# 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:

 Type of Variation Formula Example Wording Direct or Proportional Variation $$y=kx$$    or     $$\displaystyle \frac{{{{y}_{1}}}}{{{{x}_{1}}}}=\frac{{{{y}_{2}}}}{{{{x}_{2}}}}$$ The value of $$y$$ varies directly with $$x$$, $$y$$ is directly proportional to $$x$$. $$k$$ is called the constant of proportionality, and is typically positive. Special Case: Direct Square variation:   $$y=k{{x}^{2}}$$. Inverse or Indirect Variation $$\displaystyle y=\frac{k}{x}$$     or      $$xy=k$$ The value of $$y$$ varies inversely with $$x$$, $$y$$ is inversely proportional to $$x$$, $$y$$ is indirectly proportional to $$x$$. Special case: Indirect Square variation:   $$\displaystyle y=\frac{k}{{{{x}^{2}}}}$$. Joint Variation Like direct variation, but involves more than one variable. Example:   $$y=kx{{z}^{2}}$$ Example:  $$y$$ varies jointly with $$x$$ and the square of $$z$$. Combined Variation Involves a combination of direct variation or joint variation, and indirect variation. Example:  $$\displaystyle y=\frac{{kxw}}{{{{z}^{2}}}}$$ Example:  $$y$$ varies jointly as $$x$$ and $$w$$ and inversely as the square of $$z$$. Partial Variation Two variables are related by the sum of two or more variables (one of which may be a constant). Example:  $$y={{k}_{1}}x+{{k}_{2}}$$ Example:  $$y$$ is partly constant and partly varies directly with $$x$$.

# 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.

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 $$y$$.  Direct Variation Problem Formula Method Proportion Method 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 $$x=8$$”.) $$\begin{array}{l}y=kx\\20=k2\\k=10\end{array}$$ $$\begin{array}{l}y=kx\\y=10x\\y=10(8)\\y=80\end{array}$$ Since $$x$$ and $$y$$ vary directly, we know that $$y=kx$$. Since the problem was stated that $$y$$ varies directly with $$x$$, we place the $$y$$ first. Solve for $$k$$, using the values of $$x$$ and $$y$$ that we know ($$x=2,\,\,y=20$$). We see that $$k=10$$. Now use $$y=10x$$. Plug in the new $$x$$, which is 8. We get the new $$y=80$$. \displaystyle \begin{align}\frac{{{{y}_{1}}}}{{{{x}_{1}}}}&=\frac{{{{y}_{2}}}}{{{{x}_{2}}}}\,\\\frac{{20}}{2}&=\frac{y}{8}\,\\\\2y&=160\\y&=80\end{align} Set up a proportion with the $$y$$’s on top, and the $$x$$’s on bottom (think of setting slopes equal to each other 🙂 ). When we see the word “when” in the original problem (“$$y=20$$ when $$x=2$$”), it means that that $$x$$ goes with that $$y$$. Then cross multiply to get the new $$y$$. 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 Problem Formula Method Proportion Method 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 was2500.    How much money will be raised if 1000 people attend this year? \begin{align}y&=kx\\2500&=k100\\k&=25\end{align}     \begin{align}y&=25x\\y&=25(1000)\\y&=25000\end{align}   Since the amount of money is directly proportional (varies directly) to the number who attend, we know that $$y=kx$$, where $$y=$$ the amount of money raised and $$x=$$ the number of attendees. (Since the problem states that the amount of money is directly proportional to the number of attendees, we put the amount of money first, or as the $$y$$).   Fill in the numbers from the problem, and solve for $$k$$. We see that $$k=25$$. We have $$y=25x$$. Plug in the new $$x$$, which is 1000.   We get the new $$y=25000$$. If 1000 people attend, 25,000 would be raised! \displaystyle \begin{align}\frac{{\text{ }\!\!\\!\!\text{ }\!\!\\!\!\text{ }}}{{\text{attendees}}}&=\frac{{\text{ }\!\!\\!\!\text{ }\!\!\\!\!\text{ }}}{{\text{attendees}}}\\\frac{{2500}}{{100}}&=\frac{y}{{1000}}\\\\100y&=2500000\\y&=25000\end{align} Set up a proportion with the $$y$$’s on top (amount of money), and the $$x$$’s on bottom (number of attendees). Cross multiply to get the new amount of money ($$y$$). We get the new $$y=25000$$. If 1000 people attend,25,000 will be raised!

Direct Variation Word Problem:

Here’s another; let’s use the proportion method:

 Direct Variation Problem Proportion Method 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? \displaystyle \begin{align}\frac{{{{y}_{1}}}}{{{{x}_{1}}}}&=\frac{{{{y}_{2}}}}{{{{x}_{2}}}}\\\frac{{10}}{1}&=\frac{y}{{20}}\\\\y&=200\end{align}   Set up a proportion with the $$y$$’s on top (representing gallons), and the $$x$$’s on bottom (representing number of loads). Remember that “per load” means “for 1 load”.   Then, cross multiply to get the new $$y$$. Brady will save 200 gallons if she washes 20 loads of laundry.

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:

 Direct Square Variation Problem Formula Method Proportion Method If $$y$$ varies directly with the square of $$x$$, and if $$y=4$$ when $$x=3$$, what is $$y$$ when $$x=2$$? \displaystyle \begin{align}y&=k{{x}^{2}}\\4&=k\cdot {{3}^{2}}\,\,\\k&=\frac{4}{9}\,\end{align}        \displaystyle \begin{align}y&=\frac{4}{9}{{x}^{2}}\\y&=\frac{4}{9}\cdot {{2}^{2}}\,\,\,\,\,\,\\y&=\frac{{16}}{9}\end{align}   Since $$y$$ is directly proportional (varies directly) to the square of $$x$$, we know that $$y=k{{x}^{2}}$$. Plug in the first numbers we have for $$x$$ and $$y$$ to see that  $$\displaystyle k=\frac{{4}}{9}$$.   We have $$\displaystyle y=\frac{4}{9}{{x}^{2}}$$. Plug in the new $$x$$, which is 2, and get the new $$y$$, which is $$\displaystyle \frac{{16}}{9}$$. \displaystyle \begin{align}\frac{{{{y}_{1}}}}{{{{{\left( {{{x}_{1}}} \right)}}^{2}}}}&=\frac{{{{y}_{2}}}}{{{{{\left( {{{x}_{2}}} \right)}}^{2}}}}\\\frac{4}{{{{3}^{2}}}}&=\frac{y}{{{{2}^{2}}}}\\y=\frac{{4\cdot {{2}^{2}}}}{{{{3}^{2}}}}&=\frac{{16}}{9}\end{align}   Set up a proportion with the $$y$$’s on top, and $${{x}^{2}}$$’s on the bottom. Plug in the numbers we have, and then cross multiply to get the new $$y$$.   We then get the new $$\displaystyle y=\frac{{16}}{9}$$.

# 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:

 Average Speed of car ($$x$$) Time to get to school ($$y$$) (minutes) $$x$$ times $$y$$ 25 10 $$25\times 10=250$$ 30 8.33 $$\displaystyle 30\times 8.33~\approx 250$$ 35 7.14 $$\displaystyle 35\times 7.14~\approx 250$$ 40 6.25 $$40\times 6.25=250$$

(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.

 Indirect Variation Formula Graph $$\displaystyle y=\frac{k}{x}\text{ }\,\,\text{or }\,\text{ }xy=k$$   $$\displaystyle {{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}$$   In our case, $$k=250$$ $$\displaystyle xy=250\text{ }\,\,\text{ or }\,\,\text{ }y=\frac{{250}}{x}$$ 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 $$y$$:

 Indirect Variation Problem Formula Method Product Rule Method The value of $$y$$ varies inversely (or indirectly) with $$x$$, and $$y=4$$ when $$x=3$$.    Find $$x$$ when $$y=6$$.     The problem may also be worded like this: Let $${{x}_{1}}=3$$, $${{y}_{1}}=4$$, and $${{y}_{2}}=6$$. $$y$$ varies inversely as $$x$$. Find $${{x}_{2}}$$. \displaystyle \begin{align}{{y}_{1}}&=\frac{k}{{{{x}_{1}}}}\\\,4&=\frac{k}{3}\\\,k&=12\end{align}           \displaystyle \begin{align}{{y}_{2}}&=\frac{{12}}{{{{x}_{2}}}}\\6&=\frac{{12}}{{{{x}_{2}}}}\\6{{x}_{2}}=&12;\,\,\,{{x}_{2}}=2\end{align}   Since $$x$$ and $$y$$ vary inversely, we know that $$xy=k$$, or $$\displaystyle y=\frac{k}{x}$$.   First fill in the $$x$$ and $$y$$ values with $${{x}_{1}}$$ and $${{y}_{1}}$$ from the problem. Remember that the variables with the same subscript, such as $${{x}_{1}}$$ and $${{y}_{1}}$$, stay together. Then solve for $$k$$, which is 12.   Then put the $${{y}_{2}}$$ value in for $$y$$. Solve for $${{x}_{2}}$$, which is 2. (If the $${{x}_{2}}$$ value were given, you’d put that in for $$x$$, and solve for $${{y}_{2}}$$).   The formula way may take a little more time, but you may be asked to do it this way, especially if you need to find $$k$$, and the equation of variation, which is $$\displaystyle y=\frac{{12}}{x}$$. $$\begin{array}{l}\,\,\,\,\,{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}\\\\\left( 3 \right)\left( 4 \right)={{x}_{2}}\left( 6 \right)\\\,\,\,\,\,\,\,\,\,\,12=6{{x}_{2}}\\\,\,\,\,\,\,\,\,\,\,{{x}_{2}}=2\end{array}$$   We know that when you multiply the $$x$$’s and $$y$$’s (with the same subscript) we get a constant, which is $$k$$. You can see that $$k=12$$ in this problem.   Substitute in all the numbers that we are given and solve for the number we want – in this case, $${{x}_{2}}$$.   This way is easier than the formula method, but, again, you will probably be asked to know both ways.

Inverse Variation Word Problem:

Here’s another; let’s use the product method:

 Inverse Variation Problem Product Rule Method 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? $$\begin{array}{c}{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}\\\\\left( 5 \right)\left( {15} \right)={{x}_{2}}\left( 3 \right)\\75=3{{x}_{2}}\\{{x}_{2}}=25\end{array}$$   We know that when you multiply the $$x$$’s and $$y$$’s we get a constant, which is $$k$$. The number of tickets Allie can buy times the price of each ticket is $$k$$. Let the $$x$$’s be the price of the tickets.   Substitute in all the numbers that we are given and solve for the number we want. Allie can buy 25 tickets that cost $3. This makes sense, since we can see that she only can spend$75 (which is $$k$$!)

“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.

 “Work” Inverse Variation Problem Product Rule Method 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.) $$\displaystyle \begin{array}{c}{{x}_{1}}{{y}_{1}}{{z}_{1}}={{x}_{2}}{{y}_{2}}{{z}_{2}}\\\\\left( {16} \right)\left( 7 \right)\left( {48} \right)=\left( {14} \right)\left( {12} \right){{z}_{2}}\\5376=168{{z}_{2}}\\{{z}_{2}}=32\end{array}$$   Since each woman is working at the same rate, we know that when we multiply the number of women $$(x)$$ by the number of the hours a day $$(y)$$ by the number of days they work $$(z)$$, it should always be the same (a constant). (Try it yourself with some easy numbers).   Substitute in all the numbers that we are given and solve for the number we want (days). It would take 32 days for 14 women that work 12 hours a day to paint the mural. In this case, our $$k$$ is 5376, which represents the number of hours it would take one woman alone to paint 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:

Function Direct, Inverse, or Neither Variation
$$y=3x-2$$ Neither:  Direct Variation line must go through $$(0,0)$$. (Note that this is Partial Variation, as shown below).
$$8y=-x$$ Direct:  This is the same as $$\displaystyle y=-\frac{1}{8}x;\,\,\,\,\,\,k=-\frac{1}{8}$$.
 $$x$$ 2 1 0.5 $$y$$ 4 8 16
Inverse:  The product of the $$x$$’s and $$y$$’s is always 8; $$k=8$$.
$$\displaystyle x=\frac{{\frac{4}{5}}}{y}$$ Inverse:  This is the same as $$\displaystyle xy=\frac{4}{5};\,\,\,\,\,k=\frac{4}{5}$$.
$$y=40$$ Neither:  No $$x$$ in the function.
 $$x$$ 0 2 4 $$y$$ 4 6 8
Neither:  Even though this would be a line, there is no $$k$$ such that $$y=kx$$. Also, direct variation line must go through $$(0,0)$$.

# 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 Problem Formula Method Suppose $$x$$ 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 $$z=4$$. \begin{align}x&=ky\sqrt{z}\\-18&=k\left( 2 \right)\sqrt{9}\\-18&=6k\\k&=-3\end{align}                  \begin{align}x&=ky\sqrt{z}\\x&=-3y\sqrt{z}\\10&=-3y\sqrt{4}\\10&=-3y\left( 2 \right)\\y&=\frac{{10}}{{-6}}=-\frac{5}{3}\end{align}   Again, we can set it up almost word for word from the word problem. For the words “varies jointly”, just basically use the “$$=$$” sign, and everything else will fall in place. Solve for $$k$$ first by plugging in the given variables; we get $$k=-3$$. Plug in the new values of $$x$$ and $$z$$ to get the new $$y$$.   We see that $$\displaystyle y=-\frac{5}{3}$$. Really not that bad!

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 $$k$$:

 Joint Variation Problem Math and Notes The area of a triangle is jointly related to both its height and base.   If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area? $$\displaystyle \begin{array}{c}A=kbh\,\,\,\,\,\,\,\text{(original)}\\\,A=k\left( {1.4b} \right)\left( {.9h} \right)\,\,\,\,\,\,\text{(new)}\\\\\,A=k\left( {1.4} \right)\left( {.9} \right)bh\\A=k\left( {1.26} \right)bh\end{array}$$   Remember that when we increase a number by 40%, we are actually multiplying it by 1.4, since we have to add 40% to the original amount. Similarly, when we decrease a number by 10%, we are multiplying it by .9, since we are decreasing the original amount by 10%.   Reduce the original values by the new values, and find the new “multiplier”; we see that there will be a 26% increase in the area ($$A$$ would be multiplied by 1.26, or be 26% greater.) You can put real numbers to verify this, using the formula $$\displaystyle A=\frac{1}{2}bh$$.

Another Joint Variation Word Problem:

 Joint Variation Problem Math and Notes 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 $${{m}^{3}}$$ and girth of 2 meters? $$\begin{array}{l}V=k\text{(height)(girth}{{\text{)}}^{2}}\\V=kh{{g}^{2}}\\\\144=k(20){{(1.5)}^{2}}=45k\\144=45k;\,\,k=3.2\\\\V=kh{{g}^{2}};\,\,V=3.2h{{g}^{2}};\,\,1000=3.2h\cdot {{2}^{2}}\\h=78.125\end{array}$$   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 $$k=3.2$$. Plug in the new values to get the new height. The new height is 78.125 meters.

## 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:

 Combined Variation Problem Math and Notes (a)   $$y$$ varies jointly as $$x$$ and $$w$$ and inversely as the square of $$z$$. Find 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 $$z=4$$. \begin{align}y&=\frac{{kxw}}{{{{z}^{2}}}}\\100&=\frac{{k(2)(4)}}{{{{{(20)}}^{2}}}}=\frac{{8k}}{{400}}\\8k&=(100)(400)\\k&=\frac{{(100)(400)}}{8}=5000\end{align}           \begin{align}y&=\frac{{5000xw}}{{{{z}^{2}}}}\text{ }\,\,\,\leftarrow \text{ answer to (a)}\\\\y&=\frac{{5000(1)(5)}}{{{{4}^{2}}}}\\y&=\,\,\,\frac{{25000}}{{16}}=\,\,\,1562.5\text{ }\,\,\,\leftarrow \text{ answer to (b)}\end{align}   This looks really complicated, and you may get “word problems” like this, but all we do is fill in all the variables we know, and then solve for $$k$$. We know that “the square of $$z$$” is a fancy way of saying $${{z}^{2}}$$.   Remember that what follows the “varies jointly as” is typically on the top of any fraction (this is like a direct variation), and what follows “inversely as” is typically on the bottom of the fraction. And always put $$k$$ on the top! Now that we have the $$k$$, we have the answer to (a) above by plugging it in the original equation. Get the new $$y$$ when we have “new” $$x$$, $$w$$, and $$z$$ values.   For (b), when $$x=1$$, $$w=5$$, and $$z=4$$, $$y=1562.5$$.   (Just plug in).

Combined Variation Word 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 Problem Math and Notes The average number of phone calls per day between two cities used to be approximately jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities.   At a certain point in time, the population of Charlotte was about 1,500,000 and the population of Nashville was about 1,200,000; the distance between the two cities is about 400 miles. During this time, the average number of calls between the cities was about 200,000.    (a)  Find the $$\boldsymbol {k}$$ and write the equation of variation.   (b)  Also, the average number of daily phone calls between Charlotte and Indianapolis (which, during the time, had a population of about 1,700,000) was about 134,000. Find the approximate distance between the two cities, using the combined variation model. \displaystyle \begin{align}C&=\frac{{k\left( {{{P}_{1}}} \right)\left( {{{P}_{2}}} \right)}}{{{{d}^{2}}}}\\200000&=\frac{{k\left( {1500000} \right)\left( {1200000} \right)}}{{{{{400}}^{2}}}}\\k&=\frac{{\left( {200000} \right){{{\left( {400} \right)}}^{2}}}}{{\left( {1500000} \right)\left( {1200000} \right)}}\approx.01778\\C&\approx\frac{{.01778\left( {{{P}_{1}}} \right)\left( {{{P}_{2}}} \right)}}{{{{d}^{2}}}}\,\,\,\,\leftarrow \text{ answer to (a)}\end{align}    \displaystyle \begin{align}134000&=\frac{{.01778\left( {1500000} \right)\left( {1700000} \right)}}{{{{d}^{2}}}}\\134000{{d}^{2}}&=.01778\left( {1500000} \right)\left( {1700000} \right)\\d&\approx581.7 \, \text{miles}\,\,\,\,\,\,\leftarrow \text{ answer to (b)}\end{align}   Set it up almost word for word from the word problem. Remember to put everything on top for “jointly proportional” (including $$k$$) since these are direct variations, and everything on bottom for “inversely proportional”. Solve for $$k$$ first; we get $$k\approx.01778$$.   Plug in the new values to get the distance between the cities ($$d$$). We can actually cross multiply to get $${{d}^{2}}$$, and then take the positive square root get $$d$$.   The distance between Charlotte and Indianapolis is about 581.7 miles. In reality, the distance between these two cities is 585.6 miles, so we weren’t too far off!

Combined Variation Word Problem:

 Combined Variation Problem Math and Notes $$y$$ 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”: \displaystyle \begin{align}y&=\frac{{k{{x}^{3}}z}}{{{{r}^{2}}}}\,\,\,\,\,\,\,\,\text{(original)}\\y&=\frac{{k{{{\left( {2x} \right)}}^{3}}z}}{{{{{\left( {\frac{1}{2}r} \right)}}^{2}}}}\,\,\,\,\,\,\,\,\text{(new)}\end{align}           $$\displaystyle y=\frac{{k8{{x}^{3}}z}}{{\frac{1}{4}{{r}^{2}}}}=\frac{8}{{\frac{1}{4}}}\frac{{k{{x}^{3}}z}}{{{{r}^{2}}}}=\left( {\frac{8}{1}\cdot \frac{4}{1}} \right)\frac{{k{{x}^{3}}z}}{{{{r}^{2}}}}=32\frac{{k{{x}^{3}}z}}{{{{r}^{2}}}}$$   Set it up with everything on top for “varies jointly” (including $$k$$) since these are direct variations, and everything on bottom for “varies inversely”. Now substitute “$$2x$$” for $$x$$, since $$x$$ is doubled, and “$$\displaystyle \frac{1}{2}r$$” for $$r$$, since $$r$$ is halved.   Simplify to see that we have a 32 in front of the old variation. (Don’t forget to flip and multiply when we divide by a fraction.) The effect on $$y$$ would be 32 times greater, or multiplied by 32.

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 Math and Notes 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 $$x=5$$? \displaystyle \begin{align}x&=\frac{{{{k}_{1}}}}{y}\\3&=\frac{{{{k}_{1}}}}{{12}}\\{{k}_{1}}&=36\end{align}  \displaystyle \begin{align}x&=\frac{{36}}{y}\\5&=\frac{{36}}{y}\\y&=\frac{{36}}{5}\end{align}                     \displaystyle \begin{align}x&={{k}_{2}}z\\3&={{k}_{2}}12\\{{k}_{2}}&=\frac{1}{4}\end{align}     \displaystyle \begin{align}x&=\frac{1}{4}z\\5&=\frac{1}{4}z\\z&=20\end{align} $$y+z=27.2$$   Set up 2 variation equations, the first using $${{k}_{1}}$$ and the second $${{k}_{2}}$$ as constants. The first equation is inverse variation, and the second equation is direct variation.   Solve for $${{k}_{1}}$$ and $${{k}_{2}}$$ separately, using the fact that $$y$$ and $$z$$ are both 12 when $$x=3$$. We get $${{k}_{1}}=36$$ and $$\displaystyle {{k}_{2}}=\frac{1}{4}$$. Put the constants back in the equation and solve for $$y$$ and $$z$$ when $$x=5$$. Cross multiply to see that $$\displaystyle y=\frac{{36}}{5}=7.2$$ and $$\displaystyle z=20$$, so $$y+z=27.2$$.

# 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:

 Partial Variation Problem Solution $$y$$ is partly constant and partly varies (directly) with $$x$$.   When $$y=4,\,x=2$$, and when $$y=16,\,x=4$$.   Find an equation connecting $$y$$ and $$x$$, and find $$y$$ when $$x=6$$. Since $$y$$ varies partly with a constant and directly with $$x$$, we have $$y={{k}_{1}}x+{{k}_{2}}$$. Use a system of equations to solve for $${{k}_{1}}$$ and $${{k}_{2}}$$ by plugging in what we know: \begin{align}y&={{k}_{1}}x+{{k}_{2}}\\4&=2{{k}_{1}}+{{k}_{2}}\\16&=4{{k}_{1}}+{{k}_{2}}\end{align}            Use elimination to solve:  $$\begin{array}{l}-4=-2{{k}_{1}}-{{k}_{2}}\\\underline{{\,16=\,\,\,\,4{{k}_{1}}\,+{{k}_{2}}}}\\12\,=\,\,\,\,2{{k}_{1}};\,\,\,\,{{k}_{1}}=6\\{{k}_{2}}=16-4\left( 6 \right)=-8\end{array}$$ Now we have $$\,y=6x-8$$. When $$x=6,\,\,y=6\left( 6 \right)-8=28$$. $$y$$ partly varies directly with $$x$$ and also partly varies inversely with $$x$$.   When $$y=2,\,x=-2$$, and when $$y=7,\,x=-4$$.   Find an equation connecting $$y$$ and $$x$$, and Find $$y$$ when $$x=8$$. Since $$y$$ partly varies directly with $$x$$ and partly varies inversely with $$x$$, we have $$\displaystyle y={{k}_{1}}x+\frac{{{{k}_{2}}}}{x}$$. Use a system to solve for $${{k}_{1}}$$ and $${{k}_{2}}$$:    \displaystyle \begin{align}\,\,\,\,y&={{k}_{1}}x+\frac{{{{k}_{2}}}}{x}\\\,2&=-2{{k}_{1}}+\frac{{{{k}_{2}}}}{{-2}}\\\,\,7&=-4{{k}_{1}}+\frac{{{{k}_{2}}}}{{-4}}\end{align}         Use substitution to solve: \displaystyle \begin{align}2&=-2{{k}_{1}}+\frac{{{{k}_{2}}}}{{-2}}\\{{k}_{2}}&=-4-4{{k}_{1}}\\7&=-4{{k}_{1}}+\frac{{-4-4{{k}_{1}}}}{{-4}}\\-28&=16{{k}_{1}}-4-4{{k}_{1}}\\{{k}_{1}}&=-2;\,\,\,{{k}_{2}}=4\end{align} Now we have $$\displaystyle \,y=-2x+\frac{4}{x}$$. When $$\displaystyle x=8,\,\,\,y=-2\left( 8 \right)+\frac{4}{8}=-\frac{{31}}{2}$$. The cost of attending a fair consists of a fixed entrance cost $$f$$ of $10, and a charge for riding rides, which is proportional to the number of rides ridden. If the cost of attending the fair is$24 when 7 rides are ridden, find the cost of riding 10 rides. Since $$y$$ varies partly with a constant $$f$$ and directly with the number of rides ridden, say $$x$$, we have $$y=cx+10$$, where $$c$$ (cost of each ride) is a constant of variation. Plug in what we know: \displaystyle \begin{align}y&=cx+10\\24&=c\left( 7 \right)+10\\c&=\frac{{14}}{7}=2\end{align} The cost of riding 10 rides is $$y=2x+10=2\left( {10} \right)+10=\30$$.

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!