The Number That Stays the Same
In a proportional relationship, the quantities may change, but one important number remains constant.
Suppose notebooks cost $3 each. One notebook costs $3, two notebooks cost $6, five notebooks cost $15, and ten notebooks cost $30.
The totals change, but the cost for each notebook does not:
\[ \frac{\$3}{1}=3, \qquad \frac{\$6}{2}=3, \qquad \frac{\$15}{5}=3, \qquad \frac{\$30}{10}=3. \]That unchanging value is called the constant of proportionality.
The Constant \(k\)
The constant of proportionality is commonly represented by the letter \(k\).
If two quantities \(x\) and \(y\) are proportional, then their relationship can be written as
\[ y=kx. \]Here:
- \(x\) is the input or independent quantity;
- \(y\) is the corresponding output or dependent quantity; and
- \(k\) is the constant of proportionality.
If \(x\neq0\), dividing both sides by \(x\) gives
\[ k=\frac{y}{x}. \]This equation tells us how to find the constant: divide the corresponding \(y\)-value by the \(x\)-value.
The Constant as a Unit Rate
In many real-world situations, the constant of proportionality is also a familiar unit rate.
Suppose a worker earns $72 for 6 hours of work.
\[ k=\frac{72}{6}=12. \]The constant of proportionality is
\[ 12\text{ dollars per hour}. \]The proportional equation is therefore
\[ y=12x, \]where \(x\) is the number of hours worked and \(y\) is the amount earned.
The constant \(k=12\) is not merely an abstract number. It has meaning: each additional hour corresponds to $12 of earnings.
Finding \(k\) from a Pair of Values
If you know one nonzero pair of corresponding values in a proportional relationship, you can find \(k\).
Suppose
\[ x=4 \qquad\text{and}\qquad y=18. \]Since
\[ k=\frac{y}{x}, \]substitute the values:
\[ k=\frac{18}{4}=\frac92=4.5. \]The relationship can therefore be written as
\[ y=4.5x. \]Once \(k\) is known, any corresponding value can be generated from the same equation.
Finding the Constant in a Table
A table can reveal whether a relationship is proportional. Divide each \(y\)-value by its corresponding \(x\)-value and see whether the result remains constant.
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
| 1 | 4 | 4 |
| 2 | 8 | 4 |
| 5 | 20 | 4 |
| 9 | 36 | 4 |
Because
\[ \frac{y}{x}=4 \]for every nonzero pair, the table represents a proportional relationship with
\[ k=4. \]Its equation is
\[ y=4x. \]When the Ratio Does Not Stay Constant
Now consider a different table.
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
| 1 | 5 | 5 |
| 2 | 8 | 4 |
| 3 | 11 | \(\frac{11}{3}\) |
| 4 | 14 | 3.5 |
The ratio \(\frac{y}{x}\) changes from row to row. There is no single constant \(k\) satisfying
\[ y=kx \]for every pair.
Therefore, this relationship is not proportional.
Recognizing the Constant in an Equation
When a proportional relationship is already written in the form
\[ y=kx, \]the constant of proportionality is simply the coefficient multiplying \(x\).
For example, in
\[ y=7x, \]the constant is
\[ k=7. \]In
\[ y=\frac35x, \]the constant is
\[ k=\frac35. \]And in
\[ y=0.25x, \]the constant is
\[ k=0.25. \]Why Proportional Relationships Pass Through the Origin
Every equation of the form
\[ y=kx \]has an important feature. If
\[ x=0, \]then
\[ y=k(0)=0. \]Therefore, every direct proportional relationship contains the point
\[ (0,0). \]On a coordinate graph, this point is called the origin.
This gives us an important visual clue: the graph of a direct proportional relationship is a straight line that passes through the origin.
The Constant of Proportionality as Slope
The equation
\[ y=kx \]is also a linear equation.
Recall the slope-intercept form of a line:
\[ y=mx+b. \]A proportional relationship has no added constant term, so
\[ b=0. \]Therefore,
\[ y=kx \]can be viewed as
\[ y=mx+0. \]This means
\[ k=m. \]The constant of proportionality is the slope of the line.
Reading the Constant from Change
Suppose a proportional relationship contains the points
\[ (2,6) \qquad\text{and}\qquad (5,15). \]Using the ratio \(\frac{y}{x}\),
\[ \frac62=3 \]and
\[ \frac{15}{5}=3. \]Therefore,
\[ k=3. \]We can also calculate the slope between the two points:
\[ m=\frac{15-6}{5-2} = \frac93 = 3. \]Both methods reveal the same constant because, in a direct proportional relationship, the constant of proportionality and the slope are the same number.
What Does \(k\) Mean?
The numerical value of \(k\) matters, but its meaning depends on the quantities being compared.
Suppose
\[ d=60t, \]where \(d\) is distance in miles and \(t\) is time in hours.
Then
\[ k=60 \]means
\[ 60\text{ miles per hour}. \]If instead
\[ C=2.75p, \]where \(C\) is cost in dollars and \(p\) is the number of pounds purchased, then
\[ k=2.75 \]means
\[ \$2.75\text{ per pound}. \]Always interpret \(k\) in terms of the quantities and units in the situation.
Constants Can Be Fractions or Decimals
A constant of proportionality does not have to be a whole number.
Suppose
\[ y=\frac12x. \]Then
\[ k=\frac12. \]Every \(y\)-value is one-half of its corresponding \(x\)-value.
For example,
\[ x=10 \]gives
\[ y=\frac12(10)=5. \]Similarly, in
\[ y=1.6x, \]the constant is
\[ k=1.6. \]Fractions and decimals work in exactly the same way as whole-number constants.
Can \(k\) Be Negative?
In pure mathematics, yes. A proportional equation can have a negative constant:
\[ y=-2x. \]Here,
\[ k=-2. \]As \(x\) increases, \(y\) decreases in a consistent linear relationship.
In many everyday contexts, however, the quantities may be restricted to positive values. A negative price per pound, for example, would usually make no sense.
The algebra permits negative constants, but the interpretation depends on the situation being modeled.
Linear Does Not Always Mean Proportional
Every direct proportional relationship is linear, but not every linear relationship is proportional.
Compare
\[ y=4x \]with
\[ y=4x+7. \]The first equation is proportional because it has the form
\[ y=kx. \]It also passes through the origin.
The second equation is linear, but it contains an additional constant \(7\). When \(x=0\),
\[ y=7, \]so its graph does not pass through the origin.
Therefore,
\[ y=4x+7 \]is linear but not proportional.
One Relationship, Several Representations
A proportional relationship can be represented in several ways.
Suppose
\[ k=5. \]As an equation:
\[ y=5x. \]As ratios:
\[ \frac{5}{1} = \frac{10}{2} = \frac{20}{4}. \]As ordered pairs:
\[ (1,5),\quad(2,10),\quad(4,20). \]As a graph, these points lie on a straight line through the origin with slope 5.
These are not different relationships. They are different ways of showing the same proportional structure.
Common Errors to Avoid
Reversing the Ratio
If the relationship is defined as \(y\) compared with \(x\), then
\[ k=\frac{y}{x}, \]not
\[ \frac{x}{y}. \]Reversing the ratio gives the reciprocal and changes the meaning.
Using Only One Row without Checking
A single pair of values can produce a ratio, but several pairs are needed to confirm that the relationship remains proportional.
Assuming Every Straight Line Is Proportional
A direct proportional graph must be linear and pass through the origin.
Ignoring Units
If \(k\) represents dollars per pound, miles per hour, or another rate, include that interpretation. The units explain what the constant actually means.
Check Your Understanding
- If \(y=8x\), what is the constant of proportionality?
- If \(x=6\) and \(y=27\), find \(k\).
- Does the table below represent a proportional relationship? \[ \begin{array}{c|c} x & y\\ \hline 2 & 7\\ 4 & 14\\ 6 & 21\\ 10 & 35 \end{array} \]
- Is \(y=5x+2\) a proportional relationship? Explain.
- A machine produces 84 parts in 7 hours at a constant rate. Find the constant of proportionality and interpret its meaning.
- A proportional graph contains the point \((8,20)\). What is its slope?
Show Answers
- \[ k=8. \]
- \[ k=\frac{27}{6}=4.5. \]
- Yes. Each ratio gives \[ \frac{y}{x}=3.5. \] Therefore, \[ k=3.5. \]
- No. The equation contains an added constant term and does not pass through the origin.
- \[ k=\frac{84}{7}=12. \] The machine produces \[ 12\text{ parts per hour}. \]
- Since a proportional graph passes through \((0,0)\), \[ m=\frac{20-0}{8-0} = \frac{20}{8} = 2.5. \] Therefore, \[ k=m=2.5. \]
Key Ideas
- The constant of proportionality is the unchanging ratio between corresponding quantities.
- A direct proportional relationship can be written as \(y=kx\).
- For \(x\neq0\), the constant can be found using \(k=\frac{y}{x}\).
- In many applications, \(k\) is a unit rate.
- A proportional table has the same value of \(\frac{y}{x}\) for every nonzero pair.
- The graph of \(y=kx\) is a straight line through the origin.
- For a direct proportional relationship, the constant of proportionality is also the slope.
- Not every linear relationship is proportional.
- The meaning and units of \(k\) depend on the quantities being compared.