Math Review of Direct and Inverse Variation

Math Review of Direct and Inverse Variation

Math Review of Direct and Inverse Variation 150 150 Deborah

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.

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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.

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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.

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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).

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