Direct and Indirect VariationDate: 07/09/98 at 17:42:08 From: Eric Sparrow Subject: Word problems I am having trouble with direct and indirect variation. I don't really understand the difference. I have tried to work the problems over and over, and I still can't get it. I would appreciate it if you could help me with this problem. Thank you! Date: 07/09/98 at 19:54:50 From: Doctor Pat Subject: Re: Word problems Eric, Many simple problems can be examples of both direct and indirect variation, depending on which variable you make a constant. For example, the traditional formula distance = rate x time (or D = R x T) is an example of direct variation if we make the rate or the time a constant. Let's say you walk 5 mph. The distance you walk is then directly proportional to the time you walk. If you walk twice as long, you go twice as far, and the ratio of the distance and time is always the same. In this case the rate is the "constant of proportionality." Now let's change it to an indirect variation. To do so, we merely keep the distance constant. Suppose you live five miles from school. If you walk slower, it takes longer. As your speed goes up, the time goes down. That's why it is called indirect variation (it is often called inverse variation also). Two variables, call them x and y, are directly related if they can be expressed so that their ratio is always a constant. This means either y/x = k or y = kx (where k is a constant). The two equations are really the same. Just multiply both sides of the left one by x and you get the one on the right. The same two variables are indirectly related if their product is always a constant. This means either xy = k or x = k/y or y = k/x. If you have indirect variation, problems usually tell you the values of x and y at two different times (except one of them is left for you to find). They often look something like this: If x and y are indirectly related, and y = 6 when x = 4, find y when x = 8. The key is to recognize that we have two x,y pairs. For indirect variation we just set up an equation that says: first x * first y = last x * last y Using the numbers in the example problem, this becomes: 6 * 4 = 8 * last y which you can easily solve to get y = 3. I hope this helps. If I didn't make it clear, please write back and give examples of some of the problems, and what you do and don't understand. Good luck, - Doctor Pat, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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