Lesson No. 2: Proportion

 Two ratios that are equal are said to be in proportion.

For example the ratio 6:8 is 3: 4, and the ratio 12: 16 is also 3: 4.

So, the two ratios, 6: 8 and 12: 16 are said to be in proportion.

We say “the ratio of 6 to 8 is same as that of 12 to 16”, or in short as:

“6 is to 8 as 12 is to 16”

In general, four different numbers a, b, c and d are in proportion if

“the ratio a to b is same as the ratio of c to d”

and it is denoted as

a: b :: c : d

(and read as “a is to be is same as c is to d”)

For example, 2 is to 3 as 4 is to 6 is denoted as 2: 3 :: 4: 6

In the proportion a: b :: c : d,

a and d are called extremes and b and c are called means.

And a Very Important Rule:

For example in the proportion 2: 3 :: 4: 6,

Product of Extremes 2 and 6 is 12 and product of means 3 and 4 is also 12.

There is another kind of proportion.

Continued Proportion:

Three numbers a, b and c are said to be in continued proportion if

a : b :: b : c

In this proportion,

a is called the first proportional,

b is called the Mean proportional,

c is called the third proportional.

Again, applying the rule of “product of means is equal to product of extremes” in the proportion a: b :: b : c, we have

b2 = a × c,

so b =  √ (a × c)


The mean proportional of two numbers is the square root of their product.  

The Two Types of Proportions:

There are two types of Proportions:

Direct and


What is Direct Proportion?

Two quantities are said to be in Direct Proportion, if a change in one brings a similar change in the other.

Change includes increase and decrease.

An increase or decrease in one causes a corresponding increase or decrease in the other.

For example,

A box of pencils costs $10, then 2 same boxes will cost $20.

Here, the two quantities are number of boxes of pencils and their cost.

As the number (of boxes of pencils) “N”increases, then price “P”also increases.

So, there is a direct proportion between Number and Price. It is denoted as

N α P

What is Indirect Proportion?

Two quantities are in Indirect Proportion, if an increase or decrease in one brings an opposite change in the other quantity.

For Example,

An amount of $120 is to be equally divided between 3 men. Each one will get $40

If one man leaves the group, then the remaining 2 men will receive $60 each.

Fewer men leaves greater share for each.

So, there is an indirect proportion between Number and Share (of each)

N α 1/A

N is number of persons and A is the amount of each person’s share

A few more examples:

Example 1.

To reach your school earlier, you will have to ride faster.

So, speed and time are indirectly related.  

Greater the speed, lesser is the time taken to cover a same distance.

Example 2.

3 men, all of same efficiencies, can build a wall in 1 hour.

Then, 1 man alone will take 3 hours

Again, between Number of men and time taken to finish a same work, there is an indirect proportion.

Example 3.

To score high, you need to put greater efforts,

So, Result and Effort are directly related (Agree/Disagree?)