Overview
Direct and inverse variation refer to relationships between variables, so that when one variable changes the other variable changes by a specified amount. Both direct and inverse variation can be applied in many different ways.
Figure 1: Definitions of direct and inverse variation.
Direct Variation
Direct variation means that as one variable increases, another variable increases by a specific amount, called a constant. Suppose that when x equals 1, y equals 2; x equals 2, y equals 4; x equals 3, y equals 6; and so on. If x doubles, then y also doubles. Another way to describe this relationship is that y varies directly as x. Still another way to describe this relationship in symbol form is that y =2x. Suppose that when a = 1, b = 3; when a = 2, b = 4; when a = 3, b = 6, and so on. In symbol form, b = 3a, and b varies directly as a. In general form, y = kx, and k is called the constant of variation.
Figure 2: Direct variation has a constant rate of change. As x increases, y increases.
Applications of Direct Variation
There are many real-world examples of direct variation. Suppose that a car is traveling at a constant speed of 60 miles per hour. After 1 hour, it travels 60 miles, after 2 hours, it travels 120 miles, and so on. Similarly, suppose that a person makes $10.00 an hour. Their paycheck varies directly with the number of hours they work, so a person working 40 hours will make 400 dollars, working 80 hours will make 800 dollars, and so on. The graph of the values of direct variation will follow a straight line.
Inverse Variation
Inverse variation means that as one variable increases, the other variable decreases. In equations of inverse variation, the product of the two variables is a constant. Suppose when x equals 3, y equals 20; when x equals 6, y equals 10; and when x equals 12, y equals 5. The relationship in words is that doubling x causes y to halve. The product of x and y, xy, equals 60, so y = 60/x. Suppose that when x equals 2, y equals ½; when x equals 3; y equals 1/3; and when x equals 4; y equals ¼. The product of xy is 1, and x and y are in a reciprocal relationship. In general symbol form y = k/x, where k is a positive constant. The constant k is called the constant of proportionality.
Figure 3: In this example of inverse variation, as the speed increases (y), the time it takes to get to a destination (x) decreases.
Applications of Inverse Variation
There are also many real-world examples of inverse variation. Suppose it takes 4 hours for 20 people to do a fixed job. How long will it take 25 people? Time varies inversely as the number of people involved, so if T = k/n, T is 4, and n is 20, then k will equal 20∙4, or 80. If n is 25, and k is 80, then T equals 80/25 or 3.2. Similarly, suppose the current I is 96 amps and the resistance R is 20 ohms. What is the current when R equals 60 ohms? The current varies inversely as the resistance in the conductor, so if I = V/R, I is 96, and R is 20, then V will equal 96∙20 or 1920. When V at 1920 is divided by R at 60, then I, the current, is equal to 32 amps.
Figure 4: One of the applications of inverse variation is the relationship between the strength of an electrical current (I) to the resistance of a conductor (R).
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Oakdale, CA: visit Tutoring in Oakdale, CA