Math Lesson 4.5.4 - Types of Variation - Inverse Variation

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Welcome to our Math lesson on Types of Variation - Inverse Variation, this is the fourth 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 - Inverse Variation

Inverse variation is similar to inverse proportion discussed in tutorial 4.3. By definition, two quantities vary inversely when their product is always constant.

Mathematically, we express an inverse variation as

x × y = k

y = k/x

where x and y are the quantities involved in the variation and k is the constant of proportionality.

For example, the number of workers hired to do a job when multiplied by the time taken always provides a constant number, which indicates the size of the job to be done.

As a special case of inverse variation, we can mention inverse proportion, which is an inverse variation between two quantities in two different instants (1) and (2). In this case, we can write

x₁ × y₁ = x₂ × y₂ = k

x₁/x₂ = y₂/y₁

Obviously, the graph of an inverse variation is an uninterrupted hyperbola, while that of inverse proportion shows a finite number of points in the graph - the points whose coordinates represent the quantities involved in the proportion. Earlier we discussed both of these graphs in the example regarding the number of workers hired to do a job versus time taken, where the proportion and variation approach were used to express the data.

Example 3

Four identical pipes can fill a tank in 2 hours.

Plot the graph of the hours needed to fill the tank depending on the number of pipes available.
From the graph, find out how long it takes if we use 16 of the same pipes to fill the tank completely.

Solution 3

This is an example of inverse variation, where the relationship between the number of pipes used to fill the tank versus the time taken is
Number of pipes used × Time taken to fill the tank = Constant
We express the time taken to fill the tank by x and the number of pipes by y. hence, from the clues, we can calculate the constant k first, as
x × y = k
2h × 4pipes = k
k = 8
Therefore, the formula that expresses this inverse variation is
y = 8/x
Now, let's plot the graph by taking a number of values as shown in the table below. $Math Tutorials: Variation. Types of Variation Example$
From the graph (and table) it is clear that if there are 16 pipes available, they can fill the tank in 0.5 h. Therefore, the more pipes that are open, the shorter the time needed to fill the tank.

Another situation relevant to inverse variation is when we have to divide an amount into more than two shares where the criterion is determined by the inverse relationship between the quantities involved in the variation. In these situations we express the relationship as

a:b:c = k/d:k/e:k/f

This is because when we consider each quantity on the left to the corresponding quantity on the right separately, we obtain the following relations:

a × d = k
b × e = k
c × f = k

Let's consider an example to clarify this point.

Example 4

$420 are divided among three siblings aged 6, 8 and 12 respectively, inversely proportional to their ages. Find the amount each sibling receives.

Solution 4

First, let's calculate the constant k of proportionality. If we express the siblings as A, B and C, we obtain

A:B:C = k/12:k/8:k/6

In addition, we know that

A + B + C = $420

Therefore, we can write

k/12 + k/8 + k/6 = $420
2k/24 + 3k/24 + 4k/24 = $420
9k/24 = $420
k = $420 × 24/9
= $1,120

Hence, sibling A, who is 6 years old receives

A = $1,120/6 = $186.666

Sibling B, who is 10 years old receives

B = $1,120/8 = $140

And sibling C, who is 12 years old receives

C = $1,120/12 = $93.3

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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