Constant of Variation - Definition & Examples

(Video) Inverse Variation

by mathman1024

Here's a video by mathman1024 explaining inverse variation or inverse proportionality.

$\large{\text{Summary}}$

Inverse variation/proportionality means that the product of two variables will always stay the same. So when one variable changes, the other variable will also change such that their product can stay the same.

For example, say you're having a party and order a pizza. As the number of people increases, the number of slices of pizza per person decreases.

The formula for inverse variation is $$y=\frac{k}x$$ In this equation, $y$ and $x$ are inversely proportional to each other, and $k$ is (and will always be) their product. When $k$ is positive, as $x$ increases $y$ decreases: $$\text{when } k \text{ is positive:}\\ x\uparrow\ \Rightarrow \ y\downarrow\\ \text{or}\\ x\downarrow\ \Rightarrow\ y\uparrow$$

Solution

We're told $w$ and $z$ vary inversely. So we can set up out inverse equation as: $$z=\frac kw$$

Next, we use the given information to find $k$. When $z=2$, $w=14$. $$\begin{align} z&= \frac{k}{w}\\ 2&= \frac{k}{14}\\ {\color{#8925ae}{14}}(2)&={\color{#8925ae}{14}}\left(\frac k{14}\right)\\ 28 &= k \end{align}$$

Since we know $k$, we can plug it into our equation when $z = 7$. $$\begin{align} z&=\frac{28}{w}\\ 7 &= \frac{28}{w}\\ \frac17&=\frac w{28}\\ {\color{#8925ae}{28}}\left(\frac17\right)&={\color{#8925ae}{28}}\left(\frac w{28}\right)\\ 4&=w \end{align}$$ And there's our answer.

What are Inverse Proportions?

Inverse proportionality, or inverse variation, is a special relationship between two variables. When one variables changes, the other changes proportionally in the opposite direction.

For example, the hours you spend studying might vary inversely with the hours you spend playing video games. If you double the hours you spend studying, then you have less time to play video games and its time decreases by half. Similarly, if you study a third less than normal, you'll have more time to play games and can play three times more.

We use the following equation for direct variation. $$y=\frac kx$$ Here, $k$ is the constant of variation. It will be a number that relates the two variables together.

$k$ Doesn't Change

When solving inverse proportionality problems it's important to note that $k$ will never change in the problem.

If we rearrange the direct proportion equation, we see $k$ is a product of $y$ and $x$. $$k=xy$$ Because $k$ is fixed (never changes), if we increase $y$, we then have to decrease $x$ by some amount so that the product of the two still equals $k$.

Example: Is it Inversely Proportional

Example 1$\\$ The equation for the area of a circle is $$A=\pi r^2$$ where $\pi$ (pi) is the constant $3.14\dots$ and $r$ is the radius of the circle.

Is this an inversely proportional equation?

The answer is no! As one variable increases, so does the other. This is an example of direct variation.

Example 2$\\$ What about the following equation? $$y=\frac 4x$$ The answer is yes! If we increase one variable, the other will have to decrease. We can also write this as a product of two variables. $$xy=4$$

Example: Using Tables

Sometimes you'll be given a table and have to figure out if its inversely proportional or not. $$\begin{array}{c|c} x&y\\ \hline 2&18\\ 3&12\\ 4&10\\ 6&6 \end{array}$$

To solve these, we just find $k$ for each pair of coordinates. If $k$ is the same for each of them, then they're inversely proportional. $$\begin{array}{c|c|c} x&y&xy=k\\ \hline 2&18&2\cdot18=36\\ 3&12&3\cdot12=36\\ 4&10&4\cdot10=40\\ 6&6&6\cdot6=36 \end{array}$$ Since the third pair has a different $k$, the entire table is not inversely proportional.

Example: Word Problems

The general steps to solving these proportionality problems are:

1. Use the given information to find $k$.
2. Use $k$ to find the unknown information.

Let's look at an example:

The time, $t$, it takes to paint a house varies inversely with the number of people painting, $p$. If $2\text{ people}$ are painting, it takes $16\text{ hours}$. How long will it take if $10\text{ people}$ paint?

We're told that the time $t$ varies inversely with the number of people painting $p$. This tells us to set these variables up in a inverse proportion. Using the equation, we get: $$t=\frac kp$$

Proportionality problems usually follow the same format. They give one piece of information which you use to find $k$, and then you use that to solve the last piece of information.

In this problem we're given that $2\text{ people}$ paint the house in $16\text{ hours}$. Plug these into our equation and solve for $k$. $$\begin{align} t&=\frac kp\\ 16&=\frac k2\\ {\color{#8925ae}{2\,\times}}\,16&={\color{#8925ae}{2\,\times}}\,\frac k2\\ 32&=k \end{align}$$ We now have our constant of variation. This value doesn't change.

Because we know $k$, we can now find the unknown part of the problem. We want to find the time it takes to paint a house when $10\text{ people}$ paint. $$\begin{align} t&=\frac{32}x\\ t&=\frac{32}{10}\\ t&=3.2\text{ hours} \end{align}$$

Notice the product of both cases equals our constant of variation. $$2\times16=32=10\times3.2$$

Solution

We're told the time, $t$, is inversely proportional to the speed, $s$. We can set up our inverse proportion equation as: $$t=\frac ks$$

Now, we use our given information to solve for $k$. We know going $30\text{ mph}$ takes $5\text{ hours}$. $$\begin{align} t&=\frac ks\\ 5&=\frac k{30}\\ {\color{#8925ae}{30\,\times}}\,5&={\color{#8925ae}{30\,\times}}\,\frac{k}{30}\\ 150&=k \end{align}$$

Finally, using our newly found $k$, we can solve for the unknown. But be careful! Our time was in hours, but we're given minutes now. We can easily convert this to hours: $30\text{ minutes}=\frac12\text{ hour}$. $$\begin{align} t&=\frac{150}{s}\\ \frac12&=\frac{150}{s}\\ 2&=\frac s{150}\\ {\color{#8925ae}{150\,\times}}\,2&={\color{#8925ae}{150\,\times}}\,\frac{s}{150}\\ 300&=s \end{align}$$

In order to make our drive in $30\text{ minutes}$, you'd have to drive at $300\text{ mph}$. That's a little below the speed of a plane.

Constant of Variation

Recall that the constant of proportionality is the the $k$ in the direct variation equation $y=kx.$

Similarly, the constant of variation is the $k$ in the inverse variation equation $y=\dfrac{k}{x}.$

Example

Say $x$ varies inversely with $y.$ If $y = 3$ when $x = 7,$ find $y$ when $x = 9.$

Solution

First, let's find $k,$ the constant of proportionality. We already know that $x$ varies directly with $y$ and that $y = 3$ when $x = 7.$ We can use that to find $k.$

$$\begin{align} y&=\dfrac{k}{x}\\ 3&=\dfrac{k}{7}\\ 7\cdot 3 &= \dfrac{k}{\cancel{7}}\cdot \cancel{7}\\ 21 &= k \end{align}$$

Now that we know $k=21,$ we can find $y$ when $x=9.$ $$\begin{align} y&=\dfrac{k}{x}\\ y&=\dfrac{21}{9} \\ y&= \dfrac{7}{3}\\ \end{align}$$