Variation, Direct and Inverse

A variable is something that varies among components of a set or population, such as the height of high school students. Two types of relationships between variables are direct and inverse variation. In general, direct variation suggests that two variables change in the same direction. As one variable increases, the other also increases, and as one decreases, the other also decreases. In contrast, inverse variation suggests that variables change in opposite directions. As one increases, the other decreases and vice versa.

Direct Variation

Consider the case of someone who is paid an hourly wage. The amount of pay varies with the number of hours worked. If a person makes $12 per hour and works two hours, the pay is $24; for three hours worked, the pay is $36, and so on. If the number of hours worked doubles, say from 5 to 10, the pay doubles, in this case from $60 to $120. Also note that if the person works 0 hours, the pay is $0. This is an important component of direct variation: When one variable is 0, the other must be 0 as well.

So, if two variables vary directly and one variable is multiplied by a constant, then the other variable is also multiplied by the same constant. If one variable doubles, the other doubles; if one triples, the other triples; if one is cut in half, so is the other. Algebraically, the relationship between two variables that vary directly can be expressed as y = kx, where the variables are x and y, and k represents what is called the constant of proportionality. (Note that this relationship can also be expressed as or = with also representing a constant.) In the preceding example, the equation is y 12x, with x representing the number of hours worked, y representing the pay, and 12 representing the hourly rate, the constant of proportionality.

Graphically, the relationship between two variables that vary directly is represented by a ray that begins at the point (0, 0) and extends into the first quadrant. In other words, the relationship is linear, considering only positive values. See part (a) of the figure on the next page. The slope of the ray depends on the value of k, the constant of proportionality. The bigger k is, the steeper the graph, and vice versa.

Inverse Variation

When two variables vary inversely, one increases as the other decreases. As one variable is multiplied by a given factor, the other variable is divided by that factor, which is, of course, equivalent to being multiplied by the reciprocal (the multiplicative inverse) of the factor. For example, if one variable doubles, the other is divided by two (multiplied by one-half); if one triples, the other is divided by three (multiplied by one-third); if one is multiplied by two-thirds, the other is divided by two-thirds (multiplied by three-halves).

Consider a situation in which 100 miles are traveled. If traveling at an average rate of 5 miles per hour (mph), the trip takes 20 hours. If the average rate is doubled to 10 mph, then the trip time is halved to 10 hours. If the rate is doubled again, to 20 mph, the trip time is again halved, this time to 5 hours. If the average rate of speed is 60 mph, this is triple 20 mph. Therefore, if it takes 5 hours at 20 mph, 5 is divided by 3 to find the travel time at 60 mph. The travel time at 60 mph equals , or 1⅔ hours.

Algebraically, if x represents the rate (in miles per hour) and y represents the time it takes (in hours), this relationship can be expressed as xy = 1000 or or . In general, variables that vary inversely can be expressed in the following forms: , or .

The graph of the relationship between quantities that vary inversely is one branch of a hyperbola . See part (b) of the figure. The graph is asymptotic to both the positive x -and y -axes. In other words, if one of the quantities is 0, the other quantity must be infinite. For the example given here, if the average rate is 0 mph, it would take forever to go 100 miles; similarly, if the travel time is 0, the average rate must be infinite.

Many pairs of variables vary either directly or inversely. If variables vary directly, their quotient is constant; if variables vary inversely, their product is constant. In direct variation, as one variable increases, so too does the other; in inverse variation, as one variable increases, the other decreases.

see also Inverses; Ratio, Rate, and Proportion.

Bob Horton

Bibliography

Foster, Alan G., et al. Merrill Algebra 1: Applications and Connections. New York: Glencoe/McGraw Hill, 1995.

———. Merrill Algebra 2 with Trigonometry Applications and Connections. New York: Glencoe/McGraw Hill, 1995.

Larson, Roland E., Timothy D. Kanold, and Lee Stiff. Algebra I: An Integrated Approach. Evanston, IL: D.C. Heath and Company, 1998.