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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.
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.
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,
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.
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
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!