# Direct, Inverse, Joint and Combined Variation

When you start studying algebra, you will also study how two (or more) variables can relate to each other specifically. The cases you’ll study are:

• Direct Variation, where one variable is a constant multiple of another. For example, the number of dollars I make varies directly (or varies proportionally) to the number of hours I work. Or, the perimeter of a square varies directly with the length of a side of the square.
• Inverse or Indirect Variation, where when one of the variables increases, the other one decreases (their product is constant). For example, the temperature in my house varies indirectly (same or inversely) with the amount of time the air conditioning is running. Or, the number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of).
• Joint Variation, where at least two variables are related directly. For example, the area of a triangle is jointly related to both its height and base.
• Combined Variation, which involves a combination of direct or joint variation, and indirect variation. For example, the average number of phone calls per day between two cities has found to be jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities.
• Partial (Direct) Variation, where two variables are related by a formula, such as the formula for a straight line (with a non-zero $y$-intercept). For example, the total cost of my phone bill consists of a fixed cost per month, and also a charge per minute.

Note: Just because two variables have a direct relationship, the relationship may not necessarily be a causal relationship (causation), meaning one variable directly affects the other. There may be another variable that affects both of the variables. For example, there may be a correlation between the number of people buying ice cream and the number of people buying shorts. People buying ice cream do not cause people to buy shorts, but most likely warm weather outside is causing both to happen.

Here is a table for the types of variation we’ll be discussing:

## Direct or Proportional Variation

When two variables are related directly, the ratio of their values is always the same. If $k$, the constant ratio is positive, the variables go up and down in the same direction. (If $k$ is negative, as one variable goes up, the other goes down; this is still considered a direct variation, but is not seen often in these problems.) Note that $k\ne 0$.

Think of linear direct variation as a “$y=mx$” line, where the ratio of $y$ to $x$ is the slope ($m$). With direct variation, the $y$-intercept is always 0 (zero); this is how it’s defined. Direct variation problems are typically written:    →    $\boldsymbol {y=kx}$, where $k$ is the ratio of $y$ to $x$ (which is the same as the slope or rate).

Some problems will ask for that $k$ value (which is called the constant ratioconstant of variation or constant of proportionality – it’s like a slope!); others will just give you 3 out of the 4 values for $x$ and $y$ and you can simply set up a ratio to find the other value. I’m thinking the $k$ comes from the word “constant” in another language.

## Partial Variation

You don’t hear about Partial Variation or something being partly varied or part varied very often, but it means that two variables are related by the sum of two or more variables (one of which may be a constant). An example of part variation is the relationship modeled by an equation of a line that doesn’t go through the origin. Here are a few examples:

We’re doing really difficult problems now – but see how, if you know the rules, they really aren’t bad at all?

Learn these rules, and practice, practice, practice!

For Practice: Use the Mathway widget below to try a Variation problem. Click on Submit (the blue arrow to the right of the problem) and click on Find the Constant of Variation to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Introduction to the Graphing Display Calculator (GDC). I’m proud of you for getting this far!

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