When Two Quantities Scale Together
Suppose apples cost $2 per pound. If you buy 1 pound, the cost is $2. If you buy 2 pounds, the cost is $4. If you buy 5 pounds, the cost is $10.
The number of pounds changes, and the total cost changes with it. More importantly, the two quantities change according to a fixed relationship:
\[ \frac{\text{cost}}{\text{pounds}}=2. \]The total cost is always twice the number of pounds.
This is an example of direct proportion, also called direct variation.
What Is Direct Proportion?
Two quantities are directly proportional when one quantity is a constant multiple of the other.
If \(x\) and \(y\) are directly proportional, their relationship can be written as
\[ y=kx, \]where \(k\) is the constant of proportionality.
For every nonzero value of \(x\),
\[ \frac{y}{x}=k. \]The individual values of \(x\) and \(y\) may change, but the ratio between them remains constant.
Direct Proportion and Direct Variation
The terms direct proportion and direct variation describe the same basic mathematical structure.
You may see statements such as:
- \(y\) is directly proportional to \(x\);
- \(y\) varies directly with \(x\);
- \(y\) is a direct variation of \(x\); or
- \(y\propto x\).
The proportionality symbol
\[ \propto \]is read as “is proportional to.”
Thus,
\[ y\propto x \]means that there is some constant \(k\) for which
\[ y=kx. \]A Constant Multiplier Connects the Quantities
Consider the equation
\[ y=4x. \]Here,
\[ k=4. \]Every value of \(y\) is four times the corresponding value of \(x\).
\[ \begin{array}{c|c} x & y\\ \hline 1 & 4\\ 2 & 8\\ 3 & 12\\ 5 & 20\\ 10 & 40 \end{array} \]Although the values change from row to row,
\[ \frac{4}{1} = \frac{8}{2} = \frac{12}{3} = \frac{20}{5} = \frac{40}{10} = 4. \]The constant ratio is what makes the relationship proportional.
The Scaling Property
Direct proportion has a useful scaling behavior. If one quantity is multiplied by a factor, the corresponding quantity is multiplied by the same factor.
Suppose
\[ y=3x. \]If
\[ x=4, \]then
\[ y=12. \]Now double \(x\):
\[ 4\longrightarrow8. \]The value of \(y\) also doubles:
\[ 12\longrightarrow24. \]Triple \(x\), and \(y\) triples. Multiply \(x\) by one-half, and \(y\) is also multiplied by one-half.
More generally, if
\[ y=kx \]and \(x\) is multiplied by some factor \(r\), then
\[ k(rx)=r(kx). \]Both quantities scale by the same factor while their ratio remains unchanged.
Example: Price and Quantity
Suppose oranges cost $1.50 per pound.
Let \(p\) represent the number of pounds and \(C\) represent the total cost.
The constant of proportionality is
\[ k=1.50\text{ dollars per pound}. \]Therefore,
\[ C=1.50p. \]If you buy 6 pounds,
\[ C=1.50(6)=9. \]The total cost is
\[ \$9. \]The equation works because there is no starting charge or fixed fee. The cost is determined entirely by multiplying the number of pounds by the constant price per pound.
Example: Distance at Constant Speed
Suppose a vehicle travels at a constant speed of 60 miles per hour.
Let \(t\) represent time in hours and \(d\) represent distance in miles.
Since the vehicle travels 60 miles for every hour,
\[ d=60t. \]The constant of proportionality is
\[ k=60\text{ miles per hour}. \]After 3 hours,
\[ d=60(3)=180. \]After 5 hours,
\[ d=60(5)=300. \]As long as the speed remains constant, distance is directly proportional to elapsed time.
Example: Circumference and Diameter
Direct proportion also appears inside mathematics itself.
The circumference \(C\) of a circle is related to its diameter \(d\) by
\[ C=\pi d. \]This has the form
\[ y=kx \]with
\[ k=\pi. \]Therefore, circumference is directly proportional to diameter.
If the diameter doubles, the circumference doubles. If the diameter triples, the circumference triples.
No matter which circle we choose,
\[ \frac{C}{d}=\pi. \]The constant ratio is exactly what direct proportion predicts.
Recognizing Direct Proportion in a Table
A table represents direct proportion when the ratio
\[ \frac{y}{x} \]remains constant for every nonzero pair.
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
| 2 | 5 | 2.5 |
| 4 | 10 | 2.5 |
| 8 | 20 | 2.5 |
| 12 | 30 | 2.5 |
Since the ratio is always
\[ 2.5, \]the relationship is directly proportional with
\[ k=2.5. \]Its equation is
\[ y=2.5x. \]The Graph of Direct Proportion
Every direct proportional relationship has an equation of the form
\[ y=kx. \]This equation represents a straight line.
It also has an important feature: the line must pass through the origin.
If
\[ x=0, \]then
\[ y=k(0)=0. \]Therefore, the graph always contains
\[ (0,0). \]A straight-line graph through the origin is one of the clearest visual signatures of direct proportion.
The Constant of Proportionality Is the Slope
The familiar slope-intercept form of a linear equation is
\[ y=mx+b. \]For a direct proportional relationship,
\[ y=kx. \]This can be written as
\[ y=kx+0. \]Comparing the two equations shows that
\[ m=k \]and
\[ b=0. \]Therefore, the constant of proportionality is also the slope of the graph.
The value of \(k\) tells us how much \(y\) changes for each one-unit change in \(x\).
Finding a Direct-Proportion Equation from a Point
Suppose a direct proportional relationship contains the point
\[ (6,24). \]Since
\[ k=\frac{y}{x}, \]substitute the coordinates:
\[ k=\frac{24}{6}=4. \]The equation is therefore
\[ y=4x. \]Because the relationship is known to be proportional, one nonzero point is enough to determine \(k\).
Finding a Missing Value
Suppose \(y\) varies directly with \(x\), and
\[ y=18 \]when
\[ x=6. \]First find the constant:
\[ k=\frac{18}{6}=3. \]So the relationship is
\[ y=3x. \]To find \(y\) when \(x=11\),
\[ y=3(11)=33. \]Therefore,
\[ \boxed{y=33}. \]Direct Proportion as an Equality of Ratios
Direct proportion can also be expressed through ordinary proportions.
If the ratio \(\frac{y}{x}\) remains constant, then two corresponding pairs satisfy
\[ \frac{y_1}{x_1} = \frac{y_2}{x_2}. \]For example, suppose 4 units of \(x\) correspond to 10 units of \(y\). If 14 units of \(x\) correspond to \(y\), then
\[ \frac{10}{4} = \frac{y}{14}. \]Cross multiply:
\[ 4y=140. \]Therefore,
\[ y=35. \]We could obtain the same result by first finding
\[ k=\frac{10}{4}=2.5 \]and then using
\[ y=2.5(14)=35. \]The proportion method and the constant-of-proportionality method describe the same relationship from different viewpoints.
Increasing Together Is Not Enough
Two quantities can both increase without being directly proportional.
Consider
\[ y=x^2. \]For positive \(x\), both \(x\) and \(y\) increase. But
\[ \frac{y}{x} = \frac{x^2}{x} = x, \]which changes as \(x\) changes.
Therefore,
\[ y=x^2 \]is not a direct proportional relationship.
Direct proportion requires more than quantities moving in the same general direction. It requires a constant multiplicative relationship.
A Line Can Be Linear without Being Proportional
Compare these equations:
\[ y=3x \]and
\[ y=3x+5. \]Both are linear. Both have slope 3.
But only
\[ y=3x \]represents direct proportion.
The equation
\[ y=3x+5 \]contains a fixed starting amount. When \(x=0\),
\[ y=5. \]Its graph therefore does not pass through the origin, and \(\frac{y}{x}\) is not constant.
Why Fixed Fees Break Direct Proportion
Fixed charges are a common reason that an everyday relationship may look proportional without actually being proportional.
Suppose a delivery service charges a $6 base fee plus $2 per mile.
If \(m\) is the number of miles and \(C\) is the total cost, then
\[ C=2m+6. \]The cost increases at a constant rate of $2 per mile, but the total cost is not directly proportional to distance because of the initial $6 fee.
At zero miles,
\[ C=6, \]rather than zero.
A constant rate of change does not by itself guarantee direct proportion. The relationship must also have no nonzero starting amount.
Direct Proportion in Unit Conversions
Many unit conversions are direct proportional relationships.
For example, there are 12 inches in one foot. If \(f\) represents feet and \(i\) represents inches, then
\[ i=12f. \]Thus,
\[ k=12. \]If the number of feet doubles, the equivalent number of inches doubles. If the number of feet is multiplied by 5, the number of inches is also multiplied by 5.
However, not every unit conversion is proportional. Conversions involving an offset, such as Celsius and Fahrenheit temperatures, require an added constant and therefore do not have the form \(y=kx\).
What If \(k\) Is Negative?
In algebra, the constant \(k\) can also be negative.
For example,
\[ y=-2x \]still has the form
\[ y=kx \]with
\[ k=-2. \]Its graph is a straight line through the origin, but the line slopes downward from left to right.
In many elementary applications of proportion, quantities such as price, distance, mass, and time are positive, so positive values of \(k\) are more common. The algebraic definition itself is broader.
How to Recognize Direct Proportion
A relationship is directly proportional when its representations tell the same mathematical story.
- Equation: It can be written as \[ y=kx. \]
- Ratio: For every nonzero corresponding pair, \[ \frac{y}{x}=k. \]
- Table: The ratio \(\frac{y}{x}\) remains constant.
- Graph: The points lie on a straight line through \((0,0)\).
- Scaling: Multiplying one quantity by a factor multiplies the corresponding quantity by the same factor.
These are not separate definitions. They are different ways of recognizing the same underlying proportional structure.
Common Errors to Avoid
Assuming Any Linear Relationship Is Proportional
A proportional line must pass through the origin. An equation such as
\[ y=5x+2 \]is linear but not proportional.
Checking Differences Instead of Ratios
Direct proportion is based on a constant multiplicative relationship.
The key quantity is
\[ \frac{y}{x}, \]not merely the difference between \(x\) and \(y\).
Forgetting What \(k\) Represents
In an applied problem, \(k\) usually has units and meaning. A value such as \(k=60\) may mean 60 miles per hour, 60 dollars per item, or something entirely different depending on the context.
Assuming Quantities That Increase Together Must Be Proportional
A relationship such as
\[ y=x^2 \]may increase as \(x\) increases, but it does not preserve a constant ratio.
Check Your Understanding
- Is \[ y=7x \] a direct proportional relationship? Identify \(k\).
- Is \[ y=7x+3 \] directly proportional? Explain.
- If \(y\) varies directly with \(x\), and \(y=30\) when \(x=12\), find \(k\).
- Using the relationship from Question 3, find \(y\) when \(x=20\).
- Does the following table represent direct proportion? \[ \begin{array}{c|c} x & y\\ \hline 2 & 6\\ 4 & 12\\ 7 & 21\\ 10 & 30 \end{array} \]
- A printer produces 150 pages in 3 minutes at a constant rate. Write a direct-proportion equation relating pages \(p\) to time \(t\) in minutes.
- Is the relationship \[ y=x^2 \] directly proportional? Why or why not?
Show Answers
- Yes. It has the form \[ y=kx \] with \[ k=7. \]
- No. The added constant \(3\) means the graph does not pass through the origin.
- \[ k=\frac{30}{12}=2.5. \]
- Since \[ y=2.5x, \] then \[ y=2.5(20)=50. \]
- Yes. Each nonzero pair has \[ \frac{y}{x}=3. \] Therefore, \[ y=3x. \]
- First find the rate: \[ k=\frac{150}{3}=50. \] Therefore, \[ p=50t. \]
- No. Since \[ \frac{y}{x} = \frac{x^2}{x} = x, \] the ratio changes with \(x\) rather than remaining constant.
Key Ideas
- Direct proportion means that one quantity is a constant multiple of another.
- Direct proportion and direct variation describe the same basic relationship.
- A direct proportional relationship has the form \(y=kx\).
- For \(x\neq0\), the ratio \(\frac{y}{x}\) remains equal to \(k\).
- If one quantity is multiplied by a factor, the corresponding quantity is multiplied by the same factor.
- The graph of a direct proportional relationship is a straight line through the origin.
- The constant of proportionality \(k\) is also the slope of that line.
- A relationship may be linear without being proportional if it has a nonzero starting value.
- Quantities increasing together does not by itself prove direct proportion.
- Equations, ratios, tables, graphs, and scaling behavior are different representations of the same proportional structure.