Inverse proportionality occurs when one quantity decreases while the other rises or vice versa. This means that if the magnitude or absolute value of one item reduces, the magnitude or absolute value of the other quantity rises, and their product remains constant. This factor is also called the constant of proportionality.

An inverse connection between two quantities is represented by inverse variation. Inverse variation has various practical uses. For example, as a car’s speed rises, so does the time it takes to get to its destination. In this article, we will go through inverse variation, including its definition, formula, and solved examples.

Also, Read: Situations of Direct Variation

## Inverse Variation Definition & Example

Inverse variation occurs when the product of two non-zero values produces a constant term (constant of proportionality). In other words, inverse variation occurs when one parameter is directly proportional to the reciprocal of the other. This indicates that increasing one quantity causes a reduction in the other while decreasing one causes a rise in the other.

Assume x and y have inverse variation. It is assumed that x = 20 and the proportionality constant equals 80. The value of y will then be 80 / 20 = 4.

### Inverse Variation Formula

The sign ” ∞” is used to indicate or represent proportionality. When two quantities, x, and y, exhibit inverse variation, they are expressed as follows: x ∞ 1/y or y= 1/x

A proportionality coefficient or constant must be included to transform this expression into an equation. As a result, the inverse variation formula is as follows: x=k/y or y= k/x i.e. xy = k

The proportionality constant k is used here.

### Product Rule for Inverse Variation

Assume that the two inverse variation solutions are (x_{1}, y_{1}) and (x_{2}, y_{2})

This might also be written as x_{1} y_{1}= k_{1} and x_{2} y_{2}= k

This is the inverse variation product rule.

Using these two equations, x_{1} y_{1} = x_{2} y_{2}

This is the inverse variation product rule. Let’s look at an example to better grasp the inverse variation formula.

**Example:**

Assume x and y have an inverse relationship such that when x = 12, y = 6. Determine the value of y when x equals 30.

**Solution:**

Given: x_{1} = 12, y_{1} = 6, x_{2} = 8, y_{2}=?

By applying the inverse variation formula,

x_{1} y_{1}= x_{2} y_{2}

12 × 6 = 8 × y2

72 = 8 × y2

y_{2} = 9

Thus, the value of y is 9 when x is 8.

### Some Situations of Inverse Variation

- When there are more men on the job, it takes less time to do the work.
- When there are fewer men on the job, it takes longer to do the work.
- More speed equals less time to travel the same distance.
- With less speed, it takes longer to traverse the same distance.

Get acquainted with similar kinds of 7th Grade Math Concepts in one place and resolve all your doubts.

### Inverse Variation Real Life Examples with Solutions | Inverse Variation Problems

**1. If 48 men can finish the work in 24 days, how long will it take 36 men to do the same work?**

**Solution:**

This is an example of indirect variation.

Fewer men will take more days to do the work.

The work is getting completed in 24 days by 48 men as given.

1 individual can complete the same amount of work in 48 × 24 days

36 men can do the same amount of labour in (48 × 24)/36 = 32 days.

As a result, 36 men can do the same task in 32 days.

**2. A fort with 100 men had adequate supplies for 20 days. After two days, 20 additional men arrive at the fort. How long will the food supply last?**

**Solution:**

Because there are more soldiers, food lasts for fewer days. Because there are more soldiers, food lasts for fewer days.

This is an example of indirect variation.

Because 20 soldiers enter the fort after 2 days, the remaining food is enough for 100 soldiers and 18 days.

**3. In a hostel, there are 100 students. They have enough food for 30 days. After 5 days, 20 more students arrived at the hostel. How long will this provision be in consequence?**

**Solution:**

Assume the provision is extended for x days following the arrival of 20 new students, which would have been extended for 30 days if the number of students was 100. We know that the duration of meals is inversely related to the number of students in the hostel. Thus,

x_{1}/y_{1}= x_{2}/y_{2}

100/120=x/30

x= 25 days

Total number of days when the provision is valid = 25 + 5

= 30 days

**4. To fill a tank in 4 hours, 12 pipelines are necessary. How long would it take if you use 16 same pipes?**

**Solution:**

Let x be the desired time to fill the tank.

We know that as the number of pipes in the tank rises, the time it takes to fill it decreases. As a result, this is an example of inverse variation.

x_{1}/y_{1}= x_{2}/y_{2}

12x = 164

x = 3 hours

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### FAQs on Situations of Inverse variation

**1. What Exactly is Inverse Variation?**

Inverse variation describes the connection between two quantities in which one rises while the other falls, or vice versa. Both are non-zero numbers, and their product is the proportionality constant.

**2. What is an Inverse Variation Example?**

Assume you need a specific number of pipes to fill a tank. The time required to fill the tank decreases as the number of pipes increases. As an example of inverse variation, consider the number of pipes and the time required to fill the tank.

**3. How to Find the Inverse, Variation Constant?**

The proportionality constant can be found by multiplying the two numbers in inverse variation. This is written as xy = k.

**4. What are Examples of Inverse Proportion in Real Life?**

In inverse proportion, if one quantity grows, the other decreases, and vice versa. Let’s look at some real-world applications of inverse proportion.

- When we raise the speed of the automobile, the time it takes to get to our destination reduces.
- As one moves away from the sun, the brightness of the sun lessens.
- More vehicles on the road fewer vehicles on the road
- As a result, the inverse proportion may be used in a wide range of real-world situations.