Direct, Inverse, Joint and Combined Variation

This section covers:

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 both variables either increase or decrease together
  • Inverse or Indirect Variation, where when one of the variables increases, the other one decreases
  • Joint Variation, where more than two variables are related directly
  • Combined Variation, which involves a combination of direct or joint variation, and indirect variation

These sound like a lot of fancy math words, but it’s really not too bad.  Here are some examples of direct and inverse variation:

  • Direct:   The number of dollars I make varies directly (or you can say varies proportionally) with how much I work.
  • Direct:   The length of the side a square varies directly with the perimeter of the square.
  • Inverse:   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).
  • Inverse:  The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running.
  • Inverse:   My GPA may vary directly inversely with the number of hours I watch TV.

Direct or Proportional Variation

When two variables are related directly, the ratio of their values is always the same.  So as one goes up, so does the other, and if one goes down, so does the other.  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.

(Note that Part Variation, 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.)

Direct variation problems are typically written:

→        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 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.

(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.)

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 the either the first set of variables , or the second .

Direct Variation Word Problem:

So we might have a problem like this:

The value of y  varies directly with x, and y = 20 when x = 2.  Find y when x = 8.  (Note that this may be also be written “y is proportional to x, and y = 20 when x = 2.  Find y when x = 8.”)


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.

Formula Method:

Direct Variation with k

Proportion Method:

Direct Variation with Proportion

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:

The amount of money raised at a school fundraiser is directly proportional to the number of people who attend.  Last year, the amount of money raised for 100 attendees was $2500.   How much money will be raised if 1000 people attend this year?


Let’s do this problem using both the Formula Method and the Proportion Method:

Direct Variation Word Problem

Direct Variation Word Problem:

Washing Machine

Brady bought an energy efficient washing machine for her new apartment.  If she saves about 10 gallons of water per load, how many gallons of water will she save if she washes 20 loads of laundry?


Let’s do this with the proportion model:

Direct Variation Story Problem

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:

If y varies directly with the square of x, and if y = 4 when x = 3, what is y when x = 2?


Let’s do this with the formula method and the proportion method:

Direct Square Variation

Inverse or Indirect Variation

Inverse or Indirect Variation is refers to relationships of two variables that go in the opposite direction.  Let’s supposed 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:

Inverse Variation Table

(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, since with a negative slope, we can’t multiply the x’s and y’x to get the same number).

So the formula for inverse or indirect variation is:

→           Formula for Indirect Variation  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  \(y=\frac{k}{{{{x}^{2}}}}\text{ or }{{x}^{2}}y=k\).)

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 Expressions and Functions section here.

Inverse Variation Graph

Inverse Variation Word Problem:

So we might have a problem like this:

          The value of y varies inversely with x, and y = 4 when x = 3.  Find x when y = 6.

The problem may also be worded like this:

           Let = 3,  = 4, and  = 6.  Let y vary inversely as x.  Find .


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.

Formula Method:

Indirect Variation Formula Method

Product Rule Method:

Inverse Variation Product Method

Inverse Variation Word Problem:


For the Choir fundraiser, the number of tickets Allie can buy is inversely proportional to the price of the tickets.  She can afford 15 tickets that cost $5 each.  How many tickets can Allie buy if each cost $3?


Let’s use the product method:

Inverse Variation Word Problem

“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.

If 16 women working 7 hours day can paint a mural in 48 days, how many days will it take 14 women working 12 hours a day to paint the same mural?


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:

Work Problem

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:

Determine What Type Variation

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. 

Joint variation problem:

Supposed x varies jointly with y and the square root of z.  When x = ­–18 and y = 2, then z = 9.  Find y when x = 10 and z = 4.

Let’s set this up like we did with direct variation, find the k, and then solve for y:

Joint Variation Problem

Joint Variation Word Problem:

The area of a triangle is jointly related to the height and the base.   If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area?


We probably know the equation for the area of a triangle to be  \(A=\frac{1}{2}bh\),   (b = base and h = height) so we can think of the area having a joint variation with b and h, with  \(k=\frac{1}{2}\).  So let’s do the math for this problem; we can just keep the variable k in the problem:

Joint Variation Word Problem Area

Joint Variation Word Problem:

The volume of wood in a tree (V) varies directly as the height (h) and the square of the girth (g). If the volume of a tree is 144 cubic meters (\({{m}^{3}}\)) when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters?




 \(\begin{array}{l}V=k\text{(height)(girth}{{\text{)}}^{2}}\\V=kh{{g}^{2}}\\\\144=k(20){{(1.5)}^{2}}=45k\\144=45k;\,\,k=3.2\\\\V=kh{{g}^{2}};\,\,\,\,1000=3.2h\cdot {{2}^{2}}\\h=78.125\end{array}\)

We can set it up almost word for word from the word problem. For the words “varies directly”, just basically use the “=” sign, and everything else will fall in place.


Solve for k first; we get k = 3.2.

Now we can plug in the new values to get the new height.


So the new height is 78.125 meters.


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. 

Combined Variation Problem:

      (a)   y varies jointly as x and w and inversely as the square of zFind the equation of variation when y = 100, x = 2, w = 4, and z = 20.

      (b)   Then solve for y when x = 1, w = 5, and z = 4.

Let’s solve:

Combined Variation Word Problem

 Combined Variation Word Problem:

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.  The population of Charlotte is about 1,500,000 and the population of Nashville is about 1,200,000, and the distance between the two cities is about 400 miles.  The average number of calls between the cities is about 200,000. 

(a)   Find the k and write the equation of variation.

(b)   The average number of daily phone calls between Charlotte and Indianapolis (which has a population of about 1,700,000) is about 134,000.  Find the distance between the two cities.


This one looks really tough, but it’s really not that bad if you take it one step at a time:

Distance Between Cities Combined Variation Problem


Combined Variation Word Problem:

y varies jointly with  \({{x}^{3}}\)  and z, and varies inversely with  \({{r}^{2}}\).  What is the effect on y when x is doubled and r is halved?


Since we want x to double and r to be halved, we can just put in the new “values” and see what happens to y.  Make sure to put them in parentheses, and “push the exponents through”:

Joint Variation Word Problem Doubling

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:

Variation Problem with 2 Constants

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!

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

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