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

11/12/2014

D ir ect, Inver se, Joi nt and Combi ned Var i ati on - She Loves M ath

irect, Inverse, Joint and Combined Variation his section overs: Direct or

Proportional Variation Inverse or Indirect Variation Joint 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 you’ll study are: Direct Variation, Variation, where both variables either increase or decrease together Inverse or Indirect or Indirect Variation, Variation, where when one of the variables increases, the other one decreases Joint Variation, Variation, where more than two variables are related directly Combined Variation, Variation, which involves a combination of direct or joint variation, and and i ndirect variation hese sound like a lot of fancy math words, but it’s really not too bad. Here are some examples f direct direct and inverse inversev ariation: Direct: Direct: The number of dollars I make varies directly (or you can say varies proportionally) proportionally) with how much I work. Direct: Direct: The length of the side a square varies directly with the perimeter of the square. Inverse: 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: Inverse: The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running. Inverse: Inverse: My GPA may vary directly inversely with the number of hours I watch TV.

irect or Proportional Variation http://www.shelovesm ath.com /al gebr a/begi nning- al gebr a/di r ect- inver se- and- j oint- var iation/

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

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 = m x ” line, here the ratio of y to x i s the slope (m). With direct variation, the y -intercept is always 0 (zero); this is ow it’s defined. irect variation problems are typically written: y = k x

where k is the ratio of y to x ( which is the same as the slope or rate).

ome problems will ask for that k value (which is called the constant of variation or constant of roportionality – it’s like a slope!); others will just give you 3 out of the 4 values for x and y and you can imply set up a ratio to find the other value. I’m thinking the k comes from the word “constant” in another anguage. emember the example of making $10 an hour at the mall ( y = 10 x )? 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 either he top or bottom of the ratio, or on the same side. This will look like the following. Don’t 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 y varies directly with x , and y = 20 when x = 2. Find y when x = 8. (Note that this may be also e written “y is proportional to x , and y = 20 when x = 2. Find y when 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 for x and y . ormula Method:

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

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roportion Method:

t’s really that easy. Can you see why the proportion method can be the preferred method, unless you are sked to find the k constant in the formula? gain, if the problem asks for the equation that models this situation, it would be “ y = 10x“.

irect Variation Word Problem: 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: et’s do this problem using both the Formula Method and the Proportion Method:


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irect Variation Word Problem:

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: et’s do this with the proportion model:

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

nverse or Indirect Variation http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/

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nverse or Indirect Variation is refers to relationships of two variables that go in the opposite direction. et’s 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 the x v ariable goes up, the y goes down, and when you multiply the x w ith the y , we lways get the same number? (Note that this is different than a negative slope, since with a negative lope, we can’t multiply the x ’s and y ’x to get the same number). o the formula for inverse or indirect variation is:

where k is 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 Functions section here .


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nverse Variation Word Problem: o 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. he problem may also be worded like this: Let

= 3,

= 4, and

= 6. Let y vary inversely as 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 for x and 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 proportional to 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: et’s use the product method:


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Work” Inverse Proportion Word Problem: ere’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 quations Section. 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 compare x’s to unique y’s – we’ll discuss ater in the Introduction to Functions section) and determine if they are direct, inverse, or neither:


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oint Variation and Combined Variation oint variation is just like direct variation, but involves more than one other variable. Let’s do a joint ariation problem: upposed x varies jointly with y and the square root of z. When x = –18 and y = 2, then z = . Find y when x = 10 and z = 4. et’s set this up like we did with direct variation, find the k, and then solve for y :

ombined variation involves 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 what’s missing. Here is the type of problem you may get: (a) y varies jointly as x and w and inversely as the square of z. Find the equation of variation hen y = 100, x = 2, w = 4, and z = 20. http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/

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

ombined Variation Word Problem: he volume of wood in a tree ( V ) varies directly as the height ( h) and inversely as 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 be jointly proportional to the opulations of the cities, and inversely proportional to 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 it’s 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: “If x v aries inversely with y and varies directly with z, and if y and z are both 12 when x = 3, what is the alue of y + z when x = 5”. I found that I had to solve it setting up two variation equations with two ifferent k ‘s (otherwise you can’t really get an answer). So watch the wording of the problems. ere is how I did this problem:

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


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

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