Direct, Inverse, Joint and Combined Variation - She Loves Math

7/25/2019 Direct, Inverse, Joint and Combined Variation - She Loves Math

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http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/

irect, Inverse, Joint and Combined Variation

his section

overs:

Direct or

Proportional Variation

Inverse or Indirect VariationJoint and Combined Variation

Direct Variation Word Problems

Inverse Variation Word Problems

Combined Variation Word Problems

hen you start studying algebra, you will also study how two (or more) variables can relate to each other

pecifically. The cases youll study are:

Direct Variation, where both variables either increase or decrease togetherInverse orIndirect 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, andindirect variation

hese sound like a lot of fancy math words, but its really not too bad. Here are some examples

f directand inversevariation:

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.

irect or Proportional Variation


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http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/ 2

hen two variables are related directly, the ratio of their values is always the same. So as one goes up, so

oes the other, and if one goes down, so does the other. Think of linear direct variation as a y= mx line,

here the ratio of yto xis the slope(m). With direct variation, the y-intercept is always 0 (zero); this is

ow its defined.

irect variation problems are typically written:

y= kx where kis the ratio of yto x(which is the same as the slopeor rate).

ome problems will ask for thatkvalue (which is called the constant of variationor constant of

roportionality its like a slope!); others will just give you 3 out of the 4 values for xand yand you can

imply set up a ratio to find the other value. Im thinking the kcomes from the word constant in another

anguage.

emember the example of making $10 an hour at the mall (y= 10x)? This is an example of direct

ariation,since the ratio of how much you make to how many hours you work is always constant.

e can also set up direct variation problems in a ratio, as long as we have the same variable in eitherhe top or bottomof the ratio, or on the same side. This will look like the following. Dont let this scare

ou; the subscripts just refer to the either the first set of variables , or the second .

irect Variation Word Problem:

o we might have a problem like this:

he value of yvaries directly withx, and y= 20 when x= 2. Find ywhenx= 8. (Note that this may be also

e written yis proportional tox, and y= 20 whenx= 2. Find ywhen x= 8.)

olution:

e can solve this problem in one of two ways, as shown. We do these methods when we are given any

hree of the four values forxand y.

ormula Method:


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roportion Method:

ts really that easy. Can you see why the proportion method can be the preferred method, unless you are

sked to find the kconstant in the formula?

gain, if the problem asks for the equation that models this situation, it would be y = 10x.


he amount of money raised at a school fundraiser is directly proportional to the number of people who

ttend. Last year, the amount of money raised for 100 attendees was $2500. How much money will be

aised if 1000 people attend this year?

olution:

ets do this problem using both the Formula Methodand the Proportion Method:


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rady bought an energy efficient washing machine for her new apartment. If she saves about 10 gallons of

ater per load, how many gallons of water will she save if she washes 20 loads of laundry?

olution:

ets do this with the proportion model:

ee how similar these types of problems are to the Proportionsproblems we did earlier?

nverse or Indirect Variation


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nverseor IndirectVariation is refers to relationships of two variables that go in the opposite direction.

ets supposed you are comparing how fast you are driving (average speed) to how fast you get to your

chool. You might have measured the following speeds and times:

Note that means approximately equal to).

o you see how when thexvariable goes up, the ygoes down, and when you multiply the xwith the y, we

lways get the same number? (Note that this is different than a negative slope, since with a negative

lope, we cant multiply thexs and yx to get the same number).

o the formula for inverse or indirect variation is:

where kis always the same number.

ere is a sample graph for inverse or indirect variation. This is actually a type of Rational

unction(function with a variable in the denominator) that we will talk about in the Rational Expressions

nd Functionssection here .


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nverse Variation Word Problem:

o we might have a problem like this:

The value of y varies inverselywith x, and y= 4 when x= 3. Findxwhen y= 6.

he problem may also be worded like this:

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

olution:

e can solve this problem in one of two ways, as shown. We do these methods when we are given any

hree of the four values forxand y.

ormula Method:


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roduct Rule Method:

nverse Variation Word Problem:

or the Choir fundraiser, the number of tickets Allie can buy is inversely proportionalto the price of the

ickets. She can afford 15 tickets that cost $5 each. How many tickets can Allie buy if each cost $3?

olution:

ets use the product method:


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Work Inverse Proportion Word Problem:

eres a more advanced problem that uses inverse proportions in a work word problem; well see more

work problems here in the Systems of Linear Equations Sectionand here in the Rational Functions and

quationsSection.

f 16 women working 7 hours day can paint a mural in 48 days, how many days will it take 14 women

orking 12 hours a day to paint the same mural?

olution:

he three different values are inversely proportional; for example, the more women you have, the less

ays it takes to paint the mural, and the more hours in a day the women paint, the less days they need to

omplete the mural:

ou might be asked to look at functions (equations or points that comparexsto unique ys well discuss

ater in the Introduction to Functionssection) and determine if they are direct, inverse, or neither:


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oint Variation and Combined Variation

oint variationis just like direct variation, but involves more than one other variable. Lets do a joint

ariation problem:

upposedx varies jointly with yand the square root of z. Whenx= 18 and y= 2, then z=

. Find ywhenx= 10 andz= 4.

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

ombined variationinvolves a combination of direct or joint variation, and indirect variation. Since these

quations are a little more complicated, you probably want to plug in all the variables, solve for k, and then

olve back to get whats missing. Here is the type of problem you may get:

(a) y varies jointlyas xand wand inverselyas the square of z. Find the equation of variation

hen y= 100,x= 2, w= 4, andz= 20.


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(b) Then solve for ywhenx= 1, w= 5, andz= 4.

ets solve:

ombined Variation Word Problem:

he volume of wood in a tree (V) varies directlyas the height ( h) and inverselyas the square of the girth

g). If the volume of a tree is 144 cubic meters when the height is 20 meters and the girth is 1.5

eters, what is the height of a tree with a volume of 1000 and girth of 2 meters?

olution:


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ombined Variation Word Problem:

he average number of phone calls per day between two cities has found to bejointly proportionalto the

opulations of the cities, and inversely proportionalto the square of the distance between the two cities.

he population of Charlotte is about 1,500,000 and the population of Nashville is about 1,200,000, and the

istance between the two cities is about 400 miles. The average number of calls between the cities is

bout 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.

olution:

his one looks really tough, but its really not that bad if you take it one step at a time:


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ne word of caution: I found a variation problem in an SAT book that stated something like this:Ifxvaries inversely with yand varies directly with z, and if yand zare both 12 when x= 3, what is the

alue of y+zwhenx= 5. I found that I had to solve it setting up two variation equationswith two

ifferent ks (otherwise you cant really get an answer). So watch the wording of the problems.

ere is how I did this problem:

ere doing really difficult problems now but see how, if you know the rules, they really arent bad at all?

Direct, Inverse, Joint and Combined Variation - She Loves Math

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