Math Lesson 4.5.5 - Types of Variation - Joint Variation

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Welcome to our Math lesson on Types of Variation - Joint Variation, this is the fifth 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 - Joint Variation

If there are more than two quantities involved in a variation where both types of variation are present, then we are dealing with a joint variation.

For example, the quantity x may vary inversely with a quantity y and directly with another quantity z to form a joint variation. In such cases, we can write for two different situations (1) and (2):

z₁ = k × x₁ × y₁

and

z₂ = k × x₂ × y₂

where k is a constant of proportionality as usual.

Dividing the two above equations side by side, yields

z₁/z₂ = k × x₁ × y₁/k × x₂ × y₂

Simplifying k from both terms, yields

z₁/z₂ = x₁ × y₁/x₂ × y₂

The general rule of joint variation is that the two quantities that are at the same side of the formula vary inversely, while each of them varies directly with the quantity that is present at the other side of the formula.

Example 5

10 workers can build 5 houses in 12 months. How long it takes to 15 workers to build 8 houses?

Solution 5

Increasing the number of workers brings an increase in the number of houses built during a certain time. Therefore, the number of workers varies directly with the number of houses built.

Yet, increasing the number of workers brings a decrease in the time taken to build a certain number of houses. Hence, the number of workers is inversely proportional to the time taken.

Finally, increasing the number of houses increases the time taken to build them. Therefore, the number of houses is directly proportional to the time.

Since the number of houses is directly proportional to each from the other two quantities, it must appear alone in the formula, while the other two quantities must be together on the other side of the formula, multiplied with each other. Hence, we have

No.of Houses 1/No.of Houses 2 = No.of workers 1 × Time 1/No.of workers 2 × Time 2

Substituting the known values yields

5 houses/8 houses = 10 workers × 12 months/15 workers × Time 2

Applying the cross product, yields

5 × 15 × T₂ = 8 × 10 × 12
75 × T₂ = 960
T₂ = 960/75
= 12.8 months

If there are more than three quantities involved in a variation, we must carefully analyse which of them vary directly, and which inversely. Then, the appropriate expression is written by observing the above-mentioned rules.

Example 6

18 men can build 5 houses in 3 months by working 6 hours a day. How long will it take to 27 men building 12 houses by working 8 hours a day?

Solution 6

Let's express the information in a table for a better understanding.

$Math Tutorials: Variation. Types of Variation Example$

Let's focus on the missing quantity z and check what kind of variation it has with the other quantities.

More people working = less months needed to build a fixed number of houses. Thus, x and z vary inversely.
More houses built = more months needed IF the other variables remain unchanged. Thus, y and z vary directly.
More hours a day committed = less time needed to build a fixed number of houses IF the number of people is constant. Therefore, z and t vary inversely.

Combining all the above findings, we write

z = k × y/x × t

where k is the usual constant of proportionality.

Writing the above expression for the two situations (1) and (2) yields:

z₁/z₂ = k × y₁/x₁ × t₁/k × y₂/x₂ × t₂

z₁/z₂ = x₂/x₁ × y₁/y₂ × t₂/t₁

Substituting the known values, we obtain

3/z₂ = 27/18 × 5/12 × 8/6 = 1080/1296

Thus,

z₂ = 3 × 1296/1080
= 3.6 months

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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