Finding the Constant of Proportionality

In a table. Each row gives one pair of values, $x$ and $y$. In a proportional relationship, $y = kx$. In any row where $x \neq 0$, you can divide both sides by $x$ and they stay equal, so $y \div x = k$. That gives you a method: pick a row where $x$ is not 0 and divide $y$ by $x$.

One row is enough only when you already know the relationship is proportional. If you are not told, divide in every row where $x \neq 0$. The results of the divisions, called quotients, must all be the same number.

The table below shows how much flour a bakery uses for its bread. Here $x$ is the number of loaves and $y$ is the amount of flour in cups. The last column works out $y \div x$ for each row, in cups per loaf.

On a graph. The graph of a proportional relationship $y = kx$ is a straight line through $(0, 0)$. Each point on the line is a pair $(x, y)$, just like a row of a table. So the method is the same: choose a point with $x \neq 0$ and divide its $y$-value by its $x$-value.

The graph below shows the elevator from the start of the lesson. The horizontal axis shows the time $x$ in seconds, and the vertical axis shows the height $y$ in meters. Neither quantity is negative here. Line A is straight and passes through the labeled points $(0, 0)$, $(1, 2)$, $(2, 4)$ and $(4, 8)$. Each point is one pair of values: $(4, 8)$ means that after 4 seconds the elevator has risen 8 meters.

Divide at each labeled point with $x \neq 0$: $2 \div 1 = 2$, $4 \div 2 = 2$ and $8 \div 4 = 2$. The quotient is always 2, so $k = 2$ meters per second.

Often the point with $x = 1$ is not labeled, or $k$ is not a whole number. You do not need that point. On a straight line through $(0, 0)$, every point with $x \neq 0$ gives the same quotient $y \div x$, so any labeled point with $x \neq 0$ will do. Use the exact labeled coordinates, not a guess from the grid.

The next graph shows an empty tank filling with water. The horizontal axis shows the time $x$ in minutes, and the vertical axis shows the water in the tank, $y$, in liters. Line B is straight and passes through the labeled points $(0, 0)$, $(2, 3)$, $(4, 6)$ and $(6, 9)$.

In an equation. A proportional relationship can be written as $y = kx$, so $k$ is the number that multiplies $x$. The elevator's equation is $y = 2x$, and you can read $k = 2$ straight from it. This works only when the equation has the form $y = kx$: one number times $x$, with nothing added or subtracted.

The letters can change. Let $t$ be a time in hours and let $d$ be the distance in miles that a car travels in that time. In the equation $d = 55t$, the number that multiplies $t$ is 55, so $k = 55$ miles per hour.

Sometimes $k$ is hidden because the equation shows a division. Then rewrite the equation so that one number multiplies $x$. The next example shows how.

In a description in words. Some descriptions give you $k$ directly. Read this one: A sprinkler uses 9 liters of water each minute. It tells you that 1 minute goes with 9 liters. Let $x$ be the time in minutes and let $y$ be the water used in liters. Then $k = 9$ liters per minute. Words such as each and per often signal the unit rate.

Other descriptions give one pair of values that is not the unit rate. Then make three moves:

1. Name the variables and their units. Let $x$ be the quantity you start from and let $y$ be the quantity you want to find from it.
2. Write the pair of values from the description.
3. Divide $y$ by $x$. This gives $k$ because the relationship is proportional and $x \neq 0$.

The big idea. In a proportional relationship, one number $k$ links the two quantities: $y = kx$, so $k = y \div x$ for every pair with $x \neq 0$. It is the unit rate, the amount of $y$ for 1 unit of $x$. Here is where to find it:

• Table: divide $y$ by $x$ in a row where $x \neq 0$, and check that the other rows agree.
• Graph: on a straight line through $(0, 0)$, divide $y$ by $x$ at a labeled point with $x \neq 0$, or read the $y$-value of the labeled point $(1, k)$.
• Equation or words: write the equation in the form $y = kx$ and read the number that multiplies $x$. For a proportional relationship described in words, name $x$ and $y$, take one pair with $x \neq 0$, and divide $y$ by $x$.

In every view, check that $k$ times $x$ gives $y$. Next, you will use $k$ to complete a table and graph a proportional relationship.