| Direct or Proportional Variation | Partial Variation |
| Inverse or Indirect Variation | More Practice |
| 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 ratio, constant 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.
Remember the example of making $10 an hour at the mall ($ y=10x$)? This is an example of direct variation, since the ratio of how much you make to how many hours you work is always constant.
We can also set up direct variation problems in a ratio, as long as we have the same variable in either the top or bottom of the ratio, or on the same side. This will look like the following. Don’t let this scare you; the subscripts just refer to either the first set of variables $ ({{x}_{1}},{{y}_{1}})$, or the second $ ({{x}_{2}},{{y}_{2}})$: $ \displaystyle \frac{{{{y}_{1}}}}{{{{x}_{1}}}}\,\,=\,\,\frac{{{{y}_{2}}}}{{{{x}_{2}}}}$.
Notes: Partial Variation (see below), or “varies partly” means that there is an extra fixed constant, so we’ll have an equation like $ y=mx+b$, which is our typical linear equation. Also, I’m assuming in these examples that direct variation is linear; sometime I see it where it’s not, like in a Direct Square Variation where $ y=k{{x}^{2}}$. There is a word problem example of this here.
Direct Variation Word Problem:
We can solve the following direct variation problem in one of two ways, as shown. We do these methods when we are given any three of the four values for $ x$ and $ y$.
It’s really that easy. Can you see why the proportion method can be the preferred method, unless you are asked to find the $ k$ constant in the formula? Again, if the problem asks for the equation that models this situation, it would be “$ y=10x$”.
Direct Variation Word Problem:
Direct Variation Word Problem:
Here’s another; let’s use the proportion method:
See how similar these types of problems are to the Proportions problems we did earlier?
Direct Square Variation Word Problem:
Again, a Direct Square Variation is when $ y$ is proportional to the square of $ x$, or $ y=k{{x}^{2}}$. Let’s work a word problem with this type of variation and show both the formula and proportion methods:
Inverse or Indirect Variation
Inverse or Indirect Variation refers to relationships of two variables that go in the opposite direction (their product is a constant, $ k$). Let’s suppose you are comparing how fast you are driving (average speed) to how fast you get to your school. You might have measured the following speeds and times:
(Note that $ \approx $ means “approximately equal to”).
Do you see how when the $ x$ variable goes up, the $ y$ goes down, and when you multiply the $ x$ with the $ y$, we always get the same number? (Note that this is different than a negative slope, or negative $ k$ value, since with a negative slope, we can’t multiply the $ x$’s and $ y$’s to get the same number).
The formula for inverse or indirect variation is: → $ \displaystyle \boldsymbol{y=\frac{k}{x}}$ or $ \boldsymbol{xy=k}$, where $ k$ is always the same number.
(Note that you could also have an Indirect Square Variation or Inverse Square Variation, like we saw above for a Direct Variation. This would be of the form $ \displaystyle y=\frac{k}{{{{x}^{2}}}}\text{ or }{{x}^{2}}y=k$.)
Here is a sample graph for inverse or indirect variation. This is actually a type of Rational Function (function with a variable in the denominator) that we will talk about in the Rational Functions, Equations and Inequalities section.
Inverse Variation Word Problem:
We might have a problem like this; we can solve this problem in one of two ways, as shown. We do these methods when we are given any three of the four values for $ x$ and $ y$:
Inverse Variation Word Problem:
Here’s another; let’s use the product method:
“Work” Inverse Proportion Word Problem:
Here’s a more advanced problem that uses inverse proportions in a “work” word problem; we’ll see more “work problems” here in the Systems of Linear Equations Section and here in the Rational Functions and Equations section.
In the problem below, the three different values are inversely proportional; for example, the more women you have, the less days it takes to paint the mural, and the more hours in a day the women paint, the less days they need to complete the mural:
Recognizing Direct or Indirect Variation
You might be asked to look at functions (equations or points that compare $ x$’s to unique $ y$’s – we’ll discuss later in the Algebraic Functions section) and determine if they are direct, inverse, or neither:
Joint Variation and Combined Variation
Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let’s set this up like we did with direct variation, find the $ k$, and then solve for $ y$; we need to use the Formula Method:
Joint Variation Word Problem:
We know the equation for the area of a triangle is $ \displaystyle A=\frac{1}{2}bh$ ($ b=$ base and $ h=$ height), so we can think of the area having a joint variation with $ b$ and $ h$, with $ \displaystyle k=\frac{1}{2}$. Let’s do an area problem, where we wouldn’t even have to know the value for $ k$:
Another Joint Variation Word Problem:
Combined Variation
Combined variation involves a combination of direct or joint variation, and indirect variation. Since these equations are a little more complicated, you probably want to plug in all the variables, solve for $ k$, and then solve back to get what’s missing. Let’s try a problem:
Here’s another; this one looks really tough, but it’s really not that bad if you take it one step at a time:
Combined Variation Word Problem:
Combined Variation Word Problem:
One word of caution: I found a variation problem in an SAT book that stated something like this: “If $ x$ varies inversely with $ y$ and varies directly with $ z$, and if $ y$ and $ z$ are both 12 when $ x=3$, what is the value of $ y+z$ when $ x=5$”. I found that I had to solve it setting up two variation equations with two different $ k$‘s (otherwise you can’t really get an answer). So watch the wording of the problems. 🙁 Here is how I did this problem:
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!