**Proportionality** . Many times in practice we are presented with situations in which the value or quantity of one quantity depends on the value of another. The constant proportionality factor can be used to express the relationships between the quantities.

## Summary

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- 1 Historical Review
- 2 Direct proportionality
- 3 Inverse proportionality
- 1 Example

- 4 See also
- 5 External links
- 6 Sources

## Historical review

The mathematics is the study of relationships among quantities, magnitudes and properties and of logical operations used to derive quantities, magnitudes and unknown properties.

At the end of the 5th century BC, they discovered that there was no unit of length capable of measuring the side and the diagonal of a square , since one of the two quantities is immeasurable, that is, there are no two natural numbers whose quotient is equal to the ratio between the side and the diagonal. But since the Greeks only used natural numbers, they could not express this quotient numerically, since it is an irrational number .

## Direct proportionality

*Two magnitudes are directly proportional when, as one increases, the other increases in the same proportion.*

## Inverse proportionality

Inverse proportionality is a relationship between two variables in which the product between the corresponding quantities is always the same.

In other words, *two magnitudes are inversely proportional when, as one increases, the other decreases in the same proportion.*

### Example

If a car travels with a certain speed and increases it, the time it takes to reach its destination decreases. If you double speed, the time to reach your destination is cut in half.

You can also see this relationship between the width and length of the rectangles that have the same area .

If it is said that it is an area of 36 cm². Remembering that the area of a rectangle is the product of the length times the width.

In the following table you can see, with some values that:

Note that:

The width values are obtained by multiplying 36 by the reciprocals of the respective length values.

Here again it is appreciated that when the length increases, the width decreases.

When two quantities are related so that the values of one of them are obtained by multiplying by the same number the reciprocals of the corresponding values of the other quantity, they are said to be inversely proportional.

Speed and time are inversely proportional when the distance to travel is the same.

The same happens with the length and width of the rectangles of the same area.

The number by which each reciprocal is multiplied is called the inverse proportionality factor.

**Example** :

If a student needs 12 days to weed a tomato field , how many days will 4 students need to do the same job?

One way is to find the value corresponding to 1, that is, the proportionality factor.

Another way is to form a proportion and calculate ** x** . In this case,

**represents the number of days that the 4 students need to carry out the work.**

*x*Representing the data in a table:

The ratio is formed by equating the ratio between the values of a magnitude with the reciprocal of the ratio between its corresponding values as indicated by the arrows.

Answer: 4 students will need 4 days to complete the task.

**There are cases of magnitudes that** are related to each other so that the value of one depends on the value of the other, however **there is no proportionality between them** , for example:

- A person’s height increases with age, but the same does not grow with each passing year. If a child measures 1 meter at 4 years of age, that does not mean that at 8 years of age he will measure 2 meters, then, not all relationships represent proportionality.