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ICSE • Class 6 • Mathematics • Ch 8
Estimated Time: 45-60 mins
Study Progress: In Progress

Ratio and Proportion

The Council for the Indian School Certificate Examinations (CISCE) Class 6 Mathematics curriculum introduces Ratio and Proportion as the foundational gateway to comparative mathematics, commercial arithmetic, and algebraic modelling. In daily human activity, quantities can be compared in two fundamentally distinct ways: comparison by difference and comparison by division. Comparison by difference relies on subtraction to determine how much greater or smaller one quantity is than another. While useful for simple calculations, comparison by difference possesses severe limitations because it completely ignores scale. Comparison by division, on the other hand, determines how many times one quantity contains another or what fraction one quantity is of another. This comparative relation between two quantities of the same kind, expressed in the exact same unit, is known as a Ratio. A ratio is denoted mathematically by placing a colon between the two quantities, written as a : b, which is read aloud as 'a is to b'. It can also be represented as a vulgar fraction a / b, where the second term b cannot be equal to zero. In the expression a : b, the first term 'a' is formally designated as the Antecedent (derived from the Latin word meaning 'going before'), and the second term 'b' is designated as the Consequent (meaning 'following after'). A profound mathematical truth that students must master is that a ratio is a pure, dimensionless number. Because it is a comparison between two quantities measured in identical units, the units in the numerator and denominator cancel each other out completely. Consequently, writing units such as '3 metres : 4 metres' or attaching kilograms to a final ratio is mathematically invalid; the resulting ratio is simply 3 : 4. Furthermore, the order of terms in a ratio is of paramount importance: the ratio a : b is completely different from the ratio b : a, unless the two numbers a and b are identical. For example, in a classroom with 15 boys and 20 girls, the ratio of boys to girls is 15 : 20 (or 3 : 4), whereas the ratio of girls to boys is 20 : 15 (or 4 : 3). Inverting the order completely reverses the mathematical meaning of the comparison. Before any two quantities can be compared as a ratio, they must satisfy the Homogeneity Principle: they must represent measurements of the same kind and must be expressed in the exact same unit of measurement. It is impossible to form a ratio between 5 kilograms and 10 metres, because mass and distance are incompatible physical dimensions. When comparing quantities of the same kind expressed in differing units—such as 75 paise and 3 rupees, or 500 metres and 2 kilometres—one must first convert both quantities into a common, identical unit before establishing the ratio. Converting 3 rupees into 300 paise yields the ratio 75 paise : 300 paise, which simplifies directly to 1 : 4. A ratio is said to be in its Simplest Form (or lowest terms) when its antecedent and consequent are whole numbers that share no common factor other than 1; that is, their Highest Common Factor (HCF) is equal to 1. Just like fractions, equivalent ratios are formed by multiplying or dividing both the antecedent and consequent by the same non-zero constant. Beyond comparing two existing measurements, ratios serve as a powerful tool for partitioning quantities. When an inheritance, prize money, or chemical mixture must be divided among multiple recipients in a specified ratio m : n, the total quantity is conceptually divided into (m + n) equal parts. Each recipient then receives a fraction of the total quantity proportional to their share: the first party receives m / (m + n) of the whole, while the second party receives n / (m + n) of the whole. This systematic method extends seamlessly to ratios involving three or more terms, such as dividing a sum among three partners in the ratio a : b : c, where the total number of parts equals a + b + c. When two separate ratios are equal in magnitude, their equality forms a Proportion. If the ratio a : b is equal to the ratio c : d, the four quantities a, b, c, and d are said to be in proportion, which is written as a : b :: c : d, and read aloud as 'a is to b as c is to d'. In any proportion a : b :: c : d, the first and fourth terms (a and d) sit at the outer boundaries and are designated as the Extremes (or extreme terms), while the second and third terms (b and c) sit in the interior and are designated as the Means (or middle terms). The bedrock theorem governing all proportions is the Fundamental Law of Proportion (or Cross-Product Rule): the Product of the Extremes is strictly equal to the Product of the Means, expressed algebraically as a multiplied by d equals b multiplied by c. This universal rule allows students to verify immediately whether four given numbers form a proportion, and enables the effortless solution of unknown variables. A special scenario arises when three quantities a, b, and c are related such that the ratio of the first to the second equals the ratio of the second to the third: a : b :: b : c. Such quantities are said to be in Continued Proportion. In this continuous relation, the middle term b is termed the Mean Proportional between a and c, satisfying the identity b squared equals a multiplied by c, which means b equals the square root of the product of a and c. Correspondingly, the third term c is termed the Third Proportional to a and b, given by b squared divided by a. Finally, the chapter culminates in the Unitary Method, an indispensable operational procedure of practical arithmetic. The unitary method solves complex commercial problems by first calculating the value of a single unit through division, and subsequently determining the value of the required number of units through multiplication. Whether determining the price of fifteen books when the cost of six is given, calculating vehicle speed and travel time, or computing wages for variable working days, the unitary method provides a robust, intuitive, and error-proof two-step algorithmic pathway. Mastery of ratio, proportion, and the unitary method equips Class 6 students with quantitative fluency that will underpin their future studies in percentages, profit and loss, simple interest, geometry, trigonometry, and physical sciences.

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Have You Ever Wondered?

When an architect prepares a blueprint where one centimetre on paper represents two metres of a skyscraper, or when a chef adjusts a delicate recipe for four guests to feed a banquet of four hundred people without altering its taste, or when your smartphone screen scales a high-definition movie to a 16 : 9 aspect ratio without distorting a single face, mathematics is performing one of its most elegant and practical arts: proportional reasoning. In everyday life, simply subtracting quantities tells us how much more one thing is than another, but it fails to reveal the relative relationship between them. If a mouse gains one kilogram and an elephant gains one kilogram, the absolute difference in weight is identical, yet the impact on their bodies is worlds apart! To understand relative comparison, scale, balance, and universal fairness, civilization developed the twin concepts of Ratio and Proportion. From the golden spirals of ancient Greek temples to financial exchange rates and modern computer graphics, ratio and proportion form the mathematical backbone of scale and equivalence.

Why This Chapter Matters

Ratio and proportion are not merely abstract examination topics; they represent the quantitative grammar of the physical, commercial, and artistic universe. Every time an artist sketches a human portrait using classical rules of facial balance, they rely on the golden ratio and harmonic proportions to achieve aesthetic symmetry. In architecture and civil engineering, scale drawings on architectural blueprints allow multi-storey residential complexes and suspension bridges to be planned with millimeter precision, where a scale ratio such as 1 : 100 ensures that structural models behave identically to real-world constructions. In the medical and pharmaceutical sectors, doctors and pharmacists calculate life-saving medication dosages strictly based on ratios of active chemical ingredients to a patient's body mass in kilograms, where an error in ratio could mean the difference between healing and toxicity. In modern digital technology, every computer monitor, television display, digital photograph, and smartphone relies on pixel aspect ratios like 16 : 9 or 4 : 3 to render high-definition media without distorting visual geometry. In global commerce and personal finance, foreign currency exchange rates, company equity profit-sharing, interest rates, and recipe scaling all function through direct proportional reasoning and the unitary method. By thoroughly mastering the principles of antecedent and consequent, cross-product rules, continued proportion, and unitary step-wise calculations, students develop sharp logical discernment, critical estimation skills, and the mathematical confidence necessary for advanced scientific enquiry.

Before You Begin (Prerequisites)

  • Familiarity with four fundamental operations (addition, subtraction, multiplication, and division) on natural numbers and whole numbers.
  • Clear understanding of fractions, equivalent fractions, and simplifying fractions to lowest terms using the Highest Common Factor (HCF).
  • Ability to convert standard metric measurement units (rupees and paise, metres and centimetres, kilometres and metres, kilograms and grams, hours and minutes).
  • Basic concept of finding the square root of perfect square numbers (such as square root of 16 is 4, square root of 64 is 8).
  • Understanding of word problems and the ability to extract given numerical data to find an unknown quantity.

What You Will Learn (Core Objectives)

  • Distinguish clearly between comparison by difference and comparison by division, defining ratio as a dimensionless comparative relation between quantities of the same kind.
  • Identify the antecedent and consequent of a ratio, recognizing why the order of terms is non-negotiable and why ratios carry no physical units.
  • Convert given quantities into identical units using the Homogeneity Principle, reducing any ratio to its simplest form where HCF(a, b) = 1.
  • Compare two or more ratios accurately using cross-multiplication or common denominator fractional methods to identify the greater ratio.
  • Divide a given whole quantity into two or three shares in a specified ratio using the total parts fractional distribution method.
  • Define proportion as the equality of two ratios, identifying extreme terms and middle terms (means), and applying the Fundamental Law of Proportion (Product of Extremes = Product of Means).
  • Determine an unknown term in a four-term proportion and solve problems involving continued proportion, mean proportional (b = sqrt(ac)), and third proportional (c = b^2 / a).
  • Apply the two-step Unitary Method (division to find 1 unit, multiplication to find required units) to solve real-world commercial problems involving cost, time, distance, and work.

Chapter Roadmap & Progression

1 Module 1: Concept of Ratio, Compari...
2 Module 2: Simplest Form (Lowest Ter...
3 Module 3: Comparison of Ratios & Pa...
4 Module 4: Concept of Proportion, Te...
5 Module 5: Continued Proportion, Mea...
6 Module 6: The Unitary Method — Dire...
7 Module 7: Applied Word Problems, Re...

Complete Concept Guide (100% Curriculum Coverage)

Module 1: Concept of Ratio, Comparison of Quantities & Fundamental Properties

1.1 Two Ways of Comparing Quantities: Difference vs Division

In our daily lives, we constantly compare measurements—weights, lengths, test scores, prices, and ages. There are two primary mathematical methods to compare two quantities:

1. Comparison by Difference (Subtraction)

Subtracts one quantity from another: "Rohan's height is $150\text{ cm}$ and Aryan's height is $120\text{ cm}$. Rohan is $150 - 120 = 30\text{ cm}$ taller."
Limitation: Tells absolute difference, but completely hides scale and proportion.

2. Comparison by Division (Ratio)

Divides one quantity by another: "A pen costs ₹$20$ and a pencil costs ₹$5$. $\frac{20}{5} = 4$, so the pen costs $4$ times as much as the pencil."
Advantage: Reveals relative scale, multiple, or fraction regardless of magnification.

1.2 Definition of Ratio & Anatomy of Terms

A Ratio is defined as a mathematical comparison of two quantities of the same kind and in the same unit by division.

$$\text{Ratio of } a \text{ to } b = a : b = \frac{a}{b} \quad (b \neq 0)$$

Read aloud as: "$a$ is to $b$"

The Antecedent (First Term: $a$)

Derived from the Latin antecedens ("going before"). It represents the quantity being compared in the numerator position.

The Consequent (Second Term: $b$)

Derived from the Latin consequens ("following after"). It represents the standard of reference in the denominator position ($b \neq 0$).

1.3 Three Foundational Invariants of a Ratio

Invariant 1: A Ratio is a Pure Dimensionless Number (NO Units!)

Because a ratio is a comparison of two quantities expressed in identical units, the measurement units cancel out in the division.

$$\frac{6\text{ kg}}{9\text{ kg}} = \frac{6}{9} = \frac{2}{3} = 2 : 3$$ Crucial Examiner Rule: Writing $2 : 3\text{ kg}$ or $2\text{ m} : 3\text{ m}$ is strictly incorrect. A ratio has no unit of measurement.

Invariant 2: Order of Terms is Vital ($a : b \neq b : a$)

The order in which the quantities are specified in the problem statement dictates which quantity is the antecedent and which is the consequent.

In a class of $15$ boys and $20$ girls:
• Ratio of boys to girls $= 15 : 20 = 3 : 4$
• Ratio of girls to boys $= 20 : 15 = 4 : 3$
Notice that $3 : 4 \neq 4 : 3$. Reversing the terms reverses the comparison!

Invariant 3: Multi-Term Ratios ($a : b : c$)

A ratio can compare three or more quantities simultaneously. For example, if a triangle's angles are $30^\circ, 60^\circ, 90^\circ$, their ratio is $30 : 60 : 90 = 1 : 2 : 3$. In a multi-term ratio, all terms must be divided or multiplied by the exact same common factor.

Module 2: Simplest Form (Lowest Terms), Equivalent Ratios & Unit Homogenisation

2.1 Simplest Form (Lowest Terms) of a Ratio

A ratio $a : b$ is said to be in its Simplest Form (or lowest terms) when both $a$ and $b$ are integers and share no common factor other than $1$. In formal terms:

$$a : b \text{ is in simplest form} \iff \text{HCF}(a, b) = 1$$
Reduction Procedure: To reduce a ratio to simplest form, divide both the antecedent and consequent by their Highest Common Factor (HCF).
• Example: Reduce $48 : 72$ to lowest terms.
  $\text{HCF}(48, 72) = 24$. Dividing both by $24$: $(48 \div 24) : (72 \div 24) = \mathbf{2 : 3}$.

2.2 Equivalent Ratios

Just like equivalent fractions, a ratio remains unchanged if both of its terms are multiplied or divided by the same non-zero number ($m \neq 0$).

$$a : b = (a \times m) : (b \times m) = (a \div m) : (b \div m) \quad (m \neq 0)$$

For example, multiplying both terms of $2 : 5$ by $3$ gives $6 : 15$. Multiplying by $4$ gives $8 : 20$. Thus, $2 : 5, 6 : 15,$ and $8 : 20$ are all Equivalent Ratios.

2.3 The Homogeneity Principle & Unit Homogenisation

The Homogeneity Principle dictates that a ratio can only be formed between quantities that measure the same physical attribute (e.g. length with length, time with time). Furthermore, both quantities MUST be converted into the exact same unit before forming the ratio!

Given Quantities Conversion to Common Unit Raw Ratio Simplest Form
$75\text{ paise}$ to ₹$3$ ₹$3 = 3 \times 100 = 300\text{ paise}$ $75 : 300$ $1 : 4$
$500\text{ m}$ to $2\text{ km}$ $2\text{ km} = 2 \times 1000 = 2000\text{ m}$ $500 : 2000$ $1 : 4$
$45\text{ minutes}$ to $2\text{ hours}$ $2\text{ hours} = 2 \times 60 = 120\text{ min}$ $45 : 120$ $3 : 8$
$250\text{ g}$ to $1.5\text{ kg}$ $1.5\text{ kg} = 1.5 \times 1000 = 1500\text{ g}$ $250 : 1500$ $1 : 6$
$8\text{ months}$ to $1\text{ year}$ $1\text{ year} = 12\text{ months}$ $8 : 12$ $2 : 3$
Deadly Board Trap: Never write a ratio directly from mixed units! For example, finding the ratio of $75\text{ paise}$ to ₹$3$ by writing $75 : 3 = 25 : 1$ is completely wrong. You must convert ₹$3$ to $300\text{ paise}$ first to get $1 : 4$.

Module 3: Comparison of Ratios & Partitioning a Quantity in a Given Ratio

3.1 Comparing Two Ratios: Which Ratio is Greater?

To compare two ratios $a : b$ and $c : d$, express them as fractions $\frac{a}{b}$ and $\frac{c}{d}$. There are two standard algebraic techniques:

Method 1: Cross-Multiplication

Compare $\frac{a}{b}$ and $\frac{c}{d}$ by finding cross-products:

• If $a \times d > b \times c \implies a : b > c : d$
• If $a \times d < b \times c \implies a : b < c : d$
• If $a \times d = b \times c \implies a : b = c : d$

Method 2: Common Denominators (LCM)

Find $\text{LCM}(b, d)$ and convert both fractions to like fractions:

To compare $3 : 4$ and $5 : 6$:
$\frac{3}{4} = \frac{9}{12}$ and $\frac{5}{6} = \frac{10}{12}$.
Since $9 < 10 \implies 3 : 4 < 5 : 6$.

3.2 Partitioning a Whole Quantity in a Given Ratio ($m : n$)

When a total quantity $Q$ is to be shared between two parties $A$ and $B$ in the ratio $m : n$, think of the whole as being split into equal portions or "parts":

Step-by-Step Two-Way Partitioning Algorithm:

  1. Calculate Total Number of Parts: $\text{Total Parts} = m + n$
  2. Calculate Share of $A$: $\text{Share}_A = \left(\frac{m}{m + n}\right) \times Q$
  3. Calculate Share of $B$: $\text{Share}_B = \left(\frac{n}{m + n}\right) \times Q$
  4. Verification Check: $\text{Share}_A + \text{Share}_B = Q$
Worked Illustration: Divide ₹$1,200$ between Reena and Tina in the ratio $3 : 5$.
• Total Parts $= 3 + 5 = 8$
• Reena's share $= \frac{3}{8} \times 1200 = 3 \times 150 = \mathbf{\text{₹}450}$
• Tina's share $= \frac{5}{8} \times 1200 = 5 \times 150 = \mathbf{\text{₹}750}$
• Check: $450 + 750 = 1200$ ✓

3.3 Three-Way Partitioning ($a : b : c$)

The same logic applies when dividing among three recipients in the ratio $a : b : c$:

$$\text{Total Parts} = a + b + c$$ $$\text{Share}_1 = \frac{a}{a+b+c} \times Q, \quad \text{Share}_2 = \frac{b}{a+b+c} \times Q, \quad \text{Share}_3 = \frac{c}{a+b+c} \times Q$$
Example: Divide $72\text{ kg}$ of sweets among $A, B, C$ in the ratio $2 : 3 : 4$.
Total parts $= 2 + 3 + 4 = 9$.
$A = \frac{2}{9} \times 72 = 16\text{ kg}$,   $B = \frac{3}{9} \times 72 = 24\text{ kg}$,   $C = \frac{4}{9} \times 72 = 32\text{ kg}$.
Sum: $16 + 24 + 32 = 72\text{ kg}$ ✓

Module 4: Concept of Proportion, Terms & The Fundamental Law of Proportion

4.1 The Concept & Definition of Proportion

While a ratio compares two quantities, a Proportion is an equation that states that two ratios are equal.

$$a : b = c : d \iff a : b :: c : d \iff \frac{a}{b} = \frac{c}{d}$$

Read aloud as: "$a$ is to $b$ as $c$ is to $d$"

Four quantities $a, b, c, d$ are said to be in proportion only when the ratio of the first two equals the ratio of the last two. The four numbers $a, b, c, d$ are called the Terms of the proportion.

4.2 Anatomy of a Proportion: Extremes and Means

In the proportion $a : b :: c : d$:

Extreme Terms (Extremes)

The first term ($a$) and the fourth term ($d$) sit on the outer boundaries and are called the Extremes.

Middle Terms (Means)

The second term ($b$) and the third term ($c$) sit in the middle and are called the Means.

4.3 The Fundamental Law of Proportion (Cross-Product Rule)

The core mathematical theorem governing every valid proportion is:

$$\text{Product of Extremes} = \text{Product of Means}$$ $$a \times d = b \times c$$
Application 1: Testing if Four Numbers are in Proportion
Are $4, 6, 8, 12$ in proportion?
• Product of Extremes $= 4 \times 12 = 48$
• Product of Means $= 6 \times 8 = 48$
Since $\text{Product of Extremes} = \text{Product of Means} = 48$, YES, $4 : 6 :: 8 : 12$ is in proportion!
Application 2: Solving for an Unknown Term ($x$)
Find $x$ if $x : 15 :: 8 : 12$.
$$\text{Product of Extremes} = \text{Product of Means} \implies x \times 12 = 15 \times 8$$ $$12x = 120 \implies x = \frac{120}{12} = \mathbf{10}$$

Module 5: Continued Proportion, Mean Proportional & Third Proportional

5.1 Continued Proportion (Three Quantities)

Three quantities $a, b, c$ are said to be in Continued Proportion if the ratio of the first to the second is equal to the ratio of the second to the third:

$$a : b :: b : c \iff \frac{a}{b} = \frac{b}{c}$$

Applying the Fundamental Law of Proportion ($a \times c = b \times b$) produces the golden identity: $$b^2 = ac$$

Mean Proportional ($b$)

The middle term $b$ between $a$ and $c$ is called the Mean Proportional:

$$b = \sqrt{ac}$$
Worked Example: Find the mean proportional between $4$ and $16$.
$$b = \sqrt{4 \times 16} = \sqrt{64} = \mathbf{8}$$ Check: $\frac{4}{8} = \frac{8}{16} = \frac{1}{2}$ ✓

Third Proportional ($c$)

When two numbers $a$ and $b$ are given, the third term $c$ such that $a : b :: b : c$ is called the Third Proportional:

$$c = \frac{b^2}{a}$$
Worked Example: Find the third proportional to $9$ and $12$.
$$c = \frac{12^2}{9} = \frac{144}{9} = \mathbf{16}$$ Check: $\frac{9}{12} = \frac{3}{4}$ and $\frac{12}{16} = \frac{3}{4}$ ✓
Examiner Distinction: Do not confuse a fourth proportional with a third proportional!
• In a 4-term proportion ($a, b, c, d$), $d$ is the fourth proportional ($d = \frac{bc}{a}$).
• In a 3-term continued proportion ($a, b, c$), $c$ is the third proportional ($c = \frac{b^2}{a}$).

Module 6: The Unitary Method — Direct Variation

6.1 What is the Unitary Method?

The word Unitary originates from the Latin root unitas, meaning one or single unit.

The method of finding the value of one unit first (by division), and then finding the value of the required number of units (by multiplication), is called the Unitary Method.

6.2 The Two-Step Algorithmic Procedure

1 Step 1: Division (Find 1 Unit)

$$\text{Value of } 1 \text{ unit} = \frac{\text{Total Value of Given Items}}{\text{Number of Given Items}}$$

(Divide to find the cost or measurement of a single entity)
2 Step 2: Multiplication (Find Required)

$$\text{Value of Required Units} = (\text{Value of } 1 \text{ unit}) \times \text{Required Quantity}$$

(Multiply to scale up to the target quantity)

6.3 Two Types of Unitary Problems in Class 6 Exams

Type A: Finding the Cost/Value

"If $6$ notebooks cost ₹$150$, find the cost of $14$ notebooks."

Cost of $1$ notebook $= \frac{150}{6} = \text{₹}25$
Cost of $14$ notebooks $= 14 \times 25 = \mathbf{\text{₹}350}$

Type B: Finding the Quantity

"If $6$ notebooks cost ₹$150$, how many can be bought for ₹$200$?"

For ₹$1$, number of notebooks $= \frac{6}{150} = \frac{1}{25}$
For ₹$200$, notebooks $= 200 \times \frac{1}{25} = \mathbf{8\text{ notebooks}}$

Module 7: Applied Word Problems, Real-World Modelling & Advanced Multi-Step Applications

7.1 Real-World Applications of Proportional Reasoning

Proportional reasoning is the core engine behind multiple real-world science and commercial disciplines:

1. Metallurgy & Mixtures:
Alloys like brass (copper and zinc in ratio $7 : 3$) or bronze (copper and tin).
2. Cartography (Map Scales):
Scale ratio $1 : 100,000$ implies $1\text{ cm}$ on the map represents $1\text{ km}$ on Earth.
3. Uniform Speed & Travel:
Distance travelled is directly proportional to time elapsed at constant speed.
4. Financial Partnerships:
Dividing business profits in direct proportion to the capital invested by partners.

7.2 Comprehensive ICSE Board Problem-Solving Framework

The 4-Step Master Strategy for Examination Word Problems:
Step 1: Check Units First! Look carefully at all quantities. If one is in metres and another in kilometres, convert immediately before setting up any ratio.
Step 2: Identify the Problem Type: Is it a simple ratio, sharing a total quantity in parts, a 4-term proportion check ($ad = bc$), continued proportion ($b^2 = ac$), or a unitary method word problem?
Step 3: State the Formula / Relationship Explicitly: Always write the formula (e.g. $\text{Product of Extremes} = \text{Product of Means}$) to secure mandatory methodology marks.
Step 4: Check Dimensions & Plausibility: Ratios have NO units. But final answers to unitary problems DO carry physical units (₹, kg, km, hours).

Key Formulas, Identities & Theorems

Ratio Definition & Simplest Form Invariant
$$a : b = \frac{a}{b} \quad (b \neq 0); \quad a : b \text{ is in simplest form} \iff \text{HCF}(a, b) = 1$$
Equivalent Ratios Scalar Multiplicative Law
$$a : b = (a \times k) : (b \times k) = (a \div k) : (b \div k) \quad \text{for any non-zero real scalar } k \neq 0$$
The Homogeneity & Unit Homogenisation Metric
$$\text{Ratio}(Q_1, Q_2) = \frac{Q_1 \text{ (in base unit } U)}{Q_2 \text{ (in base unit } U)} = \text{Pure Number (No Units)}$$
Cross-Multiplication Ratio Comparison Rule
$$\frac{a}{b} > \frac{c}{d} \iff a \times d > b \times c; \quad \frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c$$
Quantity Partitioning Total-Parts Formula
$$\text{Share}_A = \left(\frac{m}{m + n}\right) \times Q, \quad \text{Share}_B = \left(\frac{n}{m + n}\right) \times Q \quad \text{where } \text{Total Parts} = m + n$$
Fundamental Law of Proportion (Cross-Product Rule)
$$a : b :: c : d \iff \text{Product of Extremes} = \text{Product of Means} \iff a \times d = b \times c$$
Continued Proportion, Mean & Third Proportional Identities
$$a : b :: b : c \iff b^2 = ac \implies b = \sqrt{ac} \quad (\text{Mean}); \quad c = \frac{b^2}{a} \quad (\text{Third Proportional})$$
Two-Step Unitary Method Algorithmic Equation
$$\text{Step 1: } u_1 = \frac{V_{\text{given}}}{N_{\text{given}}} \quad (\text{Division}); \quad \text{Step 2: } V_{\text{target}} = u_1 \times N_{\text{required}} \quad (\text{Multiplication})$$

Conceptual Solved Examples & Case Studies

Example 1
Step-by-Step Solution:
Solution:
Step 1: Convert both quantities to a common unit (minutes):
• First quantity $= 45\text{ minutes}$
• Second quantity $= 2\text{ hours and } 15\text{ minutes} = (2 \times 60) + 15 = 120 + 15 = 135\text{ minutes}$ [1 Mark]

Step 2: Form the ratio and reduce to lowest terms:
$$\text{Required Ratio} = 45 : 135 = \frac{45}{135}$$
Finding the Highest Common Factor (HCF) of $45$ and $135$:
$\text{HCF}(45, 135) = 45$.
Dividing numerator and denominator by $45$:
$$\frac{45 \div 45}{135 \div 45} = \frac{1}{3}$$
$$\mathbf{\text{Answer: } 1 : 3}$$ [1 Mark]
(Note: A ratio has no units. Writing $1\text{ min} : 3\text{ min}$ is strictly penalized.)
Example 2
Step-by-Step Solution:
Solution:
Step 1: Express both ratios as fractions:
$$\text{Ratio 1: } \frac{5}{8}, \quad \text{Ratio 2: } \frac{7}{12}$$ [1 Mark]

Step 2: Apply the Cross-Multiplication Rule:
To compare $\frac{a}{b} = \frac{5}{8}$ and $\frac{c}{d} = \frac{7}{12}$, compute the cross products $(a \times d)$ and $(b \times c)$:
• First Cross Product $= 5 \times 12 = 60$
• Second Cross Product $= 8 \times 7 = 56$ [1 Mark]

Step 3: Compare cross products and conclude:
Since $60 > 56$, it follows directly that:
$$\frac{5}{8} > \frac{7}{12}$$
$$\mathbf{\text{Answer: } 5 : 8 \text{ is greater than } 7 : 12}$$ [1 Mark]
(Alternative Method: By LCM of denominators: $\text{LCM}(8, 12) = 24$. Then $\frac{5}{8} = \frac{15}{24}$ and $\frac{7}{12} = \frac{14}{24}$. Since $\frac{15}{24} > \frac{14}{24}$, $5 : 8 > 7 : 12$.)
Example 3
Step-by-Step Solution:
Solution:
Step 1: Find the total number of ratio parts:
Given ratio $= 2 : 3 : 5$
$$\text{Total Parts} = 2 + 3 + 5 = 10$$ [1 Mark]

Step 2: Compute each individual share:
Total amount to be divided $= \text{₹}2,400$
• Aman's Share:
$$\text{Share}_{\text{Aman}} = \frac{2}{10} \times 2400 = 2 \times 240 = \mathbf{\text{₹}480}$$ [0.5 Mark]
• Bobby's Share:
$$\text{Share}_{\text{Bobby}} = \frac{3}{10} \times 2400 = 3 \times 240 = \mathbf{\text{₹}720}$$ [0.5 Mark]
• Chetan's Share:
$$\text{Share}_{\text{Chetan}} = \frac{5}{10} \times 2400 = 5 \times 240 = \mathbf{\text{₹}1,200}$$ [0.5 Mark]

Step 3: Verification check:
$$\text{Sum of shares} = 480 + 720 + 1200 = \text{₹}2,400 \quad (\text{Matches total amount perfectly!})$$ [0.5 Mark]
Example 4
Step-by-Step Solution:
Solution:
Part (a): Testing if $16, 24, 20, 30$ are in proportion:
For four numbers $a, b, c, d$ to be in proportion, the Fundamental Law states that:
$$\text{Product of Extremes} = \text{Product of Means}$$
Here, Extremes are $16$ and $30$; Means are $24$ and $20$.
• $\text{Product of Extremes} = 16 \times 30 = 480$
• $\text{Product of Means} = 24 \times 20 = 480$ [1 Mark]
Since $\text{Product of Extremes} = \text{Product of Means} = 480$,
$$\mathbf{\text{Yes, } 16, 24, 20, 30 \text{ are in proportion } (16 : 24 :: 20 : 30).}$$ [1 Mark]

Part (b): Solving for $x$ in $25 : 35 :: x : 49$:
Applying the Fundamental Law of Proportion:
$$\text{Product of Extremes} = \text{Product of Means}$$
$$25 \times 49 = 35 \times x$$ [0.5 Mark]
Rearranging to isolate $x$:
$$x = \frac{25 \times 49}{35}$$
Dividing $49$ and $35$ by their common factor $7$:
$$x = \frac{25 \times 7}{5}$$
Dividing $25$ by $5$:
$$x = 5 \times 7 = \mathbf{35}$$ [1 Mark]
Check: $25 : 35 = 5 : 7$ and $35 : 49 = 5 : 7$ (Both ratios equal $5 : 7$). [0.5 Mark]
Example 5
Step-by-Step Solution:
Solution:
Part (a): Mean Proportional between $9$ and $25$:
Let the mean proportional be $b$.
By definition of continued proportion, $9 : b :: b : 25$, which means:
$$b^2 = 9 \times 25$$ [0.5 Mark]
$$b^2 = 225$$
Taking the positive square root:
$$b = \sqrt{225} = \mathbf{15}$$ [1 Mark]
Verification: $\frac{9}{15} = \frac{3}{5}$ and $\frac{15}{25} = \frac{3}{5}$. Both ratios are identical! [0.5 Mark]

Part (b): Third Proportional to $16$ and $24$:
Let the third proportional be $c$.
Then $16, 24, c$ are in continued proportion, meaning:
$$16 : 24 :: 24 : c$$ [0.5 Mark]
Applying the product rule of proportion:
$$16 \times c = 24 \times 24$$
$$16c = 576$$
$$c = \frac{576}{16} = \mathbf{36}$$ [1 Mark]
Verification: $\frac{16}{24} = \frac{2}{3}$ and $\frac{24}{36} = \frac{2}{3}$. Thus, $c = 36$ is verified. [0.5 Mark]
Example 6
Step-by-Step Solution:
Solution:
Part (a): Finding the cost of $25$ rods using the Unitary Method:
• Cost of $12$ steel rods $= \text{₹}4,320$
• Step 1 (Find cost of 1 rod by division):
$$\text{Cost of } 1 \text{ rod} = \frac{4320}{12} = \text{₹}360$$ [0.5 Mark]
• Step 2 (Find cost of 25 rods by multiplication):
$$\text{Cost of } 25 \text{ rods} = 25 \times 360 = \mathbf{\text{₹}9,000}$$ [1 Mark]

Part (b): Finding the number of rods obtainable for ₹$9,000$:
• Since the cost of $1$ rod $= \text{₹}360$,
• Number of rods obtainable for ₹$9,000$:
$$\text{Number of rods} = \frac{\text{Total Money Available}}{\text{Cost of } 1 \text{ rod}} = \frac{9000}{360} = \frac{900}{36} = \mathbf{25\text{ rods}}$$ [1.5 Marks]

Part (c): Partitioning a rod of $140\text{ cm}$ in the ratio $3 : 4$:
• Given ratio $= 3 : 4$
• Total number of parts $= 3 + 4 = 7$ parts [0.5 Mark]
• Length of the first piece:
$$\text{Length}_1 = \frac{3}{7} \times 140 = 3 \times 20 = \mathbf{60\text{ cm}}$$ [0.5 Mark]
• Length of the second piece:
$$\text{Length}_2 = \frac{4}{7} \times 140 = 4 \times 20 = \mathbf{80\text{ cm}}$$ [0.5 Mark]
• Verification: $60\text{ cm} + 80\text{ cm} = 140\text{ cm}$ (Exact match) and $\frac{60}{80} = \frac{3}{4}$. [0.5 Mark]

Common Misconceptions & Examiner Traps

Common Misconception

Calculating a ratio directly from mixed units without converting them to a common unit (e.g. writing the ratio of 75 paise to ₹3 as 75 : 3 = 25 : 1).

Scientific Reality & Correction

Always convert both quantities to the same unit first: ₹3 = 300 paise, giving 75 : 300 = 1 : 4.

Common Misconception

Attaching physical units to a ratio answer (e.g. writing 2 kg : 5 kg or stating the ratio is 2 : 5 kg).

Scientific Reality & Correction

A ratio is a pure, dimensionless number and must NEVER carry any units (e.g. simply 2 : 5).

Common Misconception

Reversing the order of antecedent and consequent (e.g. writing 4 : 3 when the question asked for the ratio of girls to boys, where girls = 15 and boys = 20).

Scientific Reality & Correction

Strictly follow the order given in the question: Ratio of girls to boys = 15 : 20 = 3 : 4 (NOT 4 : 3).

Common Misconception

Confusing the Third Proportional in continued proportion with the Fourth Proportional in a standard 4-term proportion.

Scientific Reality & Correction

For three numbers in continued proportion (a, b, c), third proportional c = b^2 / a. For four numbers (a, b, c, d), fourth proportional d = bc / a.

Common Misconception

Inverting the Unitary Method steps by multiplying first instead of dividing, or dividing the wrong quantities.

Scientific Reality & Correction

Always follow the strict two-step sequence: Step 1 (Divide) to find the value of 1 unit, Step 2 (Multiply) to scale up to the required quantity.

Common Misconception

Attempting to form a ratio between incompatible physical quantities (e.g. trying to find the ratio of 5 kilograms to 10 metres).

Scientific Reality & Correction

Ratios can ONLY be established between quantities of the same physical dimension (mass with mass, length with length).

Visual Learning & Conceptual Map

CISCE / ICSE CLASS 6 Ratio and Proportion Complete Master Diagnostic Framework: Comparison, Proportion Laws, Continued Proportion & Unitary Method 1. Ratio Concept & Invariants Comparison of 2 quantities of same kind by division Ratio = a : b = a / b a = Antecedent (1st) | b = Consequent (2nd, b ≠ 0) Pure Number: Ratio has NO units (Dimensionless) Order Matters: a : b ≠ b : a (Unless a = b) Unit Rule: Quantities MUST be in identical units UNIT CONVERSION TRAP: 75 paise : ₹3 = 75 p : 300 p = 1 : 4 (NOT 75:3!) 2. Simplest Form & Share Division Lowest terms reduction & ratio partitioning Simplest Form Rule: HCF(a, b) = 1 e.g. 36 : 48 = (36 ÷ 12) : (48 ÷ 12) = 3 : 4 DIVIDING QUANTITY Q IN RATIO m : n 1. Total Equal Parts = m + n 2. First Share = [ m / (m + n) ] × Q 3. Second Share = [ n / (m + n) ] × Q Example: Divide ₹600 in 2 : 3 → Parts = 5 Share A = (2/5)×600 = ₹240 | Share B = ₹360 3. Proportion & Cross-Product Law Equality of two ratios: a : b :: c : d a : b = c : d ⟺ a / b = c / d Product of Extremes = Product of Means a × d = b × c EXTREMES (a & d) a : b :: c : d MEANS (b & c) 4. Continued Proportion Three terms relationship: a, b, c in proportion a : b :: b : c ⟺ a / b = b / c b² = a × c ⟹ b = √(a × c) b = Mean Proportional | c = Third Proportional MEAN PROPORTIONAL: Between 4 and 16 → b = √(4 × 16) = √64 = 8 Check: 4/8 = 8/16 = 1/2 ✓ THIRD PROPORTIONAL (c): Formula: c = b² / a To 9 and 12 → c = 12² / 9 = 144 / 9 = 16 ✓ 5. The Unitary Method Finding 1 unit first to calculate any quantity 1 STEP 1: DIVISION (Find 1 Unit) Cost of 1 item = Total Cost ÷ Given Items 2 STEP 2: MULTIPLICATION Cost of Required Items = (Cost of 1) × Required PRACTICAL ICSE APPLICATION: Problem: 8 pens cost ₹120. Find cost of 15 pens. 1 Pen = 120 ÷ 8 = ₹15 15 Pens = 15 × 15 = ₹225 ✓ 6. Comparison of Ratios Testing magnitude: Which ratio is greater? CROSS-MULTIPLICATION RULE To compare a / b and c / d: Compute (a × d) and (b × c) If a·d > b·c ⟹ a:b > c:d | If a·d = b·c ⟹ In Proportion WORKED COMPARISON DRILL: Compare 3 : 5 and 5 : 8 Fraction Form: 3/5 vs 5/8 Cross Product 1: 3 × 8 = 24 Cross Product 2: 5 × 5 = 25 Since 24 < 25 ⟹ 3 : 5 < 5 : 8 ✓ TargetExams ICSE Curriculum Suite • Class 6 Mathematics • Chapter 6: Ratio and Proportion Product of Extremes = Product of Means • Unitary Rule • Dimensionless Ratios

Chapter Summary & 10 Key Takeaways

Takeaway 1
A Ratio is a mathematical comparison of two quantities of the same kind and in the same unit by division, denoted as a : b or a / b (where b != 0).
Takeaway 2
In a ratio a : b, the first term 'a' is called the Antecedent, and the second term 'b' is called the Consequent.
Takeaway 3
A ratio is a pure, dimensionless number without any physical units of measurement.
Takeaway 4
The order of terms in a ratio is significant: a : b is not equal to b : a unless both terms are equal.
Takeaway 5
A ratio is in its Simplest Form (lowest terms) when its terms are coprime integers whose Highest Common Factor is 1 (HCF(a, b) = 1).
Takeaway 6
Before finding the ratio of two quantities, both must be converted into the exact same unit of measurement (Homogeneity Principle).
Takeaway 7
To divide a quantity Q in the ratio m : n, compute total parts (m + n), yielding shares [m / (m + n)] * Q and [n / (m + n)] * Q.
Takeaway 8
A Proportion is an equation stating that two ratios are equal: a : b :: c : d, where 'a' and 'd' are Extremes, and 'b' and 'c' are Means.
Takeaway 9
The Fundamental Law of Proportion (Cross-Product Rule) states: Product of Extremes = Product of Means (a * d = b * c).
Takeaway 10
The Unitary Method calculates the value of a single unit first by division, and then determines the value of the required number of units by multiplication.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Find the ratio of 250 metres to 1.5 kilometres in simplest form.
Reveal Answer & Explanation
Answer: 1.5 km = 1.5 * 1000 = 1500 m. Ratio = 250 : 1500 = 250 / 1500 = 1 / 6 = 1 : 6.
Convert 1.5 km to metres (1 km = 1,000 m).
2
Compare the ratios 3 : 5 and 4 : 7. Which ratio is greater?
Reveal Answer & Explanation
Answer: Cross products: 3 * 7 = 21 and 5 * 4 = 20. Since 21 > 20, 3 : 5 > 4 : 7 (3 : 5 is greater).
Use cross-multiplication: 3 * 7 vs 5 * 4.
3
Two numbers are in the ratio 4 : 7. If their sum is 132, find the two numbers.
Reveal Answer & Explanation
Answer: Total parts = 11. First number = (4 / 11) * 132 = 4 * 12 = 48. Second number = (7 / 11) * 132 = 7 * 12 = 84. Check: 48 + 84 = 132.
Total parts = 4 + 7 = 11.
4
Check whether 15, 20, 30, 40 are in proportion.
Reveal Answer & Explanation
Answer: Product of Extremes = 15 * 40 = 600. Product of Means = 20 * 30 = 600. Since Product of Extremes = Product of Means, YES, they are in proportion.
Calculate Product of Extremes (15 * 40) and Product of Means (20 * 30).
5
Find the mean proportional between 4 and 36.
Reveal Answer & Explanation
Answer: b^2 = 4 * 36 = 144 => b = sqrt(144) = 12.
Mean proportional b = sqrt(a * c).
6
If 7 metres of cloth costs ₹560, find the cost of 12 metres of cloth using the unitary method.
Reveal Answer & Explanation
Answer: Cost of 1 metre = 560 / 7 = ₹80. Cost of 12 metres = 12 * 80 = ₹960.
Find the cost of 1 metre by division first, then multiply by 12.
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