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(Video) Joint and Combined Variation

by mathman1024

This a video by mathman1024 explains joint variation and combined variation.

$\large{\text{Summary}}$

The joint variation equation is: $y$ varies jointly with $x$ and $z$. This means $y$ varies with $x$ and $z$. $$y=k\cdot x\cdot z$$

Alas the relationship is more complicated than a direct relation or inverse relation. Really, joint variations are combinations of both of these.

An example of a joint variation is the area of a triangle: $A=\frac12bh$. We see that the area increases as the height and base length increase. $$b\uparrow,h\uparrow\ \Rightarrow\ A\uparrow$$ And here the $\frac12$ is the constant that connects them all together. This is called the constant of proportionality.

The main phrase for combined variation is, $y$ varies directly as $x$ and inversely as $z$. This is just combining both direct and inverse proportionality. This will take the form: $$y=\frac{k\cdot x}{z}$$ $$x\uparrow\ \Rightarrow\ y\uparrow\ \text{and }\ z\uparrow\ \Rightarrow\ y\downarrow$$

Joint Variation

Many algebra problems can be solved by using a direct proportion or an inverse proportion by itself.

However, for some problems we need to combine these two types of proportions.

Joint variation is when a variable varies directly with two or more other variables. In other words, it's just a direct proportion with multiple variables. $$y=kxz$$ Here, $y$ varies directly with both $x$ and $z$. Also, $k$ is a positive constant. That means it doesn't change even if $x, y,$ and $z$ do. This constant is called the constant of proportionality.

Doubling the $x$ variable, would also double the $y$ variable . If we double the $z$, the $y$ will also double. But if we double both $x$ and $z$, then $y$ gets quadrupled.

The area of a triangle is jointly proportional to the base and height of the triangle. $$A=\frac12bh$$ Here, our constant of proportionality is $k=\frac12$.

Joint Variation Example

Joint variation problems are similar to direct and inverse proportions. Normally, you'll be given information to use to find $k$. Then, you use $k$ to find the unknown information.

Suppose the number of days it takes to watch the entirety of a show is jointly proportional to the number of episodes there are and the length of the episodes. It took you $40\text{ days}$ to watch all $73\text{ episodes}$ of "Game of Thrones" which were each an hour long. "The Office" has $201\text{ episodes}$, each $30\text{ minutes}$ long. How long will it take to watch the series?

Let's use the following variable definitions: $$\begin{align}d&=\text{ Number of days it takes to watch the show}\\ n&= \text{ Number of episodes in a series}\\ l&= \text{ Length of each episode}\end{align}$$

We're told the variables are jointly proportional, so we can use the equation

$$d=knl$$

Now, we use our given information about "Game of Thrones" to solve for $k$. We know it took $40\text{ days}$ to watch $73\text{ episodes}$ that are each $1\text{ hour}$ long. Because one time is in hours and the other is in minutes, we need to convert one into the other. I'm going to use $60\text{ minutes}$.

Now, we use our given information about Game of Thrones to solve for $k$. We know it took $40\text{ days}$ to watch $73\text{ episodes}$ that are $1\text{ hour}$ long. Because one time is in hours and the other is in minutes, we need to convert units. Specifically, both times need to be in the same units. I'm going to use $60\text{ minutes}$, which is the same as $1$ hour. You could convert all the time units to hours if you'd like.

$$\begin{align} d&=knl\\ 40&=k(73)(60)\\ 40&=4380k\\ \frac{40}{\color{#8925ae}{4380}}&=\frac{4380k}{\color{#8925ae}{4380}}\\ \frac{2}{219}&=k \end{align}$$

Using our newly found $k$, we can find out how long it will take to watch "The Office." We know there are $201\text{ episodes}$ each $30\text{ minutes}$ long. $$\begin{align} d&=\frac2{219}nl\\ d&=\frac{2}{219}(201)(30)\\ d&=\frac{12060}{219}\\ d&\approx55 \end{align}$$ It will take you about $55\text{ days}$ to watch The Office.

Combined Variation

Combined proportionality (or combined variation) is just the combination of direct and inverse proportionality. If a variable has a combined proportionality with two other variables, then it has a direct proportion with one and an inverse proportion with the other. $$y=\frac{kx}{z}$$ Here, $y$ varies directly with $x$ and inversely with $z$. Once again $k$ is the constant of proportionality.

Pressure is combined proportionally to force and area. It's directly proportional to force and inversely proportional to area. $$P=\frac FA$$ This formula often appears in physics and chemistry.

The constant of variation is $k=1$, so it's not written. Remember, multiplying a number by $1$ does not change its value, so we don't have to write the $k$.

Combined Variation Example

The time it takes you to complete a puzzle is directly proportional to the number of pieces and inversely proportional to how many people are working on it. You and your three friends can complete a $1{,}000\text{ piece}$ puzzle in $2\text{ hours}$. How long would it take you to complete a $2{,}500\text{ piece}$ puzzle by yourself?

As always it's a good idea to start by defining variables. This makes it easier to keep track of what you're doing. Let's use the following definitions: $$\begin{align} t&=\text{ Time taken to complete the puzzle}\\ n&=\text{ Number of people working on the puzzle}\\ p&=\text{ Number of pieces in the puzzle}\end{align}$$

We're told $t$ and $p$ are directly proportional and $t$ and $n$ are inversely proportional. This means the equation relating all the variables is $$t=k\frac pn$$

First, we use our given information to find $k$. We know a $1{,}000\text{ piece}$ puzzle takes $2\text{ hours}$ to complete if you and three friends work on it. This means $n=4\text{ people}$. $$\begin{align} t&=k\frac pn\\ 2&=k\frac{1000}{4}\\ 2&=250k\\ \frac{1}{125}&=k \end{align}$$

Now that we know $k$, we can solve the unknown information. We know there are $2{,}500\text{ pieces}$ and only you working on it: $n=1$. $$\begin{align} t&=\frac{1}{125}\frac pn\\ t&=\frac{1}{125}\frac{2500}{1}\\ t&=20 \end{align}$$ It'll take you $20\text{ hours}$ to solve the puzzle by yourself. You might want to get some of your friends to help you.

Summary

Here is a graphic summarizing what you need to remember for Joint Variation and Combined Variation:

Joint Variation

Joint variation is just like direct variation, but it involves more than $2$ variables.

Here is a quick refresher on simple direct variation if you need it. This concept will be used frequently in this explanation.

Recall that the general form for two variables ($x$ and $y$) that are directly proportional is:

Recall that $k$ is called the constant of variation. It's also sometimes called the constant of proportionality.

So, if $y$ varies jointly with $x$ and another variable, $z$, we could re-write the equation like this:

Let's look at an example to see what all this means.

We want to show that $x$ is directly proportional to BOTH $y$ and $2z$.

Here's how:

$$\begin{align}x &= (k) (2z) (y)\\ x&= k2zy\\ x&=2kyz\end{align}$$

When we rearrange all that, we get $x=2kyz$. Remember that $k$ is the constant of variation.

This means that the ratio between $x$ and both $y$ and $2z$ will ALWAYS be $2$.

Combined Variation

Combined variation involves a combination of direct and inverse variation. The equations look a little messier, but they are still manageable. You will now have some division thrown in.

Here are the general formulas for direct and inverse variation between $2$ variables ($x$ and $y$), individually.

When we've got combined variation, we need to combine these two formulas. But we won't use $x$ in both places. We instead introduce a third variable, $z$.

When $y$ varies directly with $x$ and varies inversely with $z$, the general formula for this combined variation is:

Let's try a sample problem to see how combined variation works.

The last two questions illustrate an important concept. If we know $x$, $y$, and $w$, we can use those to find the constant of proportionality, $k$. After that, if we're given the values of any two of $x, y,$ and $w$, we can find the third. For this second step we have to use the newly found value of $k$.