Proportionality Relationships: Concept, Examples And Exercises

The proportional relationships are links between two or more variables, such that when one of the numbers varies, so does the value of the other. For example, if one increases, the others may increase or decrease, but by a uniform amount.

Ancient Greek mathematicians realized that some variables were related in some very precise way. They realized that if one circle is twice the diameter of another, it will have a circumference twice the length.

Figure 1. The length of a circumference is directly proportional to its diameter D. Source: F. Zapata

And if the diameter is tripled, then the circumference of the circumference will triple as well. This means that an increase in diameter produces a proportional increase in the size of the circumference.

And so we can affirm that the length of the circumference L is proportional to its diameter D, which is expressed as follows:

L ∝ D

Where the symbol ∝ is read ” directly proportional to “. To change the symbol of proportionality to that of equality and incorporate numerical values, it is necessary to determine the link between the variables, called the constant of proportionality .

After taking many measurements, the ancient mathematicians determined that the constant of proportionality between the size L of the circumference, and the diameter D of the same, was the number 3.1416 … The ellipses indicate an infinite number of decimal places.

This value is none other than that of the famous number π (pi) and in this way we write:

L = π.D

In this way, the ratio of the length to the diameter of one circumference is the same as the ratio of the length to the diameter of another. And the best part is that now we have a way to calculate the length of any circumference just by knowing its diameter.

Article index

  • one

    Examples of proportionality relationships

  • two

    Direct proportionality and inverse proportionality

  • 3

    Other types of proportionality

  • 4

    Exercises

    • 4.1

      – Exercise 1

    • 4.2

      – Exercise 2

  • 5

    References

Examples of proportionality relationships

In science (and in everyday life too) it is very important to find relationships between variables, to know how changes in one of them affect the other. For example:

-If you need 3 cups of flour to make a dozen cookies. How many cups does it take to make 2 and a half dozen?

-Knowing that on the planet Mercury an object weighs 4 times less than on Earth, how much will a 1.5-ton car weigh on Mercury?

-How does the change in the applied force affect the acceleration of the body on which it is applied?

-If a vehicle travels with uniform rectilinear motion on a highway and we know that it travels 30 km in 10 minutes, what will be the distance traveled after 20 minutes?

-When we have a wire through which an electric current passes, how does the voltage vary between its ends if it increases?

-If the diameter of a circle is doubled, how is its area affected?

-How does distance affect the intensity of the electric field produced by a point charge?

The answer lies in proportionality relationships, but not all relationships are of the same type. Then we will find them for all the situations raised here.

Direct proportionality and inverse proportionality

Two variables x and y are in direct proportion if they are related by:

y = kx

Where k is the constant of proportionality. An example is the relationship between the amounts of flour and cookies. If we graph these variables, we obtain a straight line like the one shown in the figure:

Figure 2. To make 2.5 dozen cookies, you need 7.5 cups of flour (point C). Source: F. Zapata.

If y are the cups of flour and x are dozens of cookies, the relationship between them is:

y = 3x

For x = 1 dozen we need y = 3 cups of flour. And for x = 2.5 dozen, y = 7.5 cups of flour are required.

But we also have:

-The acceleration to that experienced by a body is proportional to the force F acting thereon, with the mass of the body, called m , the proportionality constant:

F = m a

Therefore, the greater the force applied, the greater the acceleration produced.

-In ohmic conductors, the voltage V between their ends is proportional to the current I applied. The constant of proportionality is the resistance R of the conductor:

V = RI

  When an object moves with uniform rectilinear motion, the distance d is proportional to the time t , the speed v being the constant of proportionality:

d = vt

Sometimes we find two quantities such that an increase in one produces a proportional decrease in the other. This dependency is called the inverse ratio .

For example, in the previous equation, the time t required to travel a certain distance d is inversely proportional to the speed v of the journey:

t = d / v

And so, the greater the speed v, the less time it takes the car to travel the distance d. If, for example, the speed is doubled, the time is cut in half.

When two variables x and y are in inverse proportion, we can write:

y = k / x

Being k the constant of proportionality. The graph of this dependency is:

Figure 3. 1 / x graph representing inverse proportionality. Source: Wikimedia Commons.

Other types of proportionality

In one of the examples mentioned earlier, we were wondering what happens to the area of ​​the circle when the radius increases. The answer is that the area is directly proportional to the square of the radius, where π is the constant of proportionality:

A = πR 2

In case the radius is doubled, the area will increase by a factor of 4.

And in the case of the electric field E produced by a point charge q , it is known that the intensity decreases with the inverse of the square of the distance r to the charge q :

E = k e q / r 2

But we can also affirm that the intensity of the field is directly proportional to the magnitude of the charge, being the constant of proportionality k e , the electrostatic constant.

Other proportionalities that are also presented in Science are exponential proportionality and logarithmic proportionality. In the first case, the variables x and y are related by:

y = ka x

Where a is the base, a positive number other than 0, which is usually 10 or the number e. For example, the exponential growth of bacteria has this form.

In the second case the relationship between the variables is:

y = k.log a x

Again a is the base of the logarithm, which is often 10 (decimal logarithm) or e (natural logarithm).

Exercises

– Exercise 1

Knowing that an object weighs 4 times less on the planet Mercury than on Earth, how much would a 1.5-ton car weigh on Mercury?

Solution  

Weight on Mercury = (1/4) Weight on Earth = (1/4) x 1.5 tons = 0.375 tons.

– Exercise 2

For a party some friends decide to prepare juice from fruit concentrate. The package directions say that one glass of concentrate makes 15 glasses of juice. How much concentrate does it take to make 110 glasses of juice?

Solution

Let y be the number of glasses of juice and x the number of glasses of concentrate. They are related by:

y = kx

Substituting the values ​​y = 15 and x = 1, the constant k solves:

k = y / x = 15/1 = 15

Therefore:

110 = 15 x

x = 110/15 = 7.33 glasses of fruit concentrate.

References

  1. Baldor, A. 1974. Algebra. Cultural Venezolana SA
  2. Giancoli, D. 2006. Physics: Principles with Applications. 6
    th
    . Ed Prentice Hall.
  3. Varsity Tutors. Proportionality relationships. Recovered from: varsitytutors.com
  4. Wikipedia. Proportionality Recovered from: es.wikipedia.org.
  5. Zill, D. 1984. Algebra and Trigonometry. McGraw Hill.

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