Inversely Proportional

In Mathematics and Physics, we learn about quantities that depend on one another, and such quantities are known as proportional to one another. In other words, two variables or quantities are proportional to each other, if one is varied, then the other also changes by a fixed amount at an equal proportion. This property of variables is known as proportionality and the symbol used to represent the proportionality is “∝.” 

Inversely proportional-

When the value of one quantity increases with respect to a decrease in another value or vice-versa, then they are said to be inversely proportional. It means that those two quantities behave opposite in nature. For example, let us consider speed and time that are in inverse proportion to each other. When you increase the speed, the time is decreased. The other names used for this type of proportion are inverse proportion or varying inversely or inverse variation or reciprocal proportion. Two variables say x and y, which are in inverse proportion to each other are represented as;

x ∝ 1/y or x ∝ y-1

There are two types of proportionality:

• Directly Proportional

• Inversely Proportional

Inversely Proportional definition-

Two quantities are related to each other inversely, i.e., when an increase in one quantity brings a decrease in the other quantity and vice versa then they are said to be inversely proportional. In other words, if one variable decreases, the other increases in the same proportion (It’s opposite to directly proportional). Two quantities are said to be inversely proportional when one quantity is in direct proportion to the reciprocal of the other quantity. For example, let us consider the relation between speed and time. Speed and travel time are inversely proportional because the faster we travel, the lesser is the time taken to reach a particular destination, i.e. greater the speed, the lesser the time taken.

• As speed increases, the time taken to reach the destination decreases.

• And as speed decreases, the time taken to reach the destination increases.

General Formula of Inversely Proportional-

the proportional relationship between two quantities is denoted by the symbol “∝”. Let x and y be two quantities, and let y be inversely proportional to x is the same thing as y being directly proportional to 1/x. It is mathematically written as y ∝ 1/x.

The general equation for inverse variation is y = k/x, where k is the proportionality constant. We can also write this as y × x = k. If x and y are in inverse variation or inversely proportional and x has two values x1 and x2 corresponding to y, which also has two values y1 and y2 respectively, then by the definition of inverse variation, we have x1 y1 = x2 y2 = k.

It case, it becomes x1 / x2 = y2 / y1 = k.

Graphical Representation of Inverse Proportionality-

The graph for inversely proportional looks like this.

For example, the graph of the equation y = 1/x and y = -1/x has an inversely proportional relationship

Applications of Inversely Proportional-

The concept of inverse proportionality is widely used in day-to-day life and also in solving many problems in the field of science, statistics, algebra, etc. There are many formulas in physics that have been derived using the concept of inverse proportionality. For example- Ohm’s law, speed and time relation, the wavelength of sound, and its frequency are a few.

Important Notes on Inversely Proportional-

The following points are to be remembered of inverse proportionality:

• If one quantity increases, the other quantity decreases.

• x ∝ 1/y or y ∝ 1/x.

• x × y = k, where k is the proportionality constant.

Conclusion:

When the value of one quantity increases with respect to a decrease in another value or vice-versa, then they are said to be inversely proportional. It means that those two quantities behave opposite in nature. For example, let us consider speed and time that are in inverse proportion to each other. When you increase the speed, the time is decreased. The other names used for this type of proportion are inverse proportion or varying inversely or inverse variation or reciprocal proportion. Two variables say x and y, which are in inverse proportion to each other are represented as;

x ∝ 1/y or x ∝ y-1

This topic is important to understand whether one quantity is directly proportional or inversely proportional to another quantity. As discussed above there are multiple applications of applying this topic in various aspects of branches like physics and mathematics which we have discussed in detail in the above-mentioned article.