# Inverse proportions

**What is an inverse proportion?**

Inverse proportions describes the relationship where when one thing gets bigger, something else gets smaller.

For instance, the alarm of an ambulance becomes louder as it approaches you and calmer as it goes farther away. That means if the distance between you and the ambulance is small, then the sound of the alarm of the ambulance heard will be loud.

If the distance between you and the ambulance is more, then the sound of the ambulance heard will be calm.

**E1.11: Demonstrate an understanding of inverse proportion.**

In direct proportion, if one amount increases, then the other amount also increases. If one amount decreases, then the other amount also decreases. But in inverse proportion, an increase in one amount causes a decreasing effect in the other amount, and vice versa. In such a situation, the two variables $$a$$ and $$b$$ are inversely proportional.

For instance, the time taken by a specific number of labourers to achieve a piece of work inversely changes with the total number of labourers at work. It implies the lesser the number of labourers, extra time is taken to complete work, and vice versa.

The speed of moving stuff like a train, vehicle, or boat is inversely proportional to the time to cover a specific distance. The higher the speed, the lesser the time is taken to cover the distance.

**Worked examples**

**Example 1:** If $$10$$ labourers construct a road of length $$1\;\text{km}$$ in $$10\;\text{days}$$. How many days can they construct the same road if $$10$$ more labourers join them? Justify it in case of an inverse proportion.

**Step 1: Find the time taken by a single labourer to construct the full road.**

$$10$$ labourers take $$10\;\text{days}$$

$$1$$ labourer takes $$10 \times 10=100\;\text{days}$$

**Step 2: Calculate the total number of labourers if $$10$$ more join them.**

$$10+10=20$$

**Step 3: Calculate the time taken by $$20$$ labourers to construct $$1\;\text{km}$$ road.**

$$100 \div 20=5$$

$$20$$ labourers can construct the road of length $$1\;\text{km}$$ in $$5\;\text{days}$$. With the rise in the number of labourers, there is a decrease in the time required for constructing the road. Therefore, it is a case of inverse proportion.

**Example 2:** Raul drives a car at an average speed of $$90\;\text{km/h}$$ for $$4\;\text{hours}$$ to cover a distance of $$360\;\text{km}$$. How much speed does he has to decrease if he wants to travel the same distance in $$6\;\text{hours}$$?

**Step 1: Find the average speed he has to travel if he wants to cover the distance in $$1\;\text{hour}$$.**

$$90 \times 4=360\;\text{km/h}$$

**Step 2: Calculate the average speed he has to travel if he wants to travel the distance in $$6\text{ hours}$$.**

$$360 \div 60=60\;\text{km/h}$$

**Step 3: Calculate the speed he has decreased to cover the distance in $$6\;\text{hours}$$.**

$$\text{Old speed}- \text{new speed} = 90-60 = 30\text{km/h}$$

Raule has to decrease his speed by $$30\;\text{km/h}$$ to cover the same distance in six hours.

The symbol used to denote the proportionality is $$\propto$$. For example, if we say, $$a$$ is proportional to $$b$$, then it is represented as ‘$$a\propto b$$’ and if we say, $$a$$ is inversely proportional to $$b$$, then it is denoted as $$a\propto \frac{1}{b}$$. To make the topic easier, start by understanding that inverse means ‘opposite’.