What is Ratio and Proportion?
Ratio and Proportion – Definition, Formula, Examples, and Tricks
Introduction
Ratio and proportion are two of the most important concepts in mathematics. They help us compare quantities and understand the relationship between different values. These concepts are widely used in school mathematics, Banking, Railway, UPSC, and in everyday life such as cooking, shopping, business, and map reading.
In this article, you will learn the meaning of ratio and proportion, their formulas, properties, solved examples, and real-life applications in a simple and easy-to-understand way.
What is Ratio?
A ratio is a comparison of two or more quantities by division. It shows how much of one quantity exists compared to another.
Definition
A ratio compares two quantities of the same kind.
Formula
Ratio = First Quantity : Second Quantity
or
Ratio = a : b
where:
a = First quantity
b = Second quantity (b ≠ 0)
Example 1
There are 8 boys and 12 girls in a class.
Ratio of boys to girls
= 8 : 12
Divide both numbers by 4.
= 2 : 3
So, the simplified ratio is 2 : 3.
Terms of a Ratio
In the ratio 5 : 8:
5 is called the first term (antecedent).
8 is called the second term (consequent).
Types of Ratios
1. Equivalent Ratios
Ratios that represent the same value.
Example:
2 : 3 = 4 : 6 = 6 : 9
2. Simplest Ratio
A ratio in which both terms have no common factor except 1.
Example:
18 : 24
Divide by 6.
= 3 : 4
3. Duplicate Ratio
The ratio of the squares of two numbers.
Example:
2 : 3
Duplicate ratio
= 2² : 3²
= 4 : 9
4. Triplicate Ratio
The ratio of the cubes of two numbers.
Example:
2 : 3
Triplicate ratio
= 2³ : 3³
= 8 : 27
What is Proportion?
A proportion is a statement that two ratios are equal.
Definition
If two ratios have the same value, they are said to be in proportion.
Formula
a : b = c : d
or
a/b = c/d
where b and d are not equal to zero.
Property of Proportion
If
a : b = c : d
then
a × d = b × c
This is called cross multiplication.
Example
Find the value of x.
3 : 5 = x : 20
Cross multiply
3 × 20 = 5 × x
60 = 5x
x = 12
Therefore, the value of x is 12.
Continued Proportion
Three quantities are said to be in continued proportion if:
a : b = b : c
Example:
2 : 4 = 4 : 8
Hence, 2, 4, and 8 are in continued proportion.
Direct Proportion
In direct proportion, both quantities increase or decrease together.
Example
If 1 notebook costs ₹20,
then 5 notebooks cost:
5 × ₹20 = ₹100
As the number of notebooks increases, the total cost also increases.
Inverse Proportion
In inverse proportion, when one quantity increases, the other decreases.
Example
If 5 workers complete a job in 12 days,
then 10 workers will complete it in:
(5 × 12) ÷ 10
= 6 days
More workers require fewer days.
Difference Between Ratio and Proportion
Ratio | Proportion |
|---|---|
Compares two quantities | Compares two equal ratios |
Written as a : b | Written as a : b = c : d |
Shows comparison | Shows equality of ratios |
Example: 3 : 5 | Example: 3 : 5 = 6 : 10 |
Real-Life Applications of Ratio and Proportion
Ratio and proportion are used in many daily situations, including:
Cooking recipes
Shopping discounts
Sharing profits
Mixing paints and chemicals
Scale maps
Speed, distance, and time problems
Business and finance
Population studies
Competitive examinations
Common Mistakes to Avoid
Comparing quantities with different units without converting them.
Forgetting to simplify ratios.
Confusing ratio with proportion.
Making mistakes during cross multiplication.
Writing the terms in the wrong order.
Practice Questions
1. Find the ratio of 24 to 36.
Answer: 2 : 3
2. Solve:
4 : 7 = x : 21
Answer: x = 12
3. A recipe uses sugar and flour in the ratio 2 : 5. If 10 cups of flour are used, how many cups of sugar are required?
Answer: 4 cups
Conclusion
Ratio and proportion are fundamental mathematical concepts used to compare quantities and solve real-life problems. Understanding these concepts helps improve logical thinking and problem-solving skills. By learning the formulas, properties, and solving regular practice questions.
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