Lesson 113. LESSON PRESENTATION Direct Variation. LESSON PRESENTATION Inverse Variation. Example 113.1. Example 113.2. Example 113.4. Example 113.3. Example 113.5. Direct Variation ~ Inverse Variation. Direct Variation.
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In mathematics, two quantities are said to be proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.
Proportion also refers to the equality of two ratios.
For example: Suppose at a particular athletic event there are twice as many boys as girls. We can describe this relations ship using the equation
We can use this equation to find the number of boys at the event if we know the number of girls. In this case the “2” is the constant ratio.
Given two variables x and y, y is (directly) proportional to x (y varies directly as x) if there is a non-zero constant k such that
is called the proportionality constant or constant of proportionality (or constant of variation).
Example 113.1 The mass of a substance varies directly as the volume of the substance. If the mass of 2 liters of the substance is 10 kilograms, what will be the volume of 35 kilograms of the substance?
1) We first recognize the words “varies directly” that imply the relationship
2) Read the problem again to find the values of M and V that can be used to find the value of the constant k. Use these values to solve for k.
3) Replace k in the equation with the value we found.
4) Reread the problem again to find that we are asked to find the value of V when M is 35.
Example 113.2 The distance traversed by a car traveling at a constant speed is directly proportional to the time spent traveling. If the car goes 75 kilometers in 5 hours, how far will it go in 7 hours?
Example 113.3 Under certain conditions the pressure of a gas varies directly as the temperature. When the pressure is 800 pascals, the temperature is 400 K. What is the temperature when the pressure is 400 pascals.
The temperature is 200 K when the pressure is 400 pascals.
When a problem states that one variable varies inversely as the other variable or that the value of one variable is inversely proportional to the value of the other variable, an equation of the form
Let’s compare the equations for direct and inverse variation.
Example 113.4 Under certain conditions, the pressure of a perfect gas varies inversely as the volume. When the pressure of a quantity of gas 7 pascals, the volume is 75 liters. What would be the volume if the pressure is increased to 15 pascals?
Example 113.5 To travel a fixed distance, the rate is inversely proportional to the time required. When the rate is 60 kilometers per hour, the time required is 4 hours. What would be the time required for the same distance if the rate is increased to 80 kilometers per hour?
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