CONSTANT OF PROPORTIONALITY: WHAT IS IT, CALCULATION, EXERCISES - MATHS - 2024

The constant of proportionality is a relational numerical element, used to define the pattern of similarity between 2 quantities that are altered simultaneously. It is very common to represent it as a linear function in a generic way using the expression F (X) = kX. However, this is not the only representation of a possible proportionality.

For example, the relationship between X and Y in the function Y = 3x has a constant of proportionality equal to 3. It is observed that as the independent variable X grows, so does the dependent variable Y, at three times its value previous.

The alterations applied to one variable have immediate repercussions on the other, so that there is a value known as the constant of proportionality. This serves to relate the different magnitudes that both variables acquire.

What is the constant of proportionality and types

According to the trend in the change of the variables, the proportionalities can be classified into 2 types.

Direct proportionality

Suggests a one-way relationship between two quantities. In it, if the independent variable shows some growth, the dependent variable will also grow. Similarly, any decrease in the independent variable will cause a decrease in the magnitude of Y.

For example, the linear function used in the introduction; Y = 3X, corresponds to a direct relationship of proportionality. This is because the increase in the independent variable X will cause a triple increase in the previous value taken by the dependent variable Y.

Similarly, the dependent variable will decrease three times its value when X decreases in magnitude.

The value of the constant of proportionality "K" in a direct relationship is defined as K = Y / X.

Inverse or indirect proportionality

In this type of functions, the relationship between the variables is presented in an antonymous way, where the growth or decrease of the independent variable corresponds respectively to the decrease or growth of the dependent variable.

For example, the function F (x) = k / x is an inverse or indirect relationship. Since the value of the independent variable begins to increase, the value of k will be divided by an increasing number, causing the dependent variable to decrease in value according to the proportion.

According to the value taken by K, the trend of the inverse proportional function can be defined. If k> 0, then the function will be decreasing on all real numbers. And your graph will be in the 1st and 3rd quadrant.

On the contrary, if the value of K is negative or less than zero, the function will be increasing and its graph will be found in the 2nd and 4th quadrants.

How is it calculated?

There are different contexts where the definition of the constant of proportionality may be required. In the different cases, different data about the problem will be shown, where the study of these will finally yield the value of K.

In a generic way, the aforementioned can be recapitulated. The values ​​of K correspond to two expressions depending on the type of proportionality present:

- Direct: K = Y / X

- Inverse or indirect: K = YX

According to its graph

Sometimes the graph of a function will only be partially or completely known. In these cases, it will be necessary, through graphic analysis, to determine the type of proportionality. Then it will be necessary to define a coordinate that allows to verify the values ​​of X and Y to apply to the corresponding formula of K.

The graphs referring to direct proportionalities are linear. On the other hand, the graphs of inverse proportional functions usually take the form of hyperbolas.

According to table of values

In some cases, there is a table of values ​​with the values ​​corresponding to each iteration of the independent variable. Usually this involves making the graph in addition to defining the value of K.

According to analytical expression

Returns the expression that defines the function analytically. The value of K can be solved directly, or it can also be inferred from the expression itself.

By direct or compound rule of three

In other exercise models, certain data are presented, which refer to the relationship between the values. This makes it necessary to apply the direct or compound rule of three to define other data required in the exercise.

History

The concept of proportionality has always been around. Not only in the mind and work of the great mathematicians, but in the daily life of the population, due to its practicality and applicability.

It is very common to find situations that require a proportionality approach. These are presented in each case where it is necessary to compare variables and phenomena that have certain relationships.

Through a timeline we can characterize the historical moments, in which mathematical advances regarding proportionality have been applied.

- 2nd century BC The system of storage of fractions and proportions is adopted in Greece.

- 5th century BC The proportion that relates the side and the diagonal of a square is also discovered in Greece.

- 600 BC Thales of Miletus presents his theorem regarding proportionality.

- Year 900. The decimal system previously used by India is expanded in ratios and proportions. Contribution made by the Arabs.

- XVII century. Contributions regarding proportions arrive in Euler's calculation.

- XIX century. Gauss contributes the concept of complex number and proportion.

- Twentieth century. Proportionality as a function model is defined by Azcarate and Deulofeo.

Solved exercises

Exercise 1

It is required to calculate the value of the variables x, y, z and g. Knowing the following proportional relationships:

3x + 2y - 6z + 8g = 1925

x / 3 = y / 8 = z / 3 = g / 5

We proceed to define the relative values ​​of the constant of proportionality. These can be obtained from the second relation, where the value that divides each variable indicates a relation or ratio referring to K.

X = 3k y = 2k z = 3k g = 5k

The values ​​are substituted in the first expression, where the new system will be evaluated in a single variable k.

3 (3k) + 2 (2k) - 6 (3k) + 8 (5k) = 1925

9k + 4k -18k + 40k = 1925

35k = 1925

K = 1925/35 = 55

Using this value of the constant of proportionality we can find the number that defines each of the variables.

x = 3 (55) = 165 y = 2 (55) = 110

z = 3 (55) = 165 g = 5 (55) = 275

Exercise 2

Calculate the constant of proportionality and the expression that defines the function, given its graph.

First, the graph is analyzed, its linear character being evident. This indicates that it is a function with direct proportionality and that the value of K will be obtained through the expression k = y / x

Then a determinable point is chosen from the graph, that is, one where the coordinates that compose it can be seen exactly.

For this case, the point (2, 4) is taken. From where we can establish the following relationship.

K = 4/2 = 2

So the expression is defined by the function y = kx, which for this case will be

F (x) = 2x

References

1. Math for Electricity & Electronics. Dr. Arthur Kramer. Cengage Learning, Jul 27 2012
2. Vision 2020: The Strategic Role of Operational Research. N. Ravichandran. Allied Publishers, Sept. 11 2005
3. Grammatical and Arithmetic Knowledge of Administrative Assistant of the State e-book. MAD-Eduforma
4. Reinforcement of Mathematics for curricular support and diversification: for curricular support and diversification. Mª Lourdes Lázaro Soto. Narcea Ediciones, Aug 29. 2003
5. Logistics and commercial management. Maria José Escudero Serrano. Ediciones Paraninfo, SA, 1 sept. 2013