Compared
Direct and inverse proportion both involve a constant relationship, but the constant appears differently.
For direct proportion,
\[ y=kx \]and therefore
\[ \frac{y}{x}=k. \]The ratio remains constant.
For inverse proportion,
\[ y=\frac{k}{x} \]and therefore
\[ xy=k. \]The product remains constant.
| Feature | Direct Proportion | Inverse Proportion |
|---|---|---|
| Equation | \(y=kx\) | \(y=\frac{k}{x}\) |
| Constant relationship | \(\frac{y}{x}=k\) | \(xy=k\) |
| If \(x\) doubles | \(y\) doubles | \(y\) is halved |
| Typical graph | Straight line through origin | Hyperbola |
Example: Speed and Travel Time
Suppose a journey has a fixed distance of 180 miles.
Distance is related to speed and time by
\[ d=rt, \]where \(d\) is distance, \(r\) is speed, and \(t\) is time.
Since the distance is fixed at 180 miles,
\[ rt=180. \]Solving for time gives
\[ t=\frac{180}{r}. \]Thus, for this fixed journey, travel time is inversely proportional to speed.
At 30 miles per hour,
\[ t=\frac{180}{30}=6\text{ hours}. \]At 60 miles per hour,
\[ t=\frac{180}{60}=3\text{ hours}. \]Doubling the speed halves the travel time in this idealized constant-speed model.
Example: Workers and Time
In an idealized work model, suppose a job requires 48 worker-hours.
Let \(w\) represent the number of equally productive workers and \(t\) represent the number of hours required.
Then
\[ wt=48. \]Solving for time gives
\[ t=\frac{48}{w}. \]With 4 workers,
\[ t=\frac{48}{4}=12\text{ hours}. \]With 8 workers,
\[ t=\frac{48}{8}=6\text{ hours}. \]Doubling the number of workers halves the time, assuming the workers have equal productivity and can work without interfering with one another.
Real workplaces may not behave this perfectly, but the inverse model is useful when those assumptions are reasonable.
Example: Pressure and Volume
In physics and chemistry, inverse proportion appears in Boyle's law for an ideal gas when temperature and the amount of gas remain constant.
\[ PV=k, \]where \(P\) is pressure and \(V\) is volume.
Solving for pressure gives
\[ P=\frac{k}{V}. \]Under the stated conditions, increasing the volume decreases the pressure in an inverse relationship.
This is another example of the same mathematical structure:
\[ (\text{one quantity})(\text{other quantity}) = \text{constant}. \]Finding the Constant of Inverse Proportion
If one pair of nonzero corresponding values is known, the constant \(k\) can be found by multiplying them.
Suppose \(y\) varies inversely with \(x\), and
\[ x=6, \qquad y=10. \]Since
\[ xy=k, \]substitute the known values:
\[ k=6(10)=60. \]The inverse-variation equation is therefore
\[ y=\frac{60}{x}. \]Finding a Missing Value
Suppose \(y\) varies inversely with \(x\), and
\[ y=15 \]when
\[ x=4. \]First find the constant:
\[ k=xy=4(15)=60. \]Therefore,
\[ y=\frac{60}{x}. \]To find \(y\) when \(x=10\),
\[ y=\frac{60}{10}=6. \]Thus,
\[ \boxed{y=6}. \]Solving Inverse Proportion with Equal Products
If two pairs of values belong to the same inverse proportional relationship, then their products are equal.
\[ x_1y_1=x_2y_2. \]Suppose 6 workers can complete an idealized job in 8 hours. How long would the same job take 12 equally productive workers?
Write
\[ 6(8)=12t. \]Therefore,
\[ 48=12t, \]so
\[ t=4. \]Under the assumptions of the model, 12 workers would require 4 hours.
Recognizing Inverse Proportion in a Table
To test a table for inverse proportion, multiply each pair of corresponding values.
| \(x\) | \(y\) | \(xy\) |
|---|---|---|
| 2 | 30 | 60 |
| 3 | 20 | 60 |
| 5 | 12 | 60 |
| 10 | 6 | 60 |
Because every pair satisfies
\[ xy=60, \]the table represents inverse proportion with
\[ k=60. \]Its equation is
\[ y=\frac{60}{x}. \]When the Product Does Not Stay Constant
Consider this table:
| \(x\) | \(y\) | \(xy\) |
|---|---|---|
| 2 | 20 | 40 |
| 4 | 12 | 48 |
| 5 | 10 | 50 |
| 8 | 7 | 56 |
The value of \(xy\) changes from row to row.
Therefore, there is no single constant \(k\), and the relationship is not inversely proportional.
The Graph of Inverse Proportion
The equation
\[ y=\frac{k}{x} \]defines a rational function.
Its graph is not a straight line. Instead, it forms a curve called a hyperbola.
For example, consider
\[ y=\frac{12}{x}. \]Some points on the graph are
\[ (1,12),\quad (2,6),\quad (3,4),\quad (4,3),\quad (6,2),\quad (12,1). \]Negative values also satisfy the equation:
\[ (-1,-12),\quad (-2,-6),\quad (-3,-4). \]When \(k>0\), the two branches of the graph lie in Quadrants I and III.
Asymptotes
The graph of inverse proportion has two important boundary lines called asymptotes.
For
\[ y=\frac{k}{x}, \]the vertical asymptote is
\[ x=0. \]The function is undefined there because division by zero is impossible.
The horizontal asymptote is
\[ y=0. \]As the magnitude of \(x\) becomes very large, the magnitude of \(\frac{k}{x}\) becomes increasingly small:
\[ y\longrightarrow0. \]The graph approaches the axes but never reaches either axis when \(k\neq0\).
What Happens Near Zero?
Consider
\[ y=\frac{12}{x}. \]As positive values of \(x\) get closer to zero, the corresponding \(y\)-values become very large:
\[ x=1 \quad\Rightarrow\quad y=12, \] \[ x=0.1 \quad\Rightarrow\quad y=120, \] \[ x=0.01 \quad\Rightarrow\quad y=1200. \]But \(x\) can never actually equal zero.
This behavior explains why the graph rises sharply as it approaches the vertical asymptote.
What If \(k\) Is Negative?
An inverse proportional relationship can also have a negative constant.
For example,
\[ y=-\frac{8}{x}. \]Then
\[ xy=-8. \]Some points are
\[ (1,-8), \qquad (2,-4), \qquad (-1,8), \qquad (-2,4). \]When \(k<0\), the graph lies in Quadrants II and IV.
Many elementary applications use positive quantities such as time, speed, workers, pressure, and volume, but the algebraic model allows either sign.
Opposite Movement Is Not Enough
It is tempting to assume that whenever one quantity increases while the other decreases, the relationship must be inversely proportional.
That is not true.
Consider
\[ y=10-x. \]As \(x\) increases, \(y\) decreases. But the product
\[ xy=x(10-x) \]does not remain constant.
Therefore,
\[ y=10-x \]is not an inverse proportional relationship.
Inverse proportion requires a specific multiplicative structure:
\[ xy=k. \]Inverse Proportion Is Not Negative Direct Proportion
The equation
\[ y=-3x \]does not represent inverse proportion.
It represents direct proportion with
\[ k=-3. \]Its ratio remains constant:
\[ \frac{y}{x}=-3. \]By contrast, an inverse relationship such as
\[ y=\frac{12}{x} \]has a constant product:
\[ xy=12. \]A negative slope and an inverse relationship are different mathematical ideas.
The Reciprocal Connection
The equation
\[ y=\frac{k}{x} \]can also be written as
\[ y=k\left(\frac{1}{x}\right). \]This shows why inverse proportion is closely connected with reciprocals.
If \(x\) becomes twice as large, its reciprocal becomes half as large:
\[ \frac{1}{2x} = \frac12\left(\frac1x\right). \]If \(x\) becomes five times as large, its reciprocal becomes one-fifth as large.
This reciprocal behavior produces the characteristic inverse scaling of \(y\).
Domain and Range
For the general inverse proportional function
\[ y=\frac{k}{x}, \qquad k\neq0, \]\(x\) cannot equal zero:
\[ x\neq0. \]Likewise, \(y\) cannot equal zero. If
\[ \frac{k}{x}=0, \]a nonzero numerator \(k\) would somehow have to produce zero after division, which cannot occur.
Therefore, over the real numbers,
\[ y\neq0. \]In applied problems, additional restrictions may narrow the domain and range further. For example, numbers of workers and elapsed times are ordinarily positive.
Inverse Models Depend on Assumptions
Real situations do not automatically obey inverse proportion merely because the model looks plausible.
For a fixed-distance travel model, we assume that average speed adequately describes the trip. For a worker-time model, we may assume workers are equally productive and can divide the work efficiently. For Boyle's law, certain physical conditions must remain fixed.
Mathematics describes the consequences of the model:
\[ xy=k. \]Whether the model accurately represents reality depends on the assumptions behind it.
How to Recognize Inverse Proportion
A relationship is inversely proportional when its different representations show the same underlying structure.
- Equation: It can be written as \[ y=\frac{k}{x}. \]
- Product: Corresponding values satisfy \[ xy=k. \]
- Table: Multiplying each corresponding pair gives the same nonzero constant.
- Scaling: Multiplying one quantity by a factor divides the other by the same factor.
- Graph: The graph is a hyperbola approaching the axes as asymptotes.
These are several views of the same inverse relationship.
Common Errors to Avoid
Checking the Ratio Instead of the Product
For inverse proportion, the defining constant is
\[ xy=k, \]not
\[ \frac{y}{x}=k. \]Assuming Any Decreasing Relationship Is Inverse
A relationship must have a constant product. Simply decreasing as another quantity increases is not enough.
Confusing Inverse Variation with Negative Slope
An equation such as
\[ y=-2x \]is direct variation with a negative constant, not inverse variation.
Allowing \(x=0\)
The equation
\[ y=\frac{k}{x} \]is undefined when \(x=0\).
Ignoring the Assumptions of a Model
Real-world inverse relationships often depend on idealized conditions. A mathematically correct inverse equation may cease to model the situation accurately when those conditions fail.
Check Your Understanding
- If \[ y=\frac{18}{x}, \] what is the constant of inverse proportion?
- If \(y\) varies inversely with \(x\), and \(x=5\) when \(y=8\), find \(k\).
- Using the relationship from Question 2, find \(y\) when \(x=10\).
- Does the following table represent inverse proportion? \[ \begin{array}{c|c} x & y\\ \hline 2 & 24\\ 3 & 16\\ 4 & 12\\ 6 & 8 \end{array} \]
- If \(x\) triples in an inverse proportional relationship, what happens to \(y\)?
- Is \[ y=20-x \] an inverse proportional relationship? Explain.
- A fixed trip is 240 miles long. Write an equation expressing travel time \(t\) as an inverse function of average speed \(r\).
- If 8 equally productive workers require 15 hours to complete an idealized job, how long would 12 workers require?
Show Answers
- \[ k=18. \]
- \[ k=xy=5(8)=40. \]
- Since \[ y=\frac{40}{x}, \] then \[ y=\frac{40}{10}=4. \]
- Yes. Each pair has product \[ xy=48. \] Therefore, \[ y=\frac{48}{x}. \]
- \(y\) is divided by 3.
- No. Although \(y\) decreases as \(x\) increases, the product \[ x(20-x) \] is not constant.
- Since \[ rt=240, \] the equation is \[ t=\frac{240}{r}. \]
- The total work is \[ 8(15)=120 \] worker-hours. Therefore, \[ 12t=120, \] so \[ t=10\text{ hours}. \]
Key Ideas
- Inverse proportion means that the product of two corresponding quantities remains constant.
- An inverse proportional relationship can be written as \(xy=k\) or \(y=\frac{k}{x}\).
- Inverse proportion and inverse variation describe the same basic structure.
- If one quantity is multiplied by a factor, the other is divided by that same factor.
- If \(x\) doubles, \(y\) is halved; if \(x\) triples, \(y\) becomes one-third as large.
- The graph of \(y=\frac{k}{x}\) is a hyperbola rather than a straight line.
- The coordinate axes \(x=0\) and \(y=0\) are asymptotes when \(k\neq0\).
- A decreasing relationship is not necessarily inversely proportional.
- Inverse proportion is different from direct proportion with a negative constant.
- Real-world inverse models depend on assumptions that should be checked against the situation.