What is the Symbol used for Direct and Inverse Proportion?

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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. 

Direct and Inverse Proportion Signs: 

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, 

\[\frac {X_1} {X_2}\] = \[\frac {Y_1} {Y_2}\]

Or

\[\frac {X_1} {X_2}\] = \[\frac {Y_1} {Y_2}\]

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

Let’s say X = \[\frac {15}{Y}\]

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

Take Y = 3.

Therefore,

X = \[\frac {15}{3}\]

X = 5

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

5 = \[\frac {15}{Y}\]

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 the 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:

Consider,

the distance covered in 10 minutes = a

The time taken to cover 100 km = b

(i)  Considering,

\[\frac {45}{60}\] = \[\frac {a}{10}\]

a = \[\frac {(45 \times 10)}{60}\]

a = 7.5 km

Therefore the distance covered in 10 minutes – 7.5 kilometres

(ii)  Considering,

\[\frac {45}{60}\] = \[\frac {100}{b}\]

a = \[\frac {(100 \times 60)}{40}\]

a = 150 minutes

Therefore the time is 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 X1  =  4, X2 = 10, Y1  = 8. Find the value of Y2 if the values X and Y are varying Directly.

Solution: 

Since X are Y are varying Directly with each other:

\[\frac {X_1}{X_2}\] =  \[\frac {Y_1}{Y_2}\]    

\[\frac {4}{10}\] =  \[\frac {8}{Y_2}\]

Y2 =  \[\frac {(8 \times 10)}{4}\]

Y2 = 20

How to discern Whether it is a Direct or Indirect Proportion? 

If it is the case of Direct Proportion, then equivalent fractions will be formed as the ratio between the matching quantities would stay the same if they were to be divided. However, if it is a case of Indirect or Inverse Proportion, then if one quantity increases, the other will decrease and vice versa.

Quiz Time!

Try and solve these questions:

• X is Directly Proportional to Y here. Relate X and Y if the value of X = 50 and Y = 5.

• X is Inversely Proportional to Y here. Relate X and Y if the value of X = 49 and Y = 7.

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

Application of Direct and Inverse Proportion in Daily Life 

Direct and Inverse Proportion can be quite handy and useful to one even in their everyday life because there are innumerable quantities in our day-to-day life that tend to share a Direct and an Inverse relationship. Here are a few examples of the same: 

1. Examples of Direct Proportion in everyday life:

• If you go to the grocery store to buy vegetables, then the number of vegetables you’re planning on buying (in kg) would be Directly Proportional to its price. 

• Another example of Direct Proportion is that the amount of work done in a business firm is Directly Proportional to the number of workers present in the firm. The more people present to do the work, the more work is going to get completed. 

2. Examples of Inverse Proportion in everyday life: 

• If you’re driving a car and you increase its speed, you will take less time to reach your destination and if you drive at a slower speed, you will take much more time to reach your destination. 

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