Like the Percentage section, Ratio and Proportion is also a very important chapter for students studying in school or preparing for SSC, RRB, Banking, NDA, CDS, Defence, Police and other competitive exams.
This page covers the basics of Ratio and Proportion and links to other important pages that explain all the major concepts with questions and answers.
Once you go through this Ratio and Proportion section along with its linked pages, you will develop a strong understanding of the topic and be able to solve Ratio and Proportion questions easily in competitive examinations.
What is Ratio?
Ratio means comparing two quantities of the same kind.
It is usually represented in the form of a : b or a/b. Here, a is called the antecedent (first term) and b is called the consequent (second term).
Using the concept of ratio, we can compare the heights of two persons, the marks of two students, the salaries of two employees, or the populations of two cities.
For Example 1:
If the height of an SSC building is 65 feet and the height of a UPSC building is 75 feet, then we can write their ratio as:
SSC Building : UPSC Building = 65 : 75
or
UPSC Building : SSC Building = 75 : 65
=15:13
We can also write a ratio in fraction form:
UPSC Building / SSC Building = 75 / 65
= 15/13
Thus, a ratio helps us compare one quantity with another.
Let us take another example.
Example 2:
If the height of a boy is 5 feet 5 inches and the height of a girl is 5 feet 3 inches, then we should first convert both heights into the same unit.
Boy’s height = (5 × 12) + 5 = 65 inches
Girl’s height = (5 × 12) + 3 = 63 inches
Therefore,
Boy’s height : Girl’s height = 65 : 63
Thus, the ratio of the boy’s height to the girl’s height is 65 : 63.
Note: Before forming a ratio, both quantities should be expressed in the same unit. For example, feet and inches should be converted into inches, and kilograms and grams should be converted into the same unit.
Confusion: Many students might think that if we convert the heights into feet instead of inches, the ratio may change. Let us check it.
We know that:
1 inch = 1/12 foot
Therefore,
Boy’s height = 5 feet 5 inches
= 5 + 5/12
= 65/12 feet
Girl’s height = 5 feet 3 inches
= 5 + 3/12
= 63/12 feet
Now,
Boy’s height : Girl’s height
= (65/12) : (63/12)
= 65 : 63
Therefore, the ratio remains the same.
Important: A ratio does not change when both quantities are converted into another common unit.
What is Proportion?
Proportion means when two ratios are equal, they are said to be in proportion.
For example,
20 : 40 and 1 : 2 are in proportion because:
20 : 40
= 20/40
= 1/2
Thus,
20 : 40 :: 1 : 2
When two ratios are in proportion, they are represented by the symbol “::”.
Therefore,
20 : 40 :: 1 : 2
is a proportion.
This is read as:
“20 is to 40 as 1 is to 2.”
Proportion Formula:
If
a : b :: c : d
then,
a/b = c/d
After cross multiplication,
ad = bc
This is called the fundamental property of proportion.
Here,
a and d are called the Extremes.
b and c are called the Means.
Therefore,
ad = Product of Extremes
bc = Product of Means
Types of Ratios
There are different types of ratios as given in the following examples:
1. Simple Ratio
When the comparison is made between two simple quantities of the same kind, it is called a simple ratio.
For example,
Boys : Girls = 10 : 15
= 2 : 3
2. Compound Ratio
When two or more ratios are multiplied together, the resulting ratio is called a compound ratio.
For example,
If two ratios are 2 : 3 and 4 : 5
Then the compound ratio becomes:
(2 × 4) : (3 × 5)
= 8 : 15
3. Duplicate Ratio
A ratio is called a duplicate ratio when its terms are squared.
For example,
If the ratio of two quantities is 2 : 3
Then the duplicate ratio will be:
2² : 3²
= 4 : 9
4. Triplicate Ratio
A ratio is called a triplicate ratio when its terms are cubed.
For example,
If the ratio is 2 : 3
Then the triplicate ratio will be:
2³ : 3³
= 8 : 27
5. Continued Ratio
When three or more quantities are present in a ratio, it takes the form of a continued ratio.
For example,
A : B = 2 : 3
B : C = 3 : 4
Therefore,
A : B : C = 2 : 3 : 4
Continued ratio is important for SSC, RRB, Banking and other competitive examinations because many ratio and proportion questions are directly or indirectly based on this concept.
Types of Proportion
There are two types of proportion:
Direct Proportion
In direct proportion, when one quantity increases, the other quantity also increases. Similarly, when one quantity decreases, the other quantity also decreases.
The symbol “∝” is used to represent “proportional to”.
For example,
If more manpower is deployed to a work, more work can be completed.
So,
More Manpower ∝ More Work Completed
If we book 4 train tickets instead of 2 tickets, more money will be spent.
So,
Number of Tickets ∝ Money Spent
Thus, in these examples, more manpower and more work are directly proportional to each other. Similarly, more train tickets and more money spent are also directly proportional.
Inverse Proportion
In inverse proportion, when one quantity increases, the other quantity decreases.
For example,
If more manpower is deployed to a work, less time is required to complete it.
If the speed of a train increases, less time is required to cover the same distance.
Thus, more manpower and less time are inversely proportional to each other. Similarly, higher speed and less time are also inversely proportional.
Mathematically,
Manpower ∝ 1/Time Required
This shows that as manpower increases, the time required decreases.
Ratio and Proportion Tricks
- Read the question carefully. Many times, the answer is hidden in the information given in the question itself.
- Do not hurry into mathematical calculations. First understand what quantities are being compared.
- Always check whether the quantities are expressed in the same unit. If not, convert them into a common unit before forming the ratio.
- Incorrect unit conversion can lead to wrong answers. For example, not converting feet and inches, kilograms and grams, or hours and minutes properly can make the entire calculation wrong.
- In proportion questions, use the cross multiplication property:
a : b :: c : d
⇒ ad = bc - If both quantities increase or decrease together, it is generally a Direct Proportion.
- Remember in Inverse Proportion, one quantity decreases and other increases.
Common Mistakes Students Make When Solving Ratio and Proportion Questions
These are some common mistakes students make while solving ratio and proportion questions:
- In proportion,
a : b :: c : d
Many students incorrectly use:
ac = bd
which is wrong.
The correct formula is:
ad = bc - Not reading the question carefully and missing important information given in the problem.
- Not converting quantities into the same unit before forming a ratio.
- Not applying the concept correctly, such as using inverse proportion when the question is based on direct proportion, or vice versa.
- Reversing the order of quantities while forming ratios.
- Not simplifying ratios before performing calculations.
- Not practicing enough questions of different difficulty levels.
- Not revising the formulas and concepts before examinations.
Ratio and Proportion Examples
1. In a SSC mock test 20 girls and 30 boys attended the test. Find the ratio of boys to girls in the mock test
Answer:
Ratio of Boys to Girls
= 30 : 20
= 3 : 2
2. Find the ratio of 5 kg to 500 grams
Answer:
First convert to the same unit.
5 kg = 5000 g
Ratio
= 5000 : 500
= 10 : 1
3. Are these ratios in proportion 15:21 and 5: 7
Answer:
15/21
= 5/7
Therefore,
15 : 21 :: 5 : 7
Hence, the ratios are in proportion.
4.If 4 notebooks cost ₹80, what will be the cost of 6 notebooks?
Answer:
Number of Notebooks ∝ Cost
4 : 6 = 80 : x
4/6 = 80/x
4x = 6 × 80
x = (6 × 80)/4
x = ₹120
5. If 8 workers can complete a work in 15 days, how many days will 20 workers take?
Answer:
Workers and Days are inversely proportional.
8 × 15 = 20 × x
120 = 20x
x = 6 days
FAQ on Ratio and Proportion
1. What is Ratio?
Ratio is a comparison between two quantities of the same kind. It is usually represented in the form of a : b or a/b.
2. What is the Proportion?
When two ratios are equal, they are said to be in proportion.
For example,
20 : 40 :: 1 : 2
Therefore, 20 : 40 and 1 : 2 are in proportion.
3. A pump pulls 5 KL of water in 1 hour and 10 KL of water in 2 hours. Is this an example of direct proportion?
Answer: Yes.
As the time increases, the quantity of water pumped also increases.
Therefore, water pumped and time are directly proportional.
4. In inverse proportion, if one quantity increases, the other quantity also increases. True or False?
Answer: False.
In inverse proportion, when one quantity increases, the other quantity decreases.
For example, if more workers are deployed to a work, less time is required to complete it.
5. Are Ratio and Proportion questions asked in SSC, RRB, NDA, CDS, Banking and other competitive examinations?
Answer: Yes.
Ratio and Proportion is one of the most important chapters for SSC, RRB, Banking, NDA, CDS, Defence, Police and many other competitive examinations.
6. What is the symbol used for proportion?
Answer: The symbol “::” is used to represent proportion.
For example,
20 : 40 :: 1 : 2
7. What is the symbol used for proportionality?
Answer: The symbol “∝” is used to represent “proportional to”.
For example,
Work Completed ∝ Manpower
Important Resources to Ratio And Proportion
- Percentage: Meaning, Formula, Examples & Questions for Exams
- Basic Ratio Questions & Answers
- Ratio Word Problems with Answers
- Ratio Formula with Examples and Explanations
- Proportion Questions and Answers with Solutions
- Unitary Method Questions and Answers with Solutions
- Direct Proportion Questions and Answers
- Inverse Proportion Questions & Answers with Solutions
- Ratio and Proportion Practice Questions with Answers
- Ratio and Proportion PYQs SSC, RRB, Banking & Defence Exams (Solved)