Unitary Method, Ratio and Proportion

Unitary Method, Ratio and Proportion - Complete Guide for Class 6 CBSE Maths

Introduction to Unitary Method, Ratio and Proportion

Understanding the unitary method, ratios, and proportions forms the foundation of quantitative reasoning in mathematics. These concepts help students solve real-world problems involving comparisons, scaling, and proportional relationships. This comprehensive guide covers all aspects of these topics as per the CBSE Class 6 syllabus, with detailed explanations, solved examples, and practice exercises.

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

1. To get more, we multiply - When finding the value of multiple units from one unit
2. 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

1. Same Units Required: Both quantities must be in the same unit
2. Order Matters: The ratio 3:4 is different from 4:3
3. No Units in Ratio: Ratios are pure numbers without units
4. 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

1. Find the HCF of both terms
2. 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

1. Find the ratio of 48 minutes to 1 hour in simplest form.
2. If 12:x::3:5, find the value of x.
3. The ratio of boys to girls in a class is 5:3. If there are 15 boys, how many girls are there?
4. Divide 63 in the ratio 7:2.
5. Find three equivalent ratios of 2:5.

Level 2: Intermediate Questions

6. A bus travels 126 km in 3 hours and a train travels 315 km in 5 hours. Find the ratio of their speeds.
7. Two numbers are in the ratio 11:12. If their sum is 460, find the numbers.
8. If x:y = 2:3, find the value of (3x + 2y):(9x + 5y).
9. The cost of 1 dozen eggs is ₹30. Find the cost of 8 eggs.
10. Find the fourth proportional to 25, 100, and 40.

Level 3: Advanced Questions

11. If 2x + 3y:3x + 5y = 18:29, find x:y.
12. 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.
13. Sushil's salary for 9 months is ₹21,000. Find his salary for 15 months.
14. Show that a, b, c are in proportion if (6a + 7b):(6c + 7d)::(6a - 7b):(6c - 7d).
15. 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

1. Memorize common ratios: 1:2, 2:3, 3:4, 4:5, etc.
2. Practice HCF quickly for simplifying ratios
3. Unit conversion chart: Keep mental notes of ml↔L, cm↔m, paise↔rupees
4. Check answer reasonableness: If buying more, cost should increase
5. 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, traveling, and countless other situations. By understanding the principles, practicing regularly, and applying the shortcuts provided, students can excel in both classroom assessments and competitive examinations.

Main Points

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

Note for Students: This guide aligns with the CBSE Class 6 Mathematics notes curriculum and provides comprehensive coverage of the Unitary Method, Ratio, and Proportion chapter. Use this resource alongside your textbook for thorough preparation.