Joint Variation Calculator

This calculator will allow you to find any missing quantity in a joint variation though a single click. For this, you have to identify which quantities are directly proportional to the missing quantity (and write them in the straight order) and which ones are inversely proportional to the missing quantity (and write them in the reverse order). For example, if the quantity to be calculated is a₂ where a₁ is given and if the quantity a varies directly as b, d and e and inversely as c and f, we write the resulting joint variation as

Joint Variation Calculator Parameters: when entering Indirect Variation it is important to remember that the fraction is inversed, so, if you have an input fraction of 2/5 you would enter 5/2 as the fraction is inversed.

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each joint variation calculation, the Joint Variation Calculator will automatically calculate the results and update the formula elements with each element of the joint variation calculation. You can then email or print this joint variation calculation as required for later use.

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Related Fields with Tutorials

1. Statistics
2. Physics

Theoretical description

In mathematics, variation is defined by any change in a quantity due to the change of another quantity. As a concept, variation is very similar to proportion. Here are some differences between proportion and variation:

1. A variation is a kind of extensive proportion that may include an infinity number of values.
2. Another difference between these two terms lies in the graph that visually represents the relationship between the two quantities involved. Thus, unlike proportion, a variation usually has a continuous line as a graph.
3. Finally, a proportion may involve two unrelated quantities while in variation they are always related to each other.

There are three types of variation between two quantities x and y:

1. Direct variation represents a relationship of type
   y = k ∙ x
   between the two quantities x and y, where k is a constant of proportionality. In direct variation, any change in the variable x brings a change by the same factor of the variable y. For example, the number of workers in a factory varies directly as the amount of work done. In simpler words, more workers hired, more work is done.
   The graph of direct variation is a straight line that passes through the origin of the coordinate system.
2. Inverse variation represents a relationship of type
   y = k ∙ x
   between the two quantities x and y, where k is a constant of proportionality. In inverse variation, any increase in the variable x brings a decrease by the same factor of the variable y and vice versa. For example, the number of workers in a factory varies directly as the time taken for an amount of work done. In simpler words, the more workers hired the less time is needed to complete the work.
   The graph of inverse variation is a hyperbola that gets closer to the axes by the increase in values of x and y but that never touches them.
3. Joint variation represents a combination of direct and inverse variation, so, obviously, at least three variables must be involved. We express them as x, y and z. Quantities that are in inverse variation (for example x and y) are multiplied with each other and written as a product, while the variables that are in direct variation are written in opposite sides of expression (for example, z and z or z and y). If written for two distinct situations (1) and (2) (for example initial and final), a joint variation is expressed as
   z₁/z₂ = x₁ ∙ y₁/x₂ ∙ y₂

One must be careful to identify the quantities that are in direct variation and the ones in inverse variation. Let's consider an example.

Example

Five men can repair 60 machines in 4 days. How many machines can two men repair in 15 days?

Solution

First, let's analyse the situation. We will consider the quantities two by two by assuming the third one as constant.

1. More men available, more machines repaired in a given time. Therefore the number of people involved (we express this variable as x) is directly proportional to the number of machines repaired (we express this variable as y).
2. More men available, less time taken to repair a given number of machines (we express this variable as z). Thus, x and z are inversely proportional.
3. More machines to repair, more time taken. Thus, y and z are directly proportional.

Writing this joint variation for two situations (1) and (2), we obtain

We have to calculate y₂. Thus, substituting the known values, we obtain

Simplifying by 10 the fraction on the right for easier operations, we obtain

Thus,

You can start from the quantity that is to be calculated, regardless the type of variation involved. In this case, any inverse variation is written by inverting the variables. Thus, if in the above example the number of men x₂ needed to repair 90 machines in 15 days were unknown given the same initial conditions, we would write

Substituting the values, we obtain