# Chapter 13 – Direct and Inverse Proportions

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You might have experienced the thrill of covering more distance in less time when you pedal your cycle faster. Or you may have seen that the more the number of people working in the kitchen, the lesser the time taken to get the food on the table. These are all examples of direct and inverse proportions in your life. Essentially, everything you learn in Mathematics- concepts, definitions, formulae-  find an application in your life.

Direct and inverse proportions represent the relation between a set of quantities and amounts of two variables. You can denote a direct proportion or inverse proportion with the symbol ‘∝’. For example,

If we want to express the relation between a and b, we write it as,

a ∝ b, to denote direct proportion, and

a 1/∝, to denote inverse proportion.

What is Direct Proportion?

In our example, when a and b are multiples of each other, implying that a increases with b, and decreases as b decreases, then they are said to be in direct proportion.

Direct Proportion Definition

When the ratio of a and b remains constant, with changing values of the two variables, they are said to be in direct proportion or direct variation to each other.

Direct Proportion Formula

If a ∝ b, then,

a/b= k, or

a = kb, where k is known as the constant of proportionality.

#### Constant of Proportionality

The change in value of the ratio of a and b is denoted by k, which is known as the constant of proportionality. This is expressed as,

a/b=c/d=k, or the two ratios a/b and c/d are proportional to each other and can be equated with a constant, represented by k.

#### Example of Direct Proportion

In a textile factory, a machine manufactures 200 units of cloth every hour. To find out how many units it can produce in 8 hours, we can use the concept of direct proportion. By the direct proportion definition, more the number of hours the machine works, more the number of units produced. This may be expressed as,

Number of units produced ∝ Number of hours

Therefore, if it can produce 200 units in 1 hour, then it should produce 200*8=1600 units in 8 hours, all other factors remaining constant.

What is Inverse Proportion?

You can understand the concept of inverse proportion through the example of labourers working at a construction site. To complete one floor of a building, if you deploy 100 labourers, it will take less time to complete compared to when you deploy only 50. So, with increasing number of labourers, the amount of time taken decreases. This is a classic example of inverse proportion.

Inverse Proportion Definition

When the value of one quantity, a, increases or decreases with a decrease or increase in the value of another quantity, b, then the two are said to be inversely proportional to each other. Inverse proportion is also referred to as inverse variation.

Inverse Proportion Formula

For two inversely proportional quantities, a and b, the product of their corresponding values should remain constant.

If a 1/∝ b, then,

a = k/b, where k is the constant of proportionality.

Inverse Proportion Example

If the time taken to plant 100 trees by 100 workers is 60 minutes, how much time do you think 150 workers will take to plant the same number of trees?

This is a classic case of “more the merrier” and can be worked out using the inverse proportion formula.

If a= time and b= number of workers, then

a = 60 and b = 100

This can be expressed by the equation,

60= k/100 or,

k = 100*60

k = 6000

When b= 150, then

a = 6000/150 or,

a= 40

Therefore, 150 people can plant 100 trees in 40 minutes.

Using the concept of inverse proportion, you can see how the same task can be performed in a much lesser time when there are more people involved. That also teaches you a lesson about the importance of teamwork!

More Examples on Direct and Inverse Proportions

Let us look at some more examples to build a better understanding of direct and inverse proportions.

1.     Consider the direct variation between X and Y, where the two variables are directly proportional to each other.

X=15

Y=20

Can you represent the relation between X and Y, using an equation?

Let us start with what you know.

X ∝ Y.

So, we can express X as X = kY (deriving from direct proportion formula).

15 = k * 20 or,

k = 15/20 or,

k = 0.75

So, the relation between X and Y can be represented by the following equation:

X = 0.75Y

1.     On a map, its scale is given as 1:2000000. Can you find the actual distance between two cities, if they are 5 cm apart on the map?

We know that the distance on the map between the two cities is 5 cm.

Let the actual distance between them be denoted by z. So,

1:2000000 = 5: z (deriving from direct proportion formula).

Therefore, z = 5*2000000

z = 10000000 centimetres.

So, the distance between the two cities is 10000000 centimetres of 100 kilometres.

Direct and inverse proportion class 8 syllabus can be learnt in a fun way on the MSVGo app. It is a super app for budding Mathematics and Science geniuses. If you are looking to up your game in more Math concepts like direct and inverse proportions, algebra, mensuration, and others, download the app now!

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