Inverse Proportion

When two quantities are related to each other inversely, i.e., when an increase in one quantity brings a decrease in the other and vice versa then they are said to be in inverse proportion. In inverse proportion, the product of the given two quantities is equal to a constant value. 

“Two quantities are said to be in inverse proportion if an increase in one leads to a decrease in the other quantity and a decrease in one leads to an increase in the other quantity”. In other words, if the product of both the quantities, irrespective of a change in their values, is equal to a constant value, then they are said to be in inverse proportion.

That is, if ab = k, then a and b are said to vary inversely. In this case, if b₁, b₂ are the values of b corresponding to the values a₁, a₂ of a, respectively then

a₁ b₁ = a₂ b₂ or a₁/a₂ = b₂ /b₁

The statement ‘a is inversely proportional to b is written as

a ∝ 1/b

For example, let us take the number of workers and the number of days required by them to complete a given amount of work as x and y respectively.

Are the number of workers and the number of days in inverse proportional relation? Let’s find out.

Observe the values written in the table carefully. You will find out that for each row, the product of x and y are the same. That means if there are 16 workers, they will complete the work in 3 days. 

So, here x × y = 16 × 3 = 48. Now, we decrease the number of workers, it is obvious that the less number of workers will do the same work in more time. But we see the product of x and y here, it is 12 × 4 = 48. Again, for 8 workers in 6 days, the product is 48. And same for 4 workers in 12 days. So, the product of two quantities in inverse proportion is always equal.

Inverse proportion formula help in establishing a relationship between two inversely proportional quantities. Let x and y be two quantities and assume that x is decreasing when y is increasing and vice versa.

Example:

The speed is inversely proportional to the time. As the speed increases, the time taken by us to cover the same distance decreases. Taking speed as y and time as x, we can say that y is said to be inversely proportional to x and is written mathematically as inverse proportion formula.

The inverse proportional formula is written as

y = k/x

where,

• k is the constant of proportionality.
• y increases as x decreases.
• y decreases as x increases.

Here the symbol ∝ denotes the proportional relationship between two quantities.

Example: 

Suppose x and y are in an inverse proportion such that when x = 120, y = 5. Find the value of y when x = 150 using the inverse proportion formula.

Solution:

To find: Value of y.

Given: x = 120 when y = 5.

x ∝ 1/y

x = k / y, where k is a constant,

or k = xy

Putting, x = 120 and y = 5, we get;

k = 120 × 5 = 600

Now, when x = 150, then;

150 y = 600

y = 600/150 = 4

That means when x is increased to 150 then y decreases to 4.

The value of y is 4 when x = 150.

Example: 

The time taken by a vehicle is 3 hours at a speed of 60 miles/hour. What would be the speed taken to cover the same distance at 4 hours?

Solution:

Consider speed as m and time parameter as n.

If the time taken increases, then the speed decreases. This is an inverse proportional relation, hence m ∝ 1/n.

Using the inverse proportion formula,

m = k/ n

m × n = k

At speed of 60 miles/hour, time = 3 hours, from this we get,

k = 60 × 3 = 180

Now, we need to find speed when time, n = 4.

m × n = k

m × 4 = 180

∴ m = 180/4 = 45

Therefore, the speed at 4 hours is 45 miles/hour.

Example: 

In a construction company, a supervisor claims that 7 men can complete a task in 42 days. In how many days will 14 men finish the same task?

Solution:

Consider the number of men be M and the number of days be D.

Given: M1 = 7, D1 = 42, and M2 = 14.

This is an inverse proportional relation, as if the number of workers increases, the number of days decreases.

M ∝ 1/D

Considering the first situation, M1 = k/D1

7 = k/42

k = 7 × 42 = 294

Considering the second situation, M2 = k/D2

14 = 294/ D2

D2 = 294/14 = 21

It will take 21 days for 14 men to do the same task.