What is Ratio and Proportion? – Definition, Formulas, Examples with Answers

Ratio and Proportion are mainly explained using fractions. If a fraction is expressed in the form of a:b it is called a ratio and when two ratios are equal it is said to be in proportion. Ratio and Proportion is the fundamental concept to understand various concepts in maths. We will come across this concept in our day to day lives while dealing with money or while cooking any dish. Check out Definitions, Formulas for Ratio and Proportion, and Example Questions belonging to the concept in the further modules.

Quick Links of Ratio and Proportion Topics

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What is Ratio and Proportion?

Ratio and Proportion is a crucial topic in mathematics. Find Definitions related to Ratio and Proportion along with examples here.

In Certain Situations comparison of two quantities by the division method is efficient. Comparison or Simplified form of two similar quantities is called ratio. The relation determines how many times one quantity is equal to the other quantity. In other words, the ratio is the number that can be used to express one quantity as a fraction of other ones.

Points to remember regarding Ratios

  • Ratio exists between quantities of a similar kind
  • During Comparison units of two things must be similar.
  • There should be significant order of terms
  • Comparison of two ratios is performed if the ratios are equivalent similar to fractions.

Proportion – Definition

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

Ex: For instance, a train travels at a speed of 100 km/hr and the other train travels at a speed of 500km/5 hrs the both are said to be in proportion since their ratios are equal

100 km/hr = 500 km/5 hrs

Continued Proportion

Consider two ratios a:b and c:d then in order to find the continued proportion of two given ratio terms we need to convert to a single term/number.

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

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

The first ratio becomes ca: bc

The second ratio becomes bc: bd

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

Ratio and Proportion Formulas

Ratio Formula

Let us consider, we have two quantities and we have to find the ratio of these two, then the formula for ratio is defined as

a: b ⇒ a/b

a, b be two quantities. In this a is called the first term or antecedent and b is called the second term or consequent.

Example: In the Ratio 5:6 5 is called the first term or antecedent and 6 is called the consequent.

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

Proportion Formula

Consider two ratios are in proportion a:b&c:d the b, c are called means or mean terms and a, d are known as extremes or extreme terms.

a/b = c/d or a : b :: c : d

Example: 3 : 5 :: 4 : 8 in this 3, 8 are extremes and 5, 4 are means

Properties of Proportion

Check out the important list of properties regarding the Proportion Below. They are as follows

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

Difference Between Ratio and Proportion

S.No. Ratio Proportion
1 The ratio is used to compare two similar quantities having the same units 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 The keyword to identify ratio in a problem is “to every” The keyword to identify proportion in a problem is “out of”
4 It is an expression It is an equation

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

Check out the Tricks and Tips to Solve Problems related to Ratio and Proportion. They are as under

  • If u/v = x/y, then u/x = v/y
  • If u/v = x/y, then uy = vx
  • 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), it 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 on Ratio and Proportion

1. Are the Ratios 4:5 and 5:10 said to be in Proportion?


Expressing the given ratios 4:5 we have 4/5 = 0.8

5:10 = 5/10 = 0.2

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

2. Out of the total students in a class, if the number of boys is 4 and the number of girls being 5, then find the ratio between girls and boys?


The ratio between girls and boys is 5:4. The ratio can be written in factor form as 5/4

3. Two numbers are in the ratio 3 : 4. If the sum of numbers is 42, find the numbers?


Given 3/4 is the ratio of any two numbers

Let us consider the numbers be 3x and 4x

Given, 3x+4x = 42

7x = 42

x = 42/7

x = 6

finding the numbers we have 3x = 3*6 = 18

4x = 4*6 = 24

Therefore, two numbers are 18, 24




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