CBSE Notes Class 8 Maths Chapter 11 Direct and Inverse Proportions

1.0Introduction to Proportions 

The relationship between 2 quantities where one quantity changes with another quantity is known as proportions. These are of two types - 

• Direct proportions
• Inverse proportions

A proportion is written as: 

$\frac{a}{b}=\frac{c}{d}$

This means the ratio of a & b is equal to the ratio of c & d in simple terms. 

2.0CBSE Class 8 Maths Chapter 11 Direct and Inverse Proportions - Revision Notes

Direct Proportions 

If two quantities are in proportion, and if one quantity increases, the other also increases, or if one decreases, the other also decreases, then the two quantities are said to be in direct proportions. It can be mathematically represented as: 

$\frac{x}{y}=\frac{a}{b}$ Or $x\propto y$

This means x is directly proportional to y. 

Key Point: In direct proportion, the ratio of two quantities always remains constant. 

If ${\mathrm{x}}_1\propto {\mathrm{y}}_{1}and{x}_2\propto y_2$, then $\frac{x_1}{y_1}=\frac{x_2}{y_2}$

Formula: $x=k\cdot y$

Here, k is the constant of proportionality. 

Example: 

Suppose a car is traveling at a constant speed, and it takes 2 hours to reach a distance of 60 km. Now, the same car travels for 3 hours, and the speed of the car remains constant. What will the distance be that it will cover?

Solution: 

Let the time taken by the car be x and the distance travelled be y. 

Here, we can see as the value of x increases, the value of y also increases in such a way that xy does not change (K)

In this example, 

$\frac{60}{2}=\frac{y}{3}$

$y=\frac{60\times 3}{2}$

y = 90

Here, the ratio is constant, that is, 30/1. 

Example: 

The variable x is directly proportional to y. If x increases by p%, then by what percent will y increase?

Solution: 

According to question 

$x\propto y$

The new value of x = x  + x(p/100)

$=x(\frac{100+p}{100})$

It is given that, x = y 

Hence, $y=y(\frac{100+p}{100})$

So, y also increases by p%. 

Memory Tip: “Same Direction”

• In other words, it's like "both move in the same direction." If one goes up, the other will, too.
• Imagine it like a straight line: In maths, when you draw a graph, the lines will always rise or fall together.

Inverse Proportions 

If two quantities are in proportion, and if one quantity increases and the other one decreases, and vice-versa, then this type of proportion is known as inverse proportion. The mathematical representation of inversely proportional quantities is as follows. 

$x\propto \frac{1}{y}$

This means x is inversely proportional to y. 

Key Point: In inverse proportions, the product of both quantities remains constant. 

$x_1\cdot y_1=x_2\cdot y_2$

Formula: $x\cdot y=k$

Where k is the constant of proportionality.

Example: 

Six pipes are required to fill a tank in 1 hour. How many pipes will be used if the time taken by the tank is 2 hours?

Solution: 

Let the time taken by the pipes be x. The number of pipes be y. 

Here, we can see that as the number of pipes decreases, the time taken to fill the tank also increases in such a way that the product of x and y remains constant.  

In the given example,

$6\times 1=2\times x$

$x=6÷2$

$x=3$

Hence, the number of pipes used is 3. In the following example, the product's two quantities remain the same, which is 6. 

Example: 

A car covers a distance in 40 minutes with an average speed of 60 km/h. What should be the average speed to cover the same distance in 25 minutes?

Solution: 

Let the distance covered be = x 

Converting minutes into hours = 40/60 = 2/3

We know the formula for speed is 

$Speed=\frac{Distancecovered}{Time}$

Let’s find the distance covered with the help of this formula 

$60=\frac{x}{\frac{2}{3}}$

x = 60 x 2/3 = 40 km

Hence, speed for 25minutes or (5/12) hour

Speed = 40/(5/12) = 84 km/h

Memory Tip: Opposite directions 

• In common words, "where one goes up, so the other goes down." That is, an increase of one diminishes that of the other.
• Just imagine a car’s speed and time: When the speed increases, then the time taken by the car to reach a certain distance decreases. 

3.0Key Features of CBSE Maths Notes of Class 8 Chapter 11

• The notes include memory tips at the end of each section for easy understanding of the concepts. 
• The notes are designed to make it easy for students to read and learn easily, making it ideal for self-learning. 
• These notes are aligned with the latest CBSE curriculum.