Class 6 Maths Chapter: Unitary Method, Ratio and Proportion – Introduction
The chapter Unitary Method, Ratio and Proportion is one of the most practical and application-based topics in Class 6 Mathematics. It teaches students how to compare quantities, understand relationships between numbers, and solve real-life problems involving sharing, pricing, measurements, and scaling. These concepts are widely used in everyday situations, such as finding the cost of multiple items, comparing the performance of teams, adjusting recipes, or calculating quantities required for a specific task.
The chapter begins with the concept of a ratio, which is used to compare two quantities of the same kind. Students then learn about proportion, which helps determine whether two ratios are equal. Once these ideas are understood, learners are introduced to the unitary method, a powerful technique used to find the value of one unit before calculating the value of many units. This step-by-step approach makes problem-solving logical and easy to follow. Must-read Class 6 Maths Notes and NCERT Solutions for Class 6, NCERT exemplar for class 6.
Understanding the unitary method, ratio, and proportion strengthens mathematical reasoning and analytical thinking. These concepts form the foundation for advanced topics such as percentages, profit and loss, speed, algebra, and data handling. By mastering this chapter, students develop confidence in solving practical mathematical problems and gain skills that are useful both in academics and in daily life.
What is the Unitary Method?
Definition
The unitary method is a technique where we first find the value of a single unit from a given quantity, and then calculate the value of the required number of units. This method is particularly useful for solving problems involving direct and inverse proportions.
Key Principles
- To get more, we multiply - When finding the value of multiple units from one unit
- To get less, we divide - When finding the value of one unit from multiple units
Real-World Applications
- Shopping calculations (price per item)
- Speed and distance problems
- Work and time calculations
- Currency conversion
- Recipe scaling
Examples of Unitary Method Problems with Step-by-Step Solutions
Example 1: Direct Proportion Problem
Problem: If 15 tins contain 234 kg of oil, how much oil will be in 10 such tins?
Solution:
- Step 1: Find oil in 1 tin
- 15 tins contain = 234 kg
- 1 tin contains = 234 ÷ 15 kg = 15.6 kg
- Step 2: Find oil in 10 tins
- 10 tins contain = 15.6 × 10 = 156 kg
Answer: 10 tins contain 156 kg of oil.
Example 2: Cost Calculation
Problem: If 5 bars of soap cost ₹31, find the cost of 2 dozen such bars.
Solution:
- Step 1: Find cost of 1 bar
- 5 bars cost = ₹31
- 1 bar costs = 31 ÷ 5 = ₹6.20
- Step 2: Find cost of 24 bars (2 dozen)
- 24 bars cost = 6.20 × 24 = ₹148.80
Answer: 2 dozen bars cost ₹148.80.
Example 3: Distance and Fuel Consumption
Problem: If 12 litres of petrol covers 222 km, how many kilometres can be covered with 22 litres?
Solution:
- Step 1: Find distance per litre
- 12 litres cover = 222 km
- 1 litre covers = 222 ÷ 12 = 18.5 km
- Step 2: Find distance with 22 litres
- 22 litres cover = 18.5 × 22 = 407 km
Answer: The car will travel 407 km with 22 litres of petrol.
How to Solve Inverse Proportion Using Unitary Method
Understanding Inverse Proportion
In inverse proportion, when one quantity increases, the other decreases proportionally. The product of the two quantities remains constant.
Method for Solving Inverse Proportion
Formula: If a₁ × b₁ = a₂ × b₂, then the quantities are in inverse proportion.
Example Problem
Problem: 6 workers can complete a task in 15 days. How many days will 10 workers take to complete the same task?
Solution:
- Step 1: Find work in "worker-days"
- Total work = 6 workers × 15 days = 90 worker-days
- Step 2: Calculate days for 10 workers
- 10 workers will take = 90 ÷ 10 = 9 days
Answer: 10 workers will complete the task in 9 days.
Key Insight
As the number of workers increases, the number of days decreases - this is inverse proportion.
Understanding Ratios
Definition
A ratio is a comparison between two quantities of the same kind and in the same unit. It shows how many times one quantity is contained in another.
Notation
The ratio of 'a' to 'b' is written as:
- a : b
- a/b (as a fraction)
Important Points About Ratios
- Same Units Required: Both quantities must be in the same unit
- Order Matters: The ratio 3:4 is different from 4:3
- No Units in Ratio: Ratios are pure numbers without units
- Terms: In a:b, 'a' is the antecedent and 'b' is the consequent
Examples with Unit Conversion
Example 1: Time Ratio
Find the ratio of 225 ml to 3 litres
Solution:
- Convert to same unit: 3 litres = 3000 ml
- Ratio = 225 : 3000
- Simplify: 225 : 3000 = 9 : 120 = 3 : 40
Answer: 3:40
Example 2: Money Ratio
Find the ratio of 65 paise to ₹5
Solution:
- Convert to same unit: ₹5 = 500 paise
- Ratio = 65 : 500
- Simplify: 65 : 500 = 13 : 100
Answer: 13:100
Equivalent Ratios
Definition
Ratios obtained by multiplying or dividing both terms by the same non-zero number are called equivalent ratios.
Properties
- a : b = ac : bc (multiplying both terms by c)
- a : b = (a÷c) : (b÷c) (dividing both terms by c)
Examples
- 6 : 10 = 3 : 5 (dividing by 2)
- 6 : 10 = 12 : 20 (multiplying by 2)
- All these ratios are equivalent
Simplest Form of Ratios
Definition
A ratio is in its simplest form or lowest terms when the antecedent and consequent have no common factor except 1.
Method to Simplify
- Find the HCF of both terms
- Divide both terms by their HCF
Example
Simplify the ratio 42:63
Solution:
- HCF(42, 63) = 21
- 42 ÷ 21 = 2
- 63 ÷ 21 = 3
- Simplest form = 2:3
Comparing Ratios
Method
To compare two ratios, make their denominators equal (find LCM), then compare numerators.
Example
Which is greater: 2:3 or 3:4?
Solution:
- LCM of 3 and 4 = 12
- 2/3 = (2×4)/(3×4) = 8/12
- 3/4 = (3×3)/(4×3) = 9/12
- Since 9 > 8, therefore 3/4 > 2/3
Answer: 3:4 is greater than 2:3
Understanding Proportion
Definition
The equality of two ratios is called proportion. If a:b = c:d, we say that a, b, c, d are in proportion, written as a:b::c:d.
Terms in Proportion
- Extremes: First and fourth terms (a and d)
- Means: Second and third terms (b and c)
Fundamental Property
Product of extremes = Product of means
If a:b::c:d, then a×d = b×c
Example
Find x if x:6::5:15
Solution:
- Using the property: x × 15 = 6 × 5
- 15x = 30
- x = 30 ÷ 15 = 2
Answer: x = 2
Continued Proportion
Definition
Three quantities a, b, c are in continued proportion if a:b = b:c, which means b² = ac.
Properties
- The middle term is called the mean proportional
- If a:b::b:c, then b is the mean proportional between a and c
Example
Find the third proportional to 4 and 8
Solution:
- Let third proportional be x
- Then 4:8::8:x
- 4 × x = 8 × 8
- 4x = 64
- x = 16
Answer: The third proportional is 16.
Difference Between Ratio, Proportion, and Unitary Method Explained Simply
| Aspect | Ratio | Proportion | Unitary Method |
| Definition | Comparison of two quantities | Equality of two ratios | Finding value through one unit |
| Expression | a:b or a/b | a:b::c:d | Value of 1 unit → value of n units |
| Key Property | Shows relative size | Product of extremes = Product of means | Uses multiplication/division |
| Example | Boys:Girls = 3:2 | 3:2::6:4 | 5 pens cost ₹25 → 1 pen costs ₹5 |
| Use Case | Comparing quantities | Solving for unknown terms | Practical calculation problems |
When to Use Each
- Ratio: When comparing two quantities (speed comparison, mixture problems)
- Proportion: When four quantities are related and you need to find an unknown
- Unitary Method: When solving real-world problems involving rates, prices, or scaling
Practice Questions on Ratio and Proportion for Class 6 to 8
Level 1: Basic Questions
- Find the ratio of 48 minutes to 1 hour in simplest form.
- If 12:x::3:5, find the value of x.
- The ratio of boys to girls in a class is 5:3. If there are 15 boys, how many girls are there?
- Divide 63 in the ratio 7:2.
- Find three equivalent ratios of 2:5.
Level 2: Intermediate Questions
- A bus travels 126 km in 3 hours and a train travels 315 km in 5 hours. Find the ratio of their speeds.
- Two numbers are in the ratio 11:12. If their sum is 460, find the numbers.
- If x:y = 2:3, find the value of (3x + 2y):(9x + 5y).
- The cost of 1 dozen eggs is ₹30. Find the cost of 8 eggs.
- Find the fourth proportional to 25, 100, and 40.
Level 3: Advanced Questions
- If 2x + 3y:3x + 5y = 18:29, find x:y.
- The ratio of male to female workers in a textile mill is 5:3. If there are 115 male workers, find the number of female workers.
- Sushil's salary for 9 months is ₹21,000. Find his salary for 15 months.
- Show that a, b, c are in proportion if (6a + 7b):(6c + 7d)::(6a - 7b):(6c - 7d).
- If b is the mean proportional between a and c, prove that abc(a + b + c)³ = (ab + bc + ca)³.
Tips and Shortcuts for Unitary Method in Competitive Exams
Shortcut 1: Direct Multiplication for Simple Problems
Instead of dividing first, multiply directly:
- If 5 items cost ₹100, then 8 items cost = (100/5) × 8 = ₹160
- Quick formula: (New quantity/Old quantity) × Old value
Shortcut 2: Cross-Multiplication for Proportions
For a:b::c:d, directly use: a×d = b×c
This saves time in finding unknown terms.
Shortcut 3: Percentage Method for Ratio Problems
Convert ratios to percentages for easier calculation:
- Ratio 3:2 means first part is 3/(3+2) = 60%, second is 40%
Shortcut 4: Inverse Proportion Quick Formula
For inverse proportion: a₁ × b₁ = a₂ × b₂
Directly solve: b₂ = (a₁ × b₁)/a₂
Shortcut 5: Speed Calculation
- Speed = Distance/Time
- If doubling time, distance also doubles (at same speed)
- Use this for quick mental calculations
Time-Saving Tips
- Memorize common ratios: 1:2, 2:3, 3:4, 4:5, etc.
- Practice HCF quickly for simplifying ratios
- Unit conversion chart: Keep mental notes of ml↔L, cm↔m, paise↔rupees
- Check answer reasonableness: If buying more, cost should increase
- Use estimation: Round numbers for quick verification
Common Mistakes to Avoid
- Forgetting to convert units before comparing
- Reversing the ratio order
- Dividing instead of multiplying (or vice versa)
- Not simplifying the final ratio
- Mixing up direct and inverse proportions
Formula
| Formula Name | Mathematical Representation | Explanation |
| Unitary Method (More) | Value of n units = (Value of 1 unit) × n | To get more, multiply |
| Unitary Method (Less) | Value of 1 unit = (Value of n units) ÷ n | To get less, divide |
| Ratio as Fraction | a:b = a/b | Ratio expressed as fraction |
| Equivalent Ratio | a:b = (a×k):(b×k) or (a÷k):(b÷k) | Multiply/divide both terms by same number |
| Proportion Property | If a:b::c:d, then a×d = b×c | Product of extremes = Product of means |
| Mean Proportional | If a:b::b:c, then b² = ac | b is mean proportional between a and c |
| Fourth Proportional | If a:b::c:x, then x = (b×c)/a | Finding fourth term in proportion |
| Third Proportional | If a:b::b:x, then x = b²/a | Finding third term in continued proportion |
| Inverse Proportion | a₁ × b₁ = a₂ × b₂ | Product remains constant |
| Ratio Comparison | Convert to same denominator, compare numerators | LCM method for comparison |
Conclusion
Mastering the unitary method, ratios, and proportions is essential for building strong mathematical foundations. These concepts are not just academic—they're practical tools used daily in shopping, cooking, travelling, and countless other situations. By understanding the principles, practising regularly, and applying the shortcuts provided, students can excel in both classroom assessments and competitive examinations.
Main Points
The unitary method simplifies complex calculations by finding unit values first
Ratios help compare quantities meaningfully
Proportions establish relationships between four quantities
Practice with diverse problems builds confidence and speed
Real-world applications make learning meaningful and memorable
Frequently Asked Questions (FAQs) on Class 6 Maths Chapter Unitary Method, Ratio and Proportion
The unitary method is a mathematical technique used to find the value of a single unit and then calculate the value of multiple units. It is one of the most important concepts in Class 6 Maths because it helps solve real-life problems involving cost, quantity, distance, and time. For example, if the cost of five notebooks is known, the unitary method helps find the cost of one notebook and then any required number of notebooks. This method improves logical thinking and forms the basis for advanced topics such as percentages, direct variation, and commercial mathematics.
A ratio is a comparison between two quantities of the same kind. It shows how much one quantity is related to another and is usually written using a colon symbol. Ratios are used in everyday life for comparing ages, marks, speeds, ingredients, and measurements. In Class 6 Maths, students learn how to write ratios in the simplest form and compare different quantities effectively. Understanding ratios helps build analytical skills and provides a foundation for proportion, percentages, fractions, and other important mathematical concepts studied in higher classes.
A proportion is an equality between two ratios. When two ratios represent the same relationship, they are said to be in proportion. For example, the ratios 2:4 and 4:8 are equal and therefore form a proportion. This concept helps students understand relationships between quantities and solve comparison-based problems. Proportion is widely used in business, science, engineering, and daily calculations. In Class 6, students learn to identify proportional relationships and solve simple problems involving equivalent ratios, making it an essential part of mathematical learning.
The main difference is that a ratio compares two quantities, while a proportion compares two ratios. A ratio may simply show the relationship between two values, whereas a proportion confirms that two ratios are equal. For example, 3:5 is a ratio, but 3:5 = 6:10 is a proportion. Understanding this distinction is important because many examination questions test students on identifying and applying these concepts correctly. A clear understanding of ratio and proportion helps students solve practical mathematics problems with confidence and accuracy.
The unitary method has many practical applications in daily life. It is used to calculate prices, wages, distances, quantities, and rates. For example, if the cost of ten kilograms of rice is known, the unitary method can determine the cost of one kilogram and then any required quantity. It is also useful when comparing travel expenses, shopping discounts, and resource consumption. Learning this method enables students to solve real-world mathematical problems efficiently and prepares them for more advanced applications in commerce and science.
Students should begin by carefully understanding the quantities being compared. Ratios should be simplified whenever possible, and proportions should be checked by comparing equivalent ratios. Practicing different examples helps students recognize patterns and solve questions more quickly. Drawing tables and organizing information clearly can make calculations easier. Regular revision of definitions, formulas, and word problems improves confidence. Solving worksheets and textbook exercises is one of the best ways to master ratio and proportion concepts and achieve good results in mathematics examinations.
This chapter provides a strong foundation for several advanced mathematical topics. Concepts of ratio and proportion are used in percentages, profit and loss, simple interest, algebra, trigonometry, statistics, and data interpretation. The unitary method is also essential for solving problems involving speed, work, and measurement. Students who understand these concepts thoroughly find it easier to handle complex calculations in higher classes. Mastery of this chapter improves logical reasoning, numerical skills, and problem-solving abilities, making it one of the most valuable chapters in Class 6 Mathematics.