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Pre-Algebra

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

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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).

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.

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 |