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Sequences and Series

Ratio and Proportion

Last Updated : 20 Apr, 2025

Ratio and Proportion are used for comparison in mathematics. Ratio is a comparison of two quantities while Proportion is a comparison of two ratios.

When a fraction a/b is written a:b then it is termed as a ratio b. When two ratios let's say a:b and c:d are equal a:b and c:d are said to be proportional to each other. Two proportional ratios are written as a:b::c:d.

Ratio

Ratio is a comparison of two quantities of the same unit. The ratio of two quantities is given by using the colon symbol (:). The ratio of two quantities a and b is given as :

a : b
where
a is called Antecedent.
b is called Consequent.

The ratio a:b means ak/bk where k is the common factor. k is multiplied to give equivalent fractions whose simplest form will be a/b. We can read a:b as 'a ratio b' or 'a to b'.

Ratio Properties

Some Key properties of Ratio are:

If a ratio is multiplied by the same term both in antecedent and consequent then there is no change in the actual ratio.
Example, A : B = nA : nB
If the antecedent and consequent of a ratio are divided by the same number then also there is no change in the actual ratio.
Example, A : B = A/n : B/n
If two ratios are equal then their reciprocals are also equal.
Example: If A : B = C : D then B : A = D : C
If two ratios are equal then their cross-multiplications are also equal.
Example: A : B = C : D then A × D = B × C.
The ratios for a pair of comparisons can be the same but the actual value may be different.
Example 50:60 = 5:6 and 100:120 = 5:6 hence ratio 5:6 is the same but the actual value is different.

Proportion

Proportion refers to the comparison of ratios. If two ratios are equal then they are said to be proportionate to each other. Two proportional ratios are represented by a double colon(::). If two ratios a:b and c:d are equal then they are represented as

a : b :: c : d
where
a and d are called extreme terms.
b and c are called mean terms.

Proportion Property Illustration

Proportion Properties

Key properties of Proportions are:

For two ratios in proportion i.e. A/B = C/D, A/C = B/D holds true.
For two ratios in proportion i.e. A/B = C/D, B/A = D/C holds true.
For two ratios in proportion i.e. A : B :: C : D, the product of mean terms is equal to the product of extreme terms i.e. AD = BC
For two ratios in proportion i.e. A/B = C/D, (A + B)/B = (C + D)/D is true.
For two ratios in proportion i.e. A/B = C/D, (A - B)/B = (C - D)/D is true.

Types of Proportions

There are three types of Proportions:

Direct Proportion : When two quantities increase and decrease in the same ratio then the two quantities are said to be in Direct Proportion. It means if one quantity increases/decreases then the other will also increase/decrease. It is represented as a ∝ b.

Example, if the speed of vehicle increases then the distance travelled will also increase. ( provided time is same in both scenarios ) ....

Inverse Proportion: When two quantities are inversely related to each other i.e. increase in one leads to a decrease in the other or a decrease in the other leads to an increase in the first quantity then the two quantities are said to be Inversely Proportional to each other.

Example, if the speed of vehicle increases then the time taken to travel the same distance travelled will decrease.

Continued Proportion: If the ratio a:b = b:c = c:d, then we see that the consequent of the first ratio is equal to the antecedent of the second ratio, and so on then the a:b:c:d is said to be in continued proportion.

If the consequent and antecedent are not the same for two ratios then they can be converted into continued proportion by multiplying.

For Example, in the case of a:b and c:d consequent and antecedent are not same then the continued proportion is given as ac:cb:bd.
In the continued proportion a:b:c:d., c is called the third proportion, and d is called the fourth proportion.

Check: Quiz on Ratio and Proportion

Ratio and Proportion Formulas

Let's discuss the formulas for ratio and proportion in detail.

Compound Ratios: If Two ratios are multiplied together then the new ratio formed is called the compound ratio. Example a:b and c:d are two ratios then ac:bd is a compound ratio.

Duplicate Ratios:

For a:b, a²:b² is called duplicate ratios
For a:b, √a:√b is called sub-duplicate ratios
For a:b, a³:b³ is called triplicate ratios

Proportion Formulas:
These are the formulas used to solve problems of proportion:

If a:b = c:d, then we can say that (a + c):(b + d), it is also called Addendo.
If a:b = c:d, then we can say that (a – c):(b – d), it is also called Subtrahendo.
If a:b = c:d, then we can say that (a – b):b = (c – d):d, it is also known as Dividendo.
If a:b = c:d, then we can say that (a + b):b = (c + d):d, it is also known as Componendo.
If a:b = c:d, then we can say that a:c = b:d, it is also known as Alternendo.
If a:b = c:d, then we can say that b:a = d:c, it is also called Invertendo.
If a:b = c:d, then we can say that (a + b):(a – b) = (c + d):(c – d), it is also known as Componendo and Dividendo.
If a is proportional to b, then it means a = kb where k is a constant.
If a is inversely proportional to b, then a = k/b, where k is a constant.
Dividing or multiplying a ratio by a certain number gives an equivalent ratio.

Mean Proportion: Consider two ratios a:b = b:c then as per the rule of proportion product of the mean term is equal to the product of extremes, this means b² = ac, hence b = √ac is called mean proportion.

Difference between Ratio and Proportion

The comparison between Ratio and Proportion is tabulated below:

Ratio vs. Proportion
Ratio	Proportion
Ratio is used to compare two quantities of the same unit	Proportion is used to compare two ratios
Ratio is represented using (:), a/b = a:b	Proportion is represented using (::), a:b = c:d ⇒ a:b::c:d
Ratio is an expression	Proportion is an equation that equates two ratios

Read More:

Percentage
Ratios and Percentages
Fractions

Ratio and Proportion Trick

Let us learn here about some rules and tricks to solve question-related ratios and proportions:

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)
If a/b + c = b/a +c = c/ a + b then a + b + c ≠ 0 then a = b = c

Solved Examples of Ratio and Proportion

Now let's solve some questions on the properties of Ratio and Proportion we discussed so far.

Example 1: Is the ratio 5:10 proportional to 1:2?
Solution:

5:10 divided by 5 gives 1:2. Thus, they are same to each other. So we can say that 5:10 is proportional to 1:2.

Example 2: Given a constant k, such that k:5 is proportional to 10:25. Find the value of k.
Solution:

Since k:5 is proportional to 10:25, we can write,
k / 5 = 10 / 25
k = 10/25 × 5 = 2
So, the value of k is 2.

Example 3: Divide 100 into two parts such that they are proportional to 3:5.
Solution:

Let's the value of two parts are 3k and 5k, where k is a constant.
Since the total sum of two parts is 100, we can write,
3k + 5k = 100
8k = 100
k = 12.5
So, the parts are 3k = 3 × 12.5 = 37.5 and 5k = 5 × 12.5 = 62.5

Example 4: If x² + 6y² = 5xy, then find the value of x/y.
Solution:

Given, x² + 6y² = 5xy.
Dividing the equation by y², we get
(x/y)² + 6 = 5 (x/y)
Let's x/y = t
So, we can write,
t² + 6 = 5t
t² - 5t + 6 = 0
(t - 2)(t - 3) = 0
t = 2 or t = 3
Since, t = x/y, we get
x/y = 2 or x/y = 3

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