Proportion Models Real Relationships
In mathematics, many situations are described by how two variables change together.
A proportional relationship is a particularly useful type of relationship because it has a simple algebraic form, a recognizable graph, and predictable scaling.
- In the previous article about formula manipulation, we touched upon direct and inverse proportions.
- In the give article, we will go a bit more into detail and more practicalities of those concepts.
In a relationship, one variable is often the independent variable (the input you choose or control) and the other is the dependent variable (the output that changes as a result).
If the price of fuel depends on how many liters you buy, then liters is the independent variable and price is the dependent variable.
Proportionality constant
A non-zero constant, usually called $k$, that links two proportional variables (also called the constant of variation).
Direct Proportion Means A Constant Ratio
- Two variables $x$ and $y$ are in direct proportion if, and only if, their ratio is constant for all values (with $x\neq 0$): $$\frac{y}{x}=k$$
- This is written $y\propto x$ and means $$y=kx \quad (k\neq 0)$$
- The function $y=kx$ is called a linear variation function.
How To Recognize Direct Proportion
There are three equivalent ways to identify direct proportion.
- Table test (constant ratio): compute $y/x$ for several pairs; it should always equal the same number $k$.
- Equation form: it can be written in the form $y=kx$ with no extra constant added.
- Graph test: it is a straight line that passes through the origin $(0,0)$.
- A straight line that does not pass through the origin is linear, but it is not proportional.
- For instance, $y=2x+3$ has a constant rate of change (slope), but the ratio $y/x$ is not constant.
Meaning Of $k$ On The Graph
- On a graph of $y=kx$, the constant $k$ is the gradient (slope): $$k=\frac{\Delta y}{\Delta x}$$
- So $k$ tells you how many units $y$ changes for each 1 unit change in $x$.
- Fuel costs \$1.50 per liter.
- If $l$ is liters and $P$ is price, then $$P(l)=1.50\,l$$
- Here $k=1.50$, and the graph is a straight line through the origin.
- If you buy $10$ L, then $P=1.50\times 10=15.0$.
Solving Direct Proportion Problems
Most direct proportion questions follow the same method.
- Write $y=kx$.
- Substitute a known pair $(x,y)$ to find $k$.
- Use the equation to find the unknown value.
- The variable $y$ varies as $x$.
- When $x=4$, $y=10$.
- Since $y\propto x$, write $y=kx$.
- Substitute: $$10=4k \Rightarrow k=\frac{10}{4}=2.5$$
- When $x=7$: $$y=2.5\times 7=17.5$$
Scaling Rules In Direct Proportion
If $y=kx$, then multiplying $x$ by a factor multiplies $y$ by the same factor.
- If $x$ is doubled, then $y$ is doubled.
- If $x$ is multiplied by $\tfrac{2}{3}$, then $y$ is multiplied by $\tfrac{2}{3}$.
- Adding a number to $x$ does not create a proportional change.
- If $y=kx$, then replacing $x$ by $x+4$ gives $y=k(x+4)=kx+4k$, which changes the relationship.
Inverse Proportion Means A Constant Product
- Two variables $x$ and $y$ are inversely proportional if multiplying one by a non-zero number results in the other being divided by the same number.
- This is written $y\propto \tfrac{1}{x}$ and means $$y=\frac{k}{x} \quad (k\neq 0,\ x\neq 0)$$
- A key equivalent condition is that the product is constant: $$xy=k$$
- Also, if $y$ is inversely proportional to $x$, then $y$ is directly proportional to $\tfrac{1}{x}$.
How The Graph Looks
- For $k>0$, the graph of $y=\tfrac{k}{x}$ is a curved reciprocal relationship (a rectangular hyperbola).
- The curve approaches the axes but does not touch them (the axes are asymptotes).
- In many real situations, you only use positive values, so you focus on the first-quadrant branch.
- Suppose the time $t$ to do a job is inversely proportional to the number of workers $w$: $$t=\frac{k}{w}$$
- If 6 workers take 10 hours, then $k=60$ and $$t=\frac{60}{w}$$
- With 12 workers, $t=\frac{60}{12}=5$ hours.
- Do not confuse inverse proportion ($y=\tfrac{k}{x}$) with a negative proportional relationship ($y=-kx$).
- In inverse proportion, the graph is curved and $xy$ stays constant.
- In negative proportion, the graph is a straight line through the origin with negative slope.
Direct Non-Linear Proportion Uses Powers
Sometimes a variable is proportional to a power of another variable.
Direct non-linear proportion
A relationship where $y$ is proportional to a positive power of $x$: $y\propto x^n$ with $n>0$, so $y=kx^n$.
If $y=kx^n$ and you multiply $x$ by a constant factor $c$, then $y$ is multiplied by $c^n$.
What The Exponent Changes
- If $n=2$, then doubling $x$ multiplies $y$ by $2^2=4$.
- If $n=3$, then tripling $x$ multiplies $y$ by $3^3=27$.
- The graph shape (for $k>0$) depends on whether $n$ is even or odd:
- even $n$ gives a U-shaped curve (symmetric about the $y$-axis)
- odd $n$ gives a curve through the origin (symmetric about the origin)
- The area $A$ of a square is proportional to the square of its side length $s$: $$A\propto s^2 \Rightarrow A=ks^2$$
- For a square, $k=1$, so $A=s^2$.
Negative Proportion Versus Inverse Proportion
- Sometimes two variables move in opposite directions in a linear way.
- This is often described as negative proportionality.
- Negative proportion is still "direct" in structure because the ratio $\frac{y}{x$ is constant, but that constant is negative.
Negative proportion
A linear proportional relationship with a negative constant of variation: $y=-kx$ where $k>0$.
Key Differences To Remember
- Negative proportion: $y=-kx$, straight line through the origin, constant ratio $\frac{y}{x}=-k$.
- Inverse proportion: $y=\tfrac{k}{x}$, curved graph, constant product $xy=k$.
- A simple real-life negative variation: if you walk east from a fixed starting point, your signed displacement west becomes more negative.
- If $x$ is distance walked east, then "west displacement" $y$ could be modeled by $y=-x$.
Using Data To Verify Proportion
When you are given a table of values, you can test for proportionality and find $k$.
Direct Proportion From A Table
Check whether $\frac{y}{x}$ is constant.
- A composter produces soil.
- Data:
- containers: 2, 3, 7, 9
- soil mass (kg): 1.6, 2.4, 5.6, 7.2
- Ratios: $$\frac{1.6}{2}=0.8,\quad \frac{2.4}{3}=0.8,\quad \frac{5.6}{7}=0.8,\quad \frac{7.2}{9}=0.8$$
- The ratio is constant, so mass is directly proportional to containers, with $k=0.8$ kg per container.
- For 5 containers: $$m=0.8\times 5=4.0\text{ kg}$$
Inverse Proportion From A Table
- Check whether $xy$ is constant.
- If the products are the same (or very close, allowing for rounding), then $y\propto \tfrac{1}{x}$.
If the relationship is inverse, plotting $y$ against $\tfrac{1}{x}$ should give a straight line through the origin, because $y=k\left(\tfrac{1}{x}\right)$.
Proportion And Function Notation
- You will often see proportionality expressed using function notation, which highlights input and output clearly.
- For example, writing $P(l)=1.50l$ emphasizes that price $P$ depends on liters $l$.
- In any proportional model:
- the dependent variable is the function value (output)
- the independent variable is the input variable
- If $y\propto x$ and $y=18$ when $x=6$, find $k$ and then find $y$ when $x=10$.
- If $y\propto \tfrac{1}{x}$ and $y=4$ when $x=3$, find $k$ and then find $y$ when $x=6$.
- Explain (in one sentence) why $y=5x+2$ is not a proportional relationship.
