Ratio and Proportion - Important formulas, Shortcuts & Tricks

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The three main fundamental pillars of Arithmetic are Percentages, Averages, Variations, Ratio and Proportion. So the aspirant should create an in-depth understanding about the concept of Ratio, Proportion and Variation'. Application of Ratio and Variation are broadly expanded in all the other Quantitative Topics. Hence it is a mandatory required topic for your serious preparation for the exams. In this article we provide important formulas, shortcuts and tricks to solve aptitude questions on ratio and proportion with help of examples.

 

The main objectives, that we are going to deal with;
  • Definition and basic concept of ratio.
  • Calculations on ratio values.
  • Application of algebraic constant.
  • Ratio applications in age problems.
  • Ratio applications in Mixtures.
  • Ratio application in Geometry.
  • Income- Expenditure- Savings problems.
  • Different ratio expressions and its effective transformations.
  • Properties of ratio.
  • Combining ratios
  • Proportion
  • Continued Proportion
  • Re-arrangements of proportionals
  • Variation
  • Joint Variation

Concept of ratio.

'Ratio is the relation between two or more same kind of quantities”. More clearly ratio is the magnitude comparison of quantities in same nature.

For example; if the salaries of Ajay and Bhavan are Rs.5000 and Rs. 6000 respectively. Then we can say that their respective salaries are in the ratio of 5:6. Here 5 and 6 are most relatively simplified representation of the original values 5000 and 6000, or 5 and 6 are the relatively prime numbers (doesn't have any common factors other than 1) which are the factors of the given values. Means, we can't simplify these values further as integers.

In a ratio expression, the order of values is very important. In the above example the required ratio is 5:6 instead 6:5 is wrong.

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

In a reverse approach, if it is given that the ratio of the salaries of Ajay and Bhavan is 5:6, then it doesn't mean that the salaries of Ajay and Bhavan are 5 rupee and 6 rupee respectively. Instead their respective salaries are a certain multiple of 5 and 6. So, from the given data, we can express the salaries of Ajay and Bhavan are in the following way;

Salary of Ajay = 5k

Salary of Bhavan = 6k, where k is a positive real number.

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

Calculation as per ratio expression:

For finding the original quantities form a given ratio expression required 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.11,000.
  • Bhavan's salary is Rs.1,000 more than that of Ajay.

How to utilize these data for finding the individual salaries as per the pre mentioned example?

As per given ratio expression, Ajay's and Bhavan's salaries are in the ratio of 5:6.

Let Ajay's salary = 5k
And Bhavan's salary = 6k

From data (i);

Sum of salaries = 5k + 6k = 11k
11k = 11,000
K = 1,000
From the assumption,
Ajay's salary = 5k
= 5 * 1000
= Rs.5,000.
Bhavan's salary = 6k
= 6*1000
= Rs.6000.
From data (ii);
Bhavan's salary - Ajay's salary = 6k - 5k
= k =1,000
Therefore, Ajay's salary = 5k
= 5 *1000
= Rs.5,000.
Bhavan's salary = 6k
= 6*1000
= Rs.6000.

Example: (practical understanding of Ratio)

In Cahndu's birthday celebration, he cut a beautiful and delicious round chocolate cake and distributed it to his family members, consisting his father, mother and sister. The ratio of the quantity of cake received by Chandu, father, mother and sister is 3:1:2:4.

Means; if he divided the cake into ten equal pieces, Chandu got 3 out of the ten equal pieces.

His father got 1 out of the ten equal pieces.

His mother got 2 out of the ten equal pieces, and his sister got 3 out of the ten equal pieces.

We can express the same concept in another way.

Sum of the ratio values = 3 + 1 + 2 + 4 = 10

Consider the given full cake into ten equal pieces.

Then the share for Chandu = 3/10 of the cake.

Share for Chandu 's father = 1/10 of the cake.

Share for Chandu 's father = 2/10 of the cake.

Share for Chandu 's sister = 4/10 of the cake.

Example:

A total of 120 toffees are distributed among three friends A, B and C in a respective ratio of 3:4:5. Find the share of each member.

Solution:

Sum of ratio values = 3 + 4 + 5 = 12

Total 120 is divided in to 12 equal parts. Each part consists 10 toffees.

Share for A = 3/12 of 120 = 30 toffees.

Share for B = 4/12 of 120 = 40 toffees.

Share for C = 5/12 of 120 = 50 toffees.

Effective mental calculation method for ratio related problems.

There are lots of various methods for finding the answer of a ratio related question. For better understanding, we will explain various concepts with using example.

Example 1:

If the number of students in three different classes A, B and C is in the respective ratio of 5:6:8. It is given that the strength in class C is 15 students lesser than the combined strength of the classes A and B together. Find the strength of class A.

For this question we can consider three different methods. Finally you can select one among them for finding the answer in the quickest way.

Method 1: Algebraic method (Application of algebraic constant)

Let the strengths of classes A, B and C are 5k, 6k and 8k respectively, where 'k' is the algebraic constant.

Combined strength of A and B is 5k + 6k = 11k

Strength of C is 8k, this is 3k lesser than the combined strength of A and B, i.e. 11k - 8k = 3k

3k = 15

Therefore, k = 5

Strength of class A = 5k = 25 students.

Method 2: Fractional approach

Let T be the total strength of the classes A, B and C.
i.e. A + B + C = T
A = 5/19 of T
B = 6/19 of T
C = 8/19 of T
A + B - C = (5/19 + 6/19 - 8/19) of T
= 3/19 of T
Given 3/19 of T = 15
Therefore T = (15 * 9) / 3 = 95
Strength of A = 5/19 of 95 = 25 students.
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