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ZingPath: Ratio, Rate, and Proportion

Directly Varying Quantities and Their Graphs

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Ratio, Rate, and Proportion

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Directly Varying Quantities and Their Graphs


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Directly varying quantities given graphs, tables, or statements are identified and problems involving direct variation are solved.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Identify direct variations given graphs, tables, or statements.
  • Write an equation that relates directly varying quantities.
  • Calculate the constant of variation (constant of proportionality).
  • Solve problems involving direct variation (proportional change).

Everything You'll Have Covered

Direct variation is a type of proportionality relation between two varying quantities. Two quantities are proportional if they are constant multiples of each other. More specifically, two variables x and y vary directly if there is a nonzero constant k such that Y = K . X . The constant k is called the constant of variation.

Examples of direct variation:

  • The cost of gasoline varies directly with the number of gallons purchased. If C represents the cost in dollars, Q the number of gallons purchased, and p the price per gallon, then C = P . Q.
  • The relationship between circumference and diameter of a general circle is given by . Therefore, circumference and diameter vary directly. The constant of proportionality is equal to
  • The length of an object, measured in feet, varies directly with its length, measured in inches. The variation equation is because there are 12 inches to a foot.

Directly varying quantities are commonly represented by statements, graphs, or tables. The form of each representation is given below. For example, given that x and y vary directly, the following statement provides enough information to find the constant of proportionality:

y is 20 when x is 10.

Graphs can also be used to represent direct variation, in which case the graph must be a straight line and pass through the origin. If the graph is a straight line, but does not pass through the origin, then the relationship it represents cannot be a direct variation.

Direct variation should not be confused with linearity. Two quantities are linearly related if they have a constant ratio of change. This constant ratio is called the rate of change, or slope:

This is different from the condition imposed by direct variation in that the quantities themselves have a constant ratio.

Finally, note that direct variation is sometimes called direct proportionality, in which case the constant of variation is called the constant of proportionality.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Students should know the concepts of ratio, and rate; be able to solve proportions and linear equations; and be able to graph linear equations.
Course Pre-Algebra
Type of Tutorial Concept Development
Key Vocabulary application of direct variation, constant of proportionality, constant of variation