Introduction:

Two quantities ‘a’ and ‘b’ are said to vary inversely to each other if the product ‘ab’ always remains constant. The product is known as constant of proportionality.

Let ‘a’ and ‘b’ be two quantities, if ab = k, where k is a constant of proportionality, then ‘a’ and ‘b’ are said to be inversely proportional to each other.

Example: If ‘a’ and ‘b’ vary inversely, then find the constant of proportionality if a = 8 and b = 10

Solution: We know, if ‘a’ and ‘b’ are inversely proportional then,

ab = k, where k is a constant

Here, given a = 8 and b = 10

Therefore, ab = 8 * 10 = 80

So, k = 80

## Examples of Inversely Proportional

1) If x and y vary inversely as each other. x = 10 when y = 6. Find y when x = 15

Solution: Given x and y are inversely proportional,

Therefore, xy = k where k is a constant

For x = 10 and y = 6,

xy = k

10 * 6 = k

60 = k

Now, for x = 15, y = ?

xy = 60

15 * y = 60

y = `(60)/(15)`

y = 4

2) If x and y vary inversely as each other and x = 5 when y = 15. Find x when y = 12

Solution: Given x and y are inversely proportional,

Therefore, xy = k where k is a constant,

For x = 5 and y = 15

x y = k

5 * 15 = k

75 = k

For y = 12, x = ?

x * 12 = 75

x = `(75)/(12)` = `(25 * 3)/(4 * 3)` = `(25)/(4)`

x = 6.25

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3) If x and y vary inversely as each other. If x = 30, find y when constant of proportionality = 900

Solution: Given x and y are inversely proportional,

so, xy = k where k is constant,

For x = 30, k = 900, y = ?

30 * y = 900

y = `(900)/(30)` = `(30 * 30)/(30)` = 30

y = 30

4) If x and y vary inversely as each other and if y = 35 find x when constant of proportionality is 7

Solution: Given x and y are inversely proportional,

Therefore, xy = k, where k is constant

For y = 35, k = 7 , x = ?

x * 35 = 7

x = `(7)/(35)` = `(7)/(7*5)` = `(1)/(5)`

x = `(1)/(5)`

## Word Problems on Inversely Proportional

1) If 52 men can do a piece of work in 35 days, in how many days 28 men will do it?

Solution: Let 28 men do the work in x days,

Clearly, less the number of men, more will be the number of days to finish the work. So, this case is an inverse proportional.

Ratio of number of men = Inverse ratio of number of days.

`=>` 52 : 28 = x : 35

`=>` `(52)/(28) = (x)/(35)`

`=>` 52 * 35 = 28 * x cross multiply the terms

`=>` x = `(52* 35)/(28)`

`=>` = 65

Hence, 28 men will do the work in 65 days.

2) Maria cycles to her school at an average speed of 12 km/hr. It takes her 20 minutes to reach the school. If she wants to reach her school in 15 minutes, what should be her average speed?

Solution: Let the required speed be x.

Clearly, less the number of men, more will be the number of days to finish the work. So, this case is an inverse proportional

Ratio of speeds = Inverse ratio of time taken

`=>` 12 : x = 15 : 20

`=>` `(12)/(x) = (15)/(20)`

`=>` 15 * x = 12 * 20 cross multiply the terms

`=>` x = `(12* 20)/(15)`

`=>` = 16

Hence, Maria average speed should be 16 km/hr.

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