Direct and Inverse Proportion

Direct Proportion and Inverse Proportion Symbol

To show how quantities are related to each other, you use direct or inverse proportion or a proportional symbol. When two quantities X and Y increase together or decrease together, they are said to be directly proportional or they are in direct proportion with each other. It is also known as a direct variation. The ratio of these values will remain constant. But when quantities X and Y are inversely proportional to each other or in the inverse proportion, one quantity decreases when the other quantity increases or when one quantity increases the other quantity decreases. It is also known as inverse variation. The ratio of these values varies inversely. 

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Direct Proportion and Inverse Proportion Signs


Proportional Symbol


X ∝ Y

This is how the directly proportional symbol is denoted 

X ∝ 1/Y

This is how the inversely proportional sign is denoted  


When two quantities X and Y are directly proportional to each other, we say “X is directly proportional to Y” or “Y is directly proportional to X”. When two quantities X and Y are inversely proportional to each other, we say that “X is inversely proportional to Y” or “Y is inversely proportional to X”. 

Properties of Direct and Indirect Proportion.

Direct Proportion

  • When one quantity increases the other quantity increases too.

  • When one quality decreases the other quantity decreases too.

  • The corresponding ratios always remain constant.

  • It is also called a direct variation.

Example:

Let’s say: X is directly proportional to Y here. Relate X and Y if the value of X = 8 and Y = 4.

Solution:

We know, X ∝ Y

Or we can also write it as X = kY, where k = is a constant proportionality. 

8 = k x 4

k = 2.

Hence the relating equation between the two variables would be: X = 2Y.

Indirect Proportion

  • When one quantity increases the other quantity decreases too.

  • When one quantity decreases the other quantity increases too.

  • The corresponding ratios always vary inversely.

  • It is also called an indirect variation.

Example: 

Let’s say: X is inversely proportional to Y here. Relate X and Y if the value of X = 815 and Y = 3.

Solution:

Let’s consider X1X2 to be the components of X and Y1Y2 to be the components of y.

Then, 

X1 / X2 = Y1 / Y2

Or

X1Y1 = X2 Y2

The statement “X is inversely proportional to Y” can be written as X ∝ 1/Y.

Let’s say X = 15 / Y

Since we have the value of one variable, the other can be figured out easily.


Take Y = 3.


Therefore,


X = 15 / 3


X = 5


Since we now know X’s value is 5, the value of Y can be found.


5 = 15 / Y


Y = 3

How to Write Direct and Indirect Proportion Equations?


Step 1: You will have to write down the proportional symbol


Step 2: With the help of constant of proportionality, convert the symbol into an equation


Step 3: Next, you will have to figure out the constant of proportionality with the information that is given to you


Step 4: Now substitute the constant value in an equation

Examples of Direct and Indirect Proportion


Example 1: 45 km/hr is the uniform speed of the train at which it is moving. Find:


(i) the distance covered by it in 10 minutes

(ii) the time required to cover 100 km

Solution:

Distance Travelled (km)

45

a

100

Time Taken (m) 

60

10

b


Consider,

the distance covered in 10 minutes = a

The time taken to cover 100 km = b


(i)  Considering,


45 / 60 = a / 10

a = (45 x 10) / 60

a = 7.5 km

Therefore the distance covered in 10 minutes - 7.5 kilometres

(ii)  Considering,

45 / 60 = 100 / b

a = (100 x 60 ) / 40

a = 150 minutes

Therefore the time taken to cover 100 kilometres - 150 minutes.


Example 2:

Let’s say: X is directly proportional to Y here. Relate X and Y if the value of X = 100 and Y = 25.

Solution:

We know, X ∝ Y

Or we can also write it as X = kY, where k = is a constant proportionality. 


100 = k x 25


k = 4.


Example 2:

The value of X=  4, X2 = 10, Y1  = 8. Find the value of Y2 if the value X and Y are varying directly.

Solution: 

Since X are Y are varying directly with each other: 


X1 / X2 = Y1 / Y2


4 / 10 =  8 / Y2


Y2 =  (8  x 10) / 4


Y2 = 20


Quiz Time

Try and solve these questions:

  1. X is directly proportional to Y here. Relate X and Y if the value of X = 50 and Y = 5.

  2. X is inversely proportional to Y here. Relate X and Y if the value of X = 49 and Y = 7.

  3. The cost of 17 books is Rs. 400. How much would be the cost of 5 books?

FAQ (Frequently Asked Questions)

1. What is Proportionality?

To show how quantities are related to each other, you use direct or inverse proportion or a proportional symbol. When two quantities X and Y increase together or decrease together, they are said to be directly proportional or they are in direct proportion with each other. When quantities X and Y are inversely proportional to each other or in the inverse proportion, one quantity decreases when the other quantity increases or when one quantity increases the other quantity decreases. It is also known as inverse variation.

2. What is direct and Indirect Proportionality?

When two quantities X and Y increase together or decrease together, they are said to be directly proportional or they are in direct proportion with each other. It is also known as a direct variation. The ratio of these values will remain constant. But when quantities X and Y are inversely proportional to each other or in the inverse proportion, one quantity decreases when the other quantity increases or when one quantity increases the other quantity decreases. It is also known as inverse variation. The ratio of these values varies inversely. 

3. What is a Direct Proportionality Symbol?

The direct proportionality symbol is donated by  “∝” and the indirect proportionality symbol is denoted by “1/∝”