Direct/Indirect VariationDate: 6/12/96 at 23:49:12 From: Anonymous Subject: Direct and Indirect variation Please help me to understand this concept. I have no clue about direct and indirect variation. Thanks! Date: 6/13/96 at 15:7:43 From: Doctor Anthony Subject: Re: Direct and Indirect variation Variation, in general, will concern two variables, say height and weight of a person, and how when one of these changes, the other might be expected to change. We have direct variation if the two variables change in the same sense, i.e. if one increases, so does the other. We have indirect variation if one going up causes the other to go down. An example of this might be speed and time to do a particular journey, so the higher the speed the shorter the time. Normally we let x be the independent variable and y the dependent variable, so that a change in x produces a change in y. For example, if x is number of motor cars on the road and y the number of accidents, then we expect an increase in x to cause an increase in y. (This obviously ceases to apply if number of cars is so large that they are all stationary in a traffic jam.) When x and y are directly proportional then doubling x will double the value of y, and if we divide these variables we get a constant result. Since if y/x = k then (2y/2x) = k where k is called the constant of proportionality We could also write this y = kx Thus if I am given the value of x, I multiply this number by k to find the value of y. Example: Given that y and x are directly proportional and y = 2 when x = 5, find the value of y when x = 15. We first find value of k, using y/x = k 2/5 = k Now use this constant value in the equation y = kx for situation when x = 15 y = (2/5)*15 = 30/5 = 6 If you want to do this quickly in your head, you could say, x has been multiplied by a factor 3 (going from 5 to 15), so y must also go up by a factor of 3. That means y goes from 2 to 6. Indirect Variation. We gave an example of inverse proportion above, namely speed and time for a particular journey. In this case if you double the speed you halve the time. So the product speed x time = constant In general, if x and y are inversely proportional then the product xy will be constant. xy = k or y = k/x Example: If it takes 4 hours at an average speed of 90 km/hr to do a certain journey, how long would it take at 120 km/hr k = speed*time = 90*4 = 360 (k in this case is the distance) Then time = k/speed = 360/120 = 3 hours. To do this in your head, you could say that speed has changed by a factor 4/3, so time must change by a factor 3/4. However, for the usual type of problem, go through the steps I outlined above. I hope these examples have made the idea of variation (both direct and inverse) reasonably clear. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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