Math Lesson 4.5.2 - The Difference between Proportion and Variation

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Welcome to our Math lesson on The Difference between Proportion and Variation, this is the second 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.

The Difference between Proportion and Variation

As we said above, the concepts of proportion and variation are very similar. However, despite the similarity, they have their differences that make these two concepts distinguishable from each other.

The first difference lies in the fact that a proportion does not necessarily involve related terms, i.e. the existence of one term of proportion does not necessarily imply the existence of the other term. For example, when we say that three sisters A, B and C share everything the family owns in the ratio 3:4:5, we can form the proportion A:B:C = 3:4:5, but this proportion does not necessarily involve the existence of money to share. They can share everything else but money. Hence, when we say that an amount of $1200 has to be shared among the three sisters in the way described above, we are not referring to a variation but to a proportion instead, as the existence of the three sisters does not necessarily imply the existence of money to share.

The second difference lies in the fact that proportion is limited to a finite number of terms while variation includes a much wider range of terms related with each other through a formula. This leads to the plotting of a graph that includes all possible ordered pairs, where the first of values in each pair is always from the first set of values, while the second term from the second set. Variation therefore is a more inclusive concept than proportion, as practically it includes an infinite number of combinations which meet the condition indicated in the formula.

Many situations we have used in the previous tutorial of this chapter to illustrate proportions, are in fact examples of variation. For example, the number of workers needed to complete a job versus the time needed to do the job can be considered either as an inverse proportion when we use a real-life approach (as the number of workers involved is practically finite), but also as a variation if we approach the situation mathematically, where the number of people involved in the process can be assumed as infinite (or when we consider not only the whole values). In the second case, we can plot a graph where the line is continuous, while if we consider the situation from the proportion viewpoint, the graph represents a set of dots representing the number of people versus the time taken to complete the job. Look at the figures below, where the relationship Number of people = 12/Time taken is shown based on the two above-mentioned approaches.

Proportion approach
Here, the number of people is discrete, i.e. it can only be a whole number. As a result, we cannot extend the graph more on the right of 12, as there cannot be less than one person working. As for the left part of the graph, it contains more points as the number of people can increase as much as we want - an action that brings a reduction in the working time. $Math Tutorials: Variation. Types of Variation Example$
Variation approach
This time, we neglect the restriction of the discrete number of people and we assume the two values as real numbers. Therefore, the graph will be an uninterrupted line (in tutorial 4.3 we called it "hyperbola"). $Math Tutorials: Variation. Types of Variation Example$

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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