Math Lesson 4.5.3 - Types of Variation - Direct Variation

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Welcome to our Math lesson on Types of Variation - Direct Variation, this is the third lesson of our suite of math lessons covering the topic of Variation. Types of Variation, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Types of Variation - Direct Variation

A direct variation is a variation that can be expressed through the general formula

y = k × x

where x and y represent possible values of the two quantities involved in the variation while k is the constant of proportionality that provides the value of each individual ratio or rate involved in variation. More specifically, the constant k represents the quotient of division between any vertical coordinate to the corresponding horizontal coordinate in the variation graph.

Obviously, direct variation is very similar to direct proportion we discussed in tutorial 4.3. The graph that geometrically represents the direct variation is a straight line that passes through the origin.

Example 1

An employee is paid $25/hour. Plot the graph indicating the variation of his daily salary from the working hours.

Solution 1

This is a situation involving a direct variation as the daily salary is directly proportional to the number of working hours (the more working hours are committed, the higher the daily salary).

If we denote the working hours by x and the daily salary by y, we obtain the variation

y = 25 · x

where 25 is the value of the constant of proportionality k given in the general formula of direct variation

y = k · x

We can stop to the number 8 in the values of working hours as an employee usually works 8 hours a day when employed full time. Thus, the graph showing this (direct) variation is

$Math Tutorials: Variation. Types of Variation Example$

From the graph, it is evident that at the end of the working day, the employee earns $200.

We can find any missing value when we know the variation formula, as in the example below.

Example 2

The quantity x and y vary directly and y = 6 when x = 18. What is the value of x when y = 102?

Solution 2

First, we find the constant k of variation (proportionality). Since the general formula of direct variation is y = k · x, we obtain

6 = k × 18
k = 6/18
= 1/3

Thus, the formula of this specific variation is

y = 1/3 × x

Now, substituting y = 102 in the above formula yields

102 = 1/3 × x
x = 102 ÷ 1/3
= 102 × 3/1
= 102 × 3
= 306

More Variation. Types of Variation Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
4.5	Variation. Types of Variation
Lesson ID	Math Lesson Title	Lesson	Video Lesson
4.5.1	Definition of Variation
4.5.2	The Difference between Proportion and Variation
4.5.3	Types of Variation - Direct Variation
4.5.4	Types of Variation - Inverse Variation
4.5.5	Types of Variation - Joint Variation
4.5.6	Summarizing the Properties of Variation

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