Understanding the Concept of Ratio in Mathematics

Many times we have to compare two values, we can do this by the method of division or finding out ratio. A ratio is an expression which is used to compare two similar quantities is termed as ratio. If a and b are two different numbers or integers, then the ratio of these two integers can be represented as a/b or a:b. 

For example: the below figure represents a 2 : 6 ratio.

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Definition of Ratio

We can say that the comparison of two quantities of the same kind by division 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. 

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Representation of Ratio

The sign used to denote a ratio is ‘:’.

For example A ratio can be written as a fraction, say 5/4, or can be represented by 5 : 4 and read “ 5 is to 4.” 

The ratio of any two same quantities p and q can be expressed as either p/q or p : q. Here p is called 'antecedent' and q is called 'consequent'.

Ratios are used to compare several quantities like length, height, width, etc. ratios are represented in two forms of notations.

• Odds Notation

• Fractional Notation

Odds notation uses ‘:’ i.e. “ is to “ in the expression. For example 2:3.

Ratios in Fractional notation are represented in fraction form. For example ⅔.

The order of the val;uews is important in ratios. When we say there are 4 red marbles and 7 blue marbles the ratio of red marbles to blue marbles will be 4:7 or 4/7 it cannot be 7 : 4 or 7/4.

How to Solve Ratios?

Let us consider an example

If the salaries Ram and Shyam are Rs.7000 and Rs. 9000 respectively. Then we can say that their respective salaries are in the ratio of 7 : 9. Here 7 and 9 are the simplified representation of the original values 7000 and 9000, or 7 and 9 are the relatively prime numbers. Means, we can't simplify these values further as integers.

In a reverse approach, if it is given that the ratio of the salaries of Ram and Shyam is 7 : 9, then it doesn't mean that the salaries of Ram and Shyam are 7 rupees and 9 rupees respectively. Instead their respective salaries are a certain multiple of 7 and 9. So, from the given data, we can express the salaries of Ram and Shyam are in the following way;

Salary of Ram = 7x

Salary of Shyam = 9x, where x is a positive real number.

And 'x' is called the 'Multiplicative Constant'.

Calculation of Given Ratio

To calculate the original values from a given ratio we require at least one constant value related to the given data, which may be in any of the following manner;

• Sum of their individual salaries is Rs.16,000.

• Shyam's salary is Rs.2,000 more than that of Ram.

As per given ratio expression, Ram's and Shyam's salaries are in the ratio of 7 : 9.

Solution:

Let Ram's salary = 7x

And Shyam's salary = 9k

Case 1: Sum of their individual salaries is Rs.16,000.

Sum of salaries = 7x + 9x = 16x

16k = 16,000

x = 1,000

Substitute the values of x

Ram's salary = 7x 

= 7 x 1000

= Rs.7,000.

Shyam's salary = 9x

= 9 x 1000

= Rs.9000.

Case 2: Shyam's salary is Rs.2,000 more than that of Ram.

Shyam's salary - Ram's salary = 2000

9x - 7x = 2000

2x = 2000

x = 1000

Therefore, Ram's salary = 7x

= 7 x 1000

= Rs.7,000.

Shayam's salary = 9k

= 9 x 1000

= Rs.9000.

In both the cases we found the actual salary of Ram and Shyam.

Types of Ratios in Math

There are various types of ratios in Maths they are as follows:

• Compounded Ratio: if in two or more ratios, we take ratio of antecedent as product of antecedents of the ratios and consequent as product of consequents, then the ratio thus formed is called compounded ratio. For example, the compound ratio of m : n and p :  q is mp : nq and that of a : b, c : d and e : f is the ratio ace : bdf.

• Duplicate Ratio: The duplicate ratio is the ratio of two equal ratios. For example duplicate ratio of  a : b is  a2 : b2

In other words,

The duplicate ratio of the ratio a : b = Compound ratio of a : b and a : b

                                               = (a × a) : (b × b)

                                               = a2 : b2

• Reciprocal Ratio: The reciprocal ratio of a:b is the ratio (1/a):(1/b), where a≠0 and b≠0. Which can also be written as 1/a : 1/b = b: a. Hence reciprocal ratio is also called as inverse ratio of the previous ratio.

For example the reciprocal ratio of 2 : 3 is ½ : ⅓ = 3 : 2

• Ratio of Equalities: If the antecedent and consequent are equal, the ratio is called ratio of equality, for example 5 : 5.

• Ratio of Inequalities: If the antecedent and consequent are not equal, the  ratio is called the ratio of inequality, for example 5:7.

Let us solve the problems involving ratios.

Solved Examples

Example 1 :The ratio of the present ages of Raj and Rohan is 4:3. Five years ago it was 7:5. Find the present age of rohan.

Solution:

Let x be the common multiple of the ratio.

Let Raj's present age = 4x

And Rohan's present age = 3x

Five years ago;

Raj's age = 4x - 5

Rohan's age = 3x - 5

So the expression formed will be 

\[\frac{(4x - 5)}{(3x - 5)} = \frac{7}{5}\]

5(4x - 5) = 7(3x - 5)

20x - 25 = 21x - 35

35 - 25 = 21x - 20x

x = 10

Substitute value of x 

Rohan's present age = 3x

= 3 x 10 = 30 years

Therefore present age of rohan is 30 years

Example 2: A total of 120 candies are distributed among three friends Raj, Ravi and Rohan in a respective ratio of 3:4:5. Find the share of each friend.

Solution:

Sum of the ratios  = 3 + 4 + 5 = 12

Total 120 is divided into 12 equal parts. Each part consists of 10 candies.

Share for Raj = 3/12 of 120 

= \[\frac{3}{12}\] x 120

= 3 x 10

= 30 candies.

Share for Ravi = \[\frac{4}{12}\] of 120 

= \[\frac{4}{12}\] x 120

= 4 x 10

= 40 candies.

Share for Rohan = \[\frac{5}{12}\] of 120

= \[\frac{5}{12}\] x 120

= 5 x 10

 = 50 candies.

Therefore , 30, 40 and 50 candies were distributed among raj, ravi and rohan respectively.