Ratio And Proportion


Ratio and proportion is one of the most important concepts in Mathematics. The concept of the ratio is used for the comparison of two quantities, whereas the concept of proportion is used to show equivalency between given ratios.

The ratio of two numbers a and b is given by a divided by b and represented by a/b or a:b

While the proportion is the relation between two ratios of a/b and c/d is represented by a:b::c:d or a/b=c/d.

Here in this article, we are going to discuss the concepts of ratio and proportion, then some tricks with examples.

Ratio and Proportion in Daily Life

Ratio and Proportion are the used concepts in Mathematics and in daily life. For instance, the concept of ratio is found in many daily life things like price (rupees/metres), speed (distance/time), etc. Similarly, you can also find concepts of proportion also in real life. 

Let's take an example; 

Weight and rate of potatoes: 1kg of potatoes costs 20 rupees, then 2kg of potatoes costs 40 rupees such that 1/20=2/40. There are many more examples which we will discuss later in this article.

Concept of Ratio

Comparison/simplification of two quantities by the method of division is known as ratio. The result of this simplification gives the number of times a quantity is equal to another, or you can say the ratio is used to express a quantity as a fraction of the other two numbers of a ratio. It can be compared when they have the same unit. ':' is the sign used to denote ratio, which is represented as a:b.


The ratio can be represented by fraction; for example, 4/7 you can use 'to' to represent the ratio which is given as '4 to 7'. 

Concept of Proportion

A proportion is said to be the comparison of two ratios. '::' and '=' are the symbols to denote proportions. When two ratios a:b and c:d are equal in value, then they are said to be a proportion.

Concept of Continued Proportion

Let a:b and c:d are two ratios, to find a continued proportion of two given ratios, we have to convert the means to a single number. The means would be converted to a single number by taking LCM. LCM of b and c by the given ratio is bc.

Then multiply the first ratio by c and the second term would be multiplied by b. So, now we have;

First ratio= ca:bc

Second ratio= bc:bd

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

Formulation of Ratio and Proportion

Now, let's learn the formulae of Ratio and Proportion using two entities a and b;

The formula for ratio is defined as

        a:b= a/b

Where 'a' is the first time or antecedent and 'b' is the second term or consequent.

Formula for proportion

Let a:b and c:d are two ratios then

          a:b::c:d      

              or

          a/b=c/d

Where 'b' and 'c' are 'means or mean terms' and 'a' and 'd' are called 'extremes or extreme terms'.

Let us consider an example of the number of employees in a company. First ratio of the number of male to female workers is given by 4:7, and the other is given by 3:10 then the proportion is given by 4:7::3:10 

Here, 7 and 3 are mean terms and 4 and 10 are extremes. 

NOTE:-  When the ratio is multiplied and divided by the same non - zero number, there will be no effect on the ratio.

Tricks for Ratio and Proportion

  • If a/b = c/d, then ad = bc

  • If a/b = c/d, then a/c = b/d

  • If a/b = c/d, then b/a = d/c

  • If a/b = c/d, then (a+b)/b = (c+d)/d

  • If a/b = c/d, then (a-b)/b = (c-d)/d

  • Componendo -Dividendo Rule = If a/b = c/d, then (a+b)/ (a-b) = (c+d)/(c-d)

  • If a/b = b/c, then a/c = a2/b2

  • If a/b = c/d, then a = c and b =d

  • If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c


Few Examples

Question 1 

There are 30 girls and 35 boys in a class. Find the ratio of the no. of boys to the total no. of students.

Solution

Ratio of number of boys to total no. of students, 

                    35/75= 7/15 = 7:15

Question 2

Determine if the following are in proportion.

  1. 15,45,40,120

  2. 32,48,70,210

    Solutions

  1. 15/45   = 1/3 

40/120 = 1/3 

Hence, 15/45 = 40/120 

Then these are in proportion.


  1. 32/48   = 2/3  

70/210 = 1/3 

Hence 32:48 and 70:210 are not equal, the. These are not in proportion.


Question 3

If the cost of cloth is rs 2170, find the cost of 5m cloth.

Solution

Cost of 7m cloth = 2170

Cost of 1m cloth = 2170/7 = 310

Cost of 5m cloth = 310*5 = 1550

     

Question 4

Weight of 72 books is 9 kg. What is the weight of 40 such books?

Solution

Weight of 72 books = 9kg

Weight of 1 book = 9/72kg = 1/8   

So, weight of 40 books = 1/8*40 = 5kg


Question 5

Determine the proportion of the given ratios. Also, write the means and extremes. 25cm:1m and rs40:rs160 

Solution

 25cm = 25/100 

             = 0.25m

0.25/1 = 1/4

40/160 = 1/4

Hence, these terms are in proportion.

Means = 1m and rs 40

Extremes = 25cm and rs 160.