In this blog, we will introduce inverse proportion.

Inverse Proportion Examples

Mani after deciding on the menu is worried about the arrangements. If she does all the arrangements & decorations by herself it will take her approximately 12 hrs. So she decided to make a chart again so that she can take help from some of his friends. Here is her chart

Person  x	1	2	3	4
Time   y	12	6	4	3

In her chart, we can clearly see that as the number of persons increases the time decreases.  With one person it takes 12 hours to complete the arrangements. 2 persons complete it in 6hrs. As the number of persons doubles the time becomes half. Whereas 3 complete in only 4 hrs and 4 in 3 hrs.

Here the ratio of increase in persons is 2/3 and the time decreases by 3/2.  And then the ratio of increase in persons is by � and decrease in time by inverse of same ratio that is 4/3. This type of relationship is called inverse variation

Taking persons as x and time as y we see that on multiplying x and y we get 12. This implies that in inverse variation xy = k where k is a constant.

It is important to note that the product xy remains constant. We can also say that x varies inversely with y and y varies inversely with x.

We can say that in inverse proportion two quantities may change in such a manner that if one quantity increases, the other quantity decreases and vice versa. There are some more examples of inverse variation.

Let us go back to Mani�s chart. Here if y₁, y₂ are the values of y corresponding to the values x₁, x₂ of x respectively then x₁y₁ = x₂y₂

(= k), or x₁/x₂ = y₁/y₂

We say that x and y are in inverse proportion.

2. Let us take another example. Anu can go to her school in four different ways.

She can walk, run, on cycle or go by car.

We can frame this table. Observe that as the speed increases, the time taken to cover the same distance decreases.

As Anu doubles her speed by running, time reduces to half.

As she increases her speed to three times by cycling, time decreases to one-third.

Similarly, as she increases her speed to 15 times, time decreases to one-fifteenth. Or, in other words, the ratio by which time decreases is the inverse of the ratio by which the corresponding speed increases.

The speed of bus = x and, the time taken = y

Here too x₁y₁ = x₂y₂ (= k),

x₁/x₂ = y₁/y₂, We say that x and y are in inverse proportion.

3. Let us take an example.

Example: 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?

Solution: Let the desired time to fill the tank be x minutes. Thus, we have the following table.

The lesser the number of pipes more will be the time required by it to fill the tank.
So, this is a case of inverse proportion.
Hence, 80 � 6 = x � 5.
That is x = 80 � 6/5.
On solving we have x = 96. Thus, the time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.

Read More:
Direct Proportion: Formula, Examples, Definition - Directly Proportional

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