### Ratio and Proportion

Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. The ratio and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science.

In our daily life, we use the concept of ratio and proportion such as in business while dealing with money or while cooking any dish, etc. Sometimes, students get confused with the concept of ratio and proportion. In this article, the students get a clear vision of these two concepts with more solved examples and problems.

Examples
Word Problems
For example, ⅘ is a ratio and the proportion statement is 20/25 = ⅘. If we solve this proportional statement, we get:

20/25 = ⅘

20 x 5 = 25 x 4

100 = 100

Therefore, the ratio defines the relationship between two quantities such as a:b, where b is not equal to 0. Example: The ratio of 2 to 4 is represented as 2:4 = 1:2. And the statement is said to proportion here. The application of proportion can be seen in direct proportion.

What is Ratio and Proportion in Maths?
The definition of ratio and proportion is described here in this section. Both concepts are an important part of Mathematics. In real life also, you may find a lot of examples such as the rate of speed (distance/time) or price (rupees/meter) of a material, etc, where the concept of the ratio is highlighted.

Proportion is an equation which defines that the two given ratios are equivalent to each other. For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.

Ratios and Proportion

Let us now learn Maths ratio and proportion concept one by one.

Definition of Ratio
In certain situations, the comparison of two quantities by the method of division is very efficient. We can say that the comparison or simplified form of two quantities of the same kind is referred to as ratio. This relation gives us how many times one quantity is equal to the other quantity. In simple words, the ratio is the number that can be used to express one quantity as a fraction of the other ones.

The two numbers in a ratio can only be compared when they have the same unit. We make use of ratios to compare two things. The sign used to denote a ratio is ‘:’.

A ratio can be written as a fraction, say 2/5. We happen to see various comparisons or say ratios in our daily life.

Key Points to Remember:

The ratio should exist between the quantities of the same kind
While comparing two things, the units should be similar
There should be significant order of terms
The comparison of two ratios can be performed, if the ratios are equivalent like the fractions

Definition of Proportion
Proportion is an equation which defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.

For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.

Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be in proportion. In simple words, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=’.

Continued Proportion
Consider two ratios to be a: b and c: d.

Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.

For the given ratio, the LCM of b & c will be bc.

Thus, multiplying the first ratio by c and second ratio by b, we have

First ratio- ca:bc

Second ratio- bc: bd

Thus, the continued proportion can be written in the form of ca: bc: bd

Direct And Inverse Proportion
Ratio To Percentage

Ratio and Proportion Formula
Now, let us learn the Maths ratio and proportion formulas here.

Ratio Formula
Assume that, we have two quantities (or two numbers or two entities) and we have to find the ratio of these two, then the formula for ratio is defined as;

a: b ⇒ a/b

where a and b could be any two quantities.

Here, “a” is called the first term or antecedent, and “b” is called the second term or consequent.

Example: In ratio 4:9, is represented by 4/9, where 4 is antecedent and 9 is consequent.

If we multiply and divide each term of ratio by the same number (non-zero), it doesn’t affect the ratio.

Example: 4:9 = 8:18 = 12:27

Proportion Formula
Now, let us assume that, in proportion, the two ratios are a:b & c:d. The two terms ‘b’ and ‘c’ are called ‘means or mean term,’ whereas the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’

a/b = c/d or a : b :: c : d
Example: Let us consider one more example of a number of students in a classroom. Our first ratio of the number of girls to boys is 3:5 and that of the other is 4:8, then the proportion can be written as:

3 : 5 :: 4 : 8 or 3/5 = 4/8

Here, 3 & 8 are the extremes, while 5 & 4 are the means.

Note: The ratio value does not affect when the same non-zero number is multiplied or divided on each term.

Important Properties of Proportion
The following are the important properties of proportion:

Addendo – If a : b = c : d, then a + c : b + d
Subtrahendo – If a : b = c : d, then a – c : b – d
Dividendo – If a : b = c : d, then a – b : b = c – d : d
Componendo – If a : b = c : d, then a + b : b = c+d : d
Alternendo – If a : b = c : d, then a : c = b: d
Invertendo – If a : b = c : d, then b : a = d : c
Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d

Difference Between Ratio and Proportion
To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.

### Difference Between Ratio and Proportion

To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.

 S.No Ratio Proportion 1 The ratio is used to compare the size of two things with the same unit The proportion is used to express the relation of two ratios 2 It is expressed using a colon (:), slash (/) It is expressed using the double colon (::) or equal to the symbol (=) 3 It is an expression It is an equation 4 Keyword to identify ratio in a problem is “to every” Keyword to identify proportion in a problem is “out of”

Fourth, Third and Mean Proportional
If a : b = c : d, then:

d is called the fourth proportional to a, b, c.
c is called the third proportion to a and b.
Mean proportional between a and b is √(ab).
Comparison of Ratios
If (a:b)>(c:d) = (a/b>c/d)

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios
If a:b is a ratio, then:

a2:b2 is a duplicate ratio
√a:√b is the sub-duplicate ratio
a3:b3 is a triplicate ratio

Ratio and Proportion Tricks
Let us learn here some rules and tricks to solve problems based on ratio and proportion topic.

If u/v = x/y, then uy = vx
If u/v = x/y, then u/x = v/y
If u/v = x/y, then v/u = y/x
If u/v = x/y, then (u+v)/v = (x+y)/y
If u/v = x/y, then (u-v)/v = (x-y)/y
If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as componendo -Dividendo Rule
If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c

Solved Questions
Question 1: Are the ratios 4:5 and 8:10 said to be in Proportion?

Solution:

4:5= 4/5 = 0.8 and 8: 10= 8/10= 0.8

Since both the ratios are equal, they are said to be in proportion.

Question 2: Are the two ratios 8:10 and 7:10 in proportion?

Solution:

8:10= 8/10= 0.8 and 7:10= 7/10= 0.7

Since both the ratios are not equal, they are not in proportion.

Question 3: Given ratio are-

a:b = 2:3

b:c = 5:2

c:d = 1:4

Find a: b: c.

Solution:

Multiplying the first ratio by 5, second by 3 and third by 6, we have

a:b = 10: 15

b:c = 15 : 6

c:d = 6 : 24

In the ratio’s above, all the mean terms are equal, thus

a:b:c:d = 10:15:6:24

Word Problems
Question 4: Out of the total students in a class, if the number of boys is 5 and the number of girls being 3, then find the ratio between girls and boys.

Solution: The ratio between girls and boys can be written as 3:5( Girls: Boys). The ratio can also be written in the form of factor like 3/5.

Question 5: Two numbers are in the ratio 2 : 3. If the sum of numbers is 60, find the numbers.

Solution: Given, 2/3 is the ratio of any two numbers.

Let the two numbers be 2x and 3x.

As per the given question, the sum of these two numbers = 60

So, 2x + 3x = 60

5x = 60

x = 12

Hence, the two numbers are;

2x = 2 x 12 = 24

and

3x = 3 x 12 = 36

24 and 36 are the required numbers.

What is the ratio with an example?
A ratio is a mathematical expression written in the form of a:b, where a and b are any integers. It expresses a fraction. For example. 2:3 = ⅔.
What is a proportion with example?
A proportion is a statement where two or more ratios are equivalent. For example, ⅔ = 4/6 = 6/9.
How to solve proportions with examples?
If a:b::c:d is a proportion, then;
a/b=c/d