In this chapter we will learn methods to solve inverse variation problems using unitary method.

Let us first review the concept of inverse variation and then move on to solve related problems.

## What is Inverse Variation?

Inverse variation is represented by below expression.

\mathtt{y\ =\ \frac{k}{x}}

Where y & are variable and k is a constant.

The expression signifies that;

⟹ **Increase in value of x will proportionally decrease the value of y**.

⟹ **Decrease in value of x will proportionally increase the value of x.**

Hence, there is inverse relationship between the value of x and y.

## Unitary method of Inverse variation

In Unitary method, **we first calculate the value of single unit** and **then move on to find the value of desired unit**.

The unitary method of inverse variation is different from direct variation.

**In inverse variation**, the value of single unit is calculated by multiplication **while in direct variation**, division is done to calculate the single unit value.

**For example**;

5 workers do a particular work in 10 days. How many days will 8 workers take to complete the same work.

**Solution**

Here number of workers & time taken are two different variables.

Both variables are inversely related as increase in workers will decrease the time to complete the job.

5 workers takes ⟹ 10 days

Calculate time taken if one worker is present.

1 worker takes ⟹ 5 x 10 = 50 days.

Note: To calculate the value of single unit we do the multiplication as the variables are inversely related.

8 worker takes ⟹ 50 / 8 days = 6.25 days.

This is how unitary method works for inverse variation.

I hope the above concepts are clear to you. Let us solve some problems related to inverse variation.

## Solved Inverse Variation Problems using Unitary method

**Example 01**

20 kids can eat Party Pizza in 8 minutes. In how much time the same pizza can be eaten by 12 kids?

**Solution**

Number of kids and time to eat the pizza are inversely related.

If more kids are present, less time would be taken to eat the whole pizza and vice-versa.

20 kids eat pizza ⟹ 8 minutes

First calculate the time to eat the pizza if one 1 kid is present.**1 kid will take** ⟹ 8 x 20 = 160 minutes

Now calculate the time for 12 kids**12 kids will take** ⟹ 160 / 12 = 13.33 minutes.

Hence, **12 kids will eat the pizza in 13.33 minutes**.

**Example 02**

30 workers can complete the work in 15 days. If the worker count is increases to 50, how many days will it take to complete the work.

**Solution**

Number of workers and time taken are in inverse relationship.

The more the workers, the less time it take to complete the job and vice-versa.

30 workers complete work ⟹ 15 days.

Calculate the time if one worker is present.**1 worker will complete work in** ⟹ 15 x 30 = 450 days.

Now calculate time if 50 workers are present.**50 workers complete work in** ⟹ 450 / 50 = 9 days

Hence, **50 workers will complete the job in 9 days**.

**Example 03**

10 pipes can fill a tank in 4 hours. How much time will 4 pipes take to fill the whole jar?

Solution

Number of pipes and time to fill tank are inversely related.

As the number of pipes increases the time taken will decrease and vice versa.

10 pipes fill tank in ⟹ 4 hours

Calculate the time taken if one pipe is present.**1 pipe fill tank in** ⟹ 4 x 10 = 40 hours.

Now calculate the time taken by 4 pipes.**4 pipe fill tank in** ⟹ 40 / 4 = 10 hours.

Hence,** 4 pipes fill tank in 10 hours**.

**Example 04**

A car with speed of 30km/hr travels between 2 cities in 6 hours. How much time will the car take if the speed is increased to 80 km/hr.

**Solution**

Speed of car and time of journey are inversely related.

Increase in speed result in less time to cover the distance and vice-versa.

30 km/hr results in ⟹ 6 hours

Calculate the time when speed is 1 km/hr**1 km/hr results in** ⟹ 6 x 30 = 180 hours

Now calculate time when speed is 80 km/hr**80 km/hr results in** ⟹ 180 / 80 = 2.25 hours

Hence, **when car travels 80 km/hr, the journey will be completed in 2.25 hours**.

**Example 05**

12 Men can eat the ration in 20 days. If the number of men increased to 15, in how many days the ration will get empty.

**Solution**

Number of men and time to empty the ration are inversely proportional.

If number of men increases, the time to empty the ration will decrease and vice-versa.

12 men eat in ⟹ 20 days

Calculate the days if 1 men is present.**1 men eats in** ⟹ 20 x 12 = 240 days

Now calculate time for 15 men.**15 man eat in** ⟹ 240 / 15 = 16 days

Hence, **15 men complete the ration in 16 days.**