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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 3

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Proportion

Welcome to Chapter 3 "Proportion" (সমানুপাত) of the West Bengal Board of Secondary Education (WBBSE) Class 7 Mathematics (Ganit Prabha / গণিত প্রভা). Structured according to the TargetExams Gold-Standard pedagogy, this master guide explores the equality of ratios, extreme and mean terms, the fundamental law of proportion ($a \times d = b \times c$), calculating missing proportional terms, continued proportion ($b^2 = ac$), and real-life direct vs. inverse variations.

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🌍 Have You Ever Wondered?

How did ancient Greek scholar Eratosthenes calculate the circumference of planet Earth using just a stick and a sun shadow?

In 240 BC, without spaceships or satellites, Eratosthenes used a simple proportion ($a : b :: c : d$) to calculate the size of our entire planet with 99% accuracy! He noted that at noon on the summer solstice, sunlight shone directly down a deep water well in Syene (no shadow). But in Alexandria, 500 miles away, a vertical stick cast a shadow at an angle of $7.2^\circ$.

He set up the proportion: $\frac{7.2^\circ}{360^\circ} = \frac{500\text{ miles}}{\text{Earth's Circumference}}$. Solving for the missing fourth term gave 25,000 miles! Proportion is not just an abstract formula—it is the universal geometric balance connecting shadows, architectural blueprints, map scales, and planetary physics.

Why This Chapter Matters

Before You Begin (Prerequisites)

Foundational understanding of ratios as simplified comparisons ($a : b$).
Identification of antecedents (first term) and consequents (second term).
Cross-multiplication rules and solving linear equations with one unknown.

What You Will Learn (Core Objectives)

Verify whether four given quantities form a proportion ($a : b :: c : d$) and identify extreme vs. mean terms.
Apply the Fundamental Law of Proportion ($\text{Product of Extremes} = \text{Product of Means}$) to compute unknown fourth, third, or first terms.
Calculate the Mean Proportional ($b = \sqrt{ac}$) and Third Proportional ($c = \frac{b^2}{a}$) in Continued Proportions.
Distinguish between Direct and Inverse Proportions in multi-step real-world word problems.
Solve geometric scale and shadow length problems with dimensional consistency.

Chapter Roadmap & Progression

1 1. The Fundamental Law of Proportio...

2 2. Computing Unknown Terms & The Fo...

3 3. Continued Proportion & The Mean...

4 4. Real-World Applications: Direct...

5 5. Geometric Proportions, Sun Shado...

Complete Concept Guide (100% Curriculum Coverage)

1. The Fundamental Law of Proportion & Term Anatomy

1. The Intuition

A proportion is an equality statement between two ratios. If $2\text{ kg}$ of apples cost ₹$160$ ($2 : 160 = 1 : 80$) and $5\text{ kg}$ cost ₹$400$ ($5 : 400 = 1 : 80$), both ratios are completely identical. Therefore, the four numbers $2, 160, 5, 400$ form a Proportion.

2. The Formal Concept & Structure

Four non-zero quantities $a, b, c, d$ are in proportion if the ratio of the first two equals the ratio of the last two:

$$\mathbf{a : b = c : d \quad \text{written as} \quad a : b :: c : d}$$

$a$ is the 1st term, $b$ is the 2nd term, $c$ is the 3rd term, and $d$ is the 4th term.
$a$ and $d$ lie at the outer borders and are called Extreme Terms (প্রান্তীয় পদ).
$b$ and $c$ lie in the middle and are called Mean Terms (মধ্যপদ).

The Fundamental Invariant Law:
$$\mathbf{\text{Product of Extremes} = \text{Product of Means}}$$ $$\mathbf{a \times d = b \times c}$$

3. Concrete Worked Example

Example: Test whether the numbers $8, 10, 16, 20$ form a proportion.

Method 1 (Ratio Comparison): $8 : 10 = \frac{8}{10} = \frac{4}{5}$ and $16 : 20 = \frac{16}{20} = \frac{4}{5}$. Since both ratios simplify to $\frac{4}{5}$, they form a proportion.

Method 2 (Product Law Verification):

Product of Extremes $= 8 \times 20 = 160$

Product of Means $= 10 \times 16 = 160$

Since $\text{Extremes Product} = \text{Means Product} = 160$, the numbers $8, 10, 16, 20$ are in proportion ($8 : 10 :: 16 : 20$).

4. Pitfall & Examiner Trap

⚠️ Trap: Reordering Terms Violates Proportion
If the order is changed to $8, 20, 10, 16$: Product of Extremes $= 8 \times 16 = 128$, but Product of Means $= 20 \times 10 = 200$. Since $128 \ne 200$, they are NOT in proportion! Order matters strictly.

5. Why This Matters in Life

HDTV resolutions and smartphone screens (such as 1920x1080 and 3840x2160) maintain a strict 16:9 proportion so movies display without black bars or stretched distortions.

2. Computing Unknown Terms & The Fourth Proportional

1. The Intuition

Because the product of extremes must equal the product of means, knowing any 3 terms in a proportion automatically unlocks the 4th unknown term with zero ambiguity.

2. The Algebraic Derivations

From the master equation $a \times d = b \times c$, we derive:

Unknown Term	Direct Solution Formula
Fourth Proportional ($d$)	$$d = \frac{b \times c}{a} = \frac{\text{2nd Term} \times \text{3rd Term}}{\text{1st Term}}$$
First Proportional ($a$)	$$a = \frac{b \times c}{d} = \frac{\text{2nd Term} \times \text{3rd Term}}{\text{4th Term}}$$
Second Proportional ($b$)	$$b = \frac{a \times d}{c} = \frac{\text{1st Term} \times \text{4th Term}}{\text{3rd Term}}$$
Third Proportional ($c$)	$$c = \frac{a \times d}{b} = \frac{\text{1st Term} \times \text{4th Term}}{\text{2nd Term}}$$

3. Concrete Worked Example

Example: Find the fourth proportional to $9, 12$, and $18$.

Let the fourth proportional be $x$.

Set up the proportion: $9 : 12 :: 18 : x$

Applying the law: $\text{Product of Extremes} = \text{Product of Means}$

$$9 \times x = 12 \times 18 \implies x = \frac{12 \times 18}{9}$$

Reducing: $x = 12 \times 2 = 24$

Answer: The fourth proportional is $\mathbf{24}$.

4. Pitfall & Examiner Trap

⚠️ Cross-Multiplication Transposition Error:
In $9 : 12 :: 18 : x$, students sometimes write $x = \frac{9 \times 18}{12}$. Remember: $x$ is an extreme term, so it multiplies with the other extreme ($9$). Thus, divide the product of means by 9!

5. Why This Matters in Life

Forex currency exchange calculations (e.g., $1\text{ USD} = 83\text{ INR} \implies 350\text{ USD} = x\text{ INR}$) execute this exact fourth-proportional formula millions of times a day.

3. Continued Proportion & The Mean Proportional ($b^2 = ac$)

1. The Intuition

What happens when the consequent of the first ratio becomes the antecedent of the second ratio? The three numbers form an unbroken geometric chain called a Continued Proportion. For example: in $4, 8, 16$, $\frac{4}{8} = \frac{8}{16} = \frac{1}{2}$.

2. The Formal Concept & Identities

Three positive numbers $a, b, c$ are in continued proportion if:

$$\mathbf{a : b :: b : c \implies \frac{a}{b} = \frac{b}{c}}$$

Cross-multiplying yields the Golden Geometric Identity:

$$\mathbf{b^2 = a \times c}$$

$b$ is called the Mean Proportional (মধ্য সমানুপাতী) between $a$ and $c$: $$\mathbf{b = \sqrt{a \times c}}$$
$c$ is called the Third Proportional (তৃতীয় সমানুপাতী) to $a$ and $b$: $$\mathbf{c = \frac{b^2}{a}}$$

3. Concrete Worked Example

Example: Find the mean proportional between $4$ and $25$, and find the third proportional to $8$ and $12$.

Part 1 (Mean Proportional):
Let mean proportional be $b$.
$b^2 = 4 \times 25 = 100 \implies b = \sqrt{100} = \mathbf{10}$

Part 2 (Third Proportional):
Let third proportional be $c$.
Set up continued proportion: $8 : 12 :: 12 : c \implies 8c = 12 \times 12 = 144 \implies c = \frac{144}{8} = \mathbf{18}$

Answers: Mean Proportional $= \mathbf{10}$, Third Proportional $= \mathbf{18}$.

4. Pitfall & Examiner Trap

⚠️ Trap: Confusing 3rd Proportional with 4th Proportional
When asked for the "Third Proportional to 8 and 12", students often search for a third number to make 4 terms. Remember: if only TWO numbers ($a, b$) are given, the middle term $b$ is repeated: $a : b :: b : c$!

5. Why This Matters in Life

Geometric sequences, acoustic speaker frequencies (octave doubling: 220Hz, 440Hz, 880Hz), and compound interest curves all evolve via continued proportions.

4. Real-World Applications: Direct vs. Inverse Proportion Setup

1. The Intuition

In word problems, determining whether two variables change in the same direction (Direct) or in opposite directions (Inverse) is the critical first step before writing down the proportion.

2. The Structural Formulation

Variation Type	Proportional Equation	Physical Reality
Direct Proportion (সরল সমানুপাত)	$x_1 : x_2 :: y_1 : y_2$	More goods = more cost; more hours = more distance
Inverse Proportion (ব্যস্ত সমানুপাত)	$x_1 : x_2 :: y_2 : y_1$	More workers = fewer days; faster speed = less time

3. Concrete Worked Example

Example: 15 farmers can harvest a field in 12 days. If the harvest must be completed in 9 days, how many extra farmers must be hired?

Step 1 (Analyze Variation): Decreasing days requires increasing farmers $\implies$ Inverse Proportion.

Step 2 (Set up Proportion): Invert the days ratio:
$12 : 9 :: x : 15$

Step 3 (Solve):
$9 \times x = 12 \times 15 \implies 9x = 180 \implies x = 20\text{ farmers}$

Step 4 (Answer the Exact Question):
Extra farmers needed $= 20 - 15 = \mathbf{5\text{ farmers}}$.

Answer: $\mathbf{5}$ extra farmers must be hired.

4. Pitfall & Examiner Trap

⚠️ Forgetting to subtract initial workers:
The total workforce is 20, but the question specifically asks for "extra" workers. Writing 20 as the final answer loses marks. Always read the final prompt carefully!

5. Why This Matters in Life

Civil engineering road construction schedules and hospital emergency staff shifts are calculated using inverse proportion balancing.

5. Geometric Proportions, Sun Shadows & Scale Modeling

1. The Intuition

Under sunlight at any given moment, the angle of the sun rays is identical everywhere in your city. This creates similar right-angled triangles, meaning: Height of an object is directly proportional to the length of its shadow.

2. The Scale Formula

$$\frac{\text{Height}_1}{\text{Shadow}_1} = \frac{\text{Height}_2}{\text{Shadow}_2} \implies \text{Height}_1 : \text{Height}_2 :: \text{Shadow}_1 : \text{Shadow}_2$$

Map Scales: $\text{Scale Factor} = \frac{\text{Map Distance}}{\text{Actual Ground Distance}}$.

3. Concrete Worked Example

Example: A vertical flagpole $6\text{ m}$ high casts a shadow of $8\text{ m}$. At the exact same time, a tall banyan tree casts a shadow of $24\text{ m}$. Find the height of the tree.

Let the height of the tree be $h\text{ m}$.

Set up direct proportion: $6 : 8 :: h : 24$

Product of Extremes $= 6 \times 24 = 144$

Product of Means $= 8 \times h \implies 8h = 144 \implies h = \frac{144}{8} = \mathbf{18\text{ m}}$

Answer: The banyan tree is $\mathbf{18\text{ meters}}$ tall.

4. Pitfall & Examiner Trap

⚠️ Mismatched Measurement Units:
If flagpole height is in meters ($6\text{ m}$) but shadow is in centimeters ($800\text{ cm}$), always harmonize all measurements into the same unit before forming the proportion.

5. Why This Matters in Life

Architects building miniature scale models of skyscrapers and Google Earth zooming functions operate strictly on this scale proportion.

Key Formulas, Identities & Theorems

Fundamental Law of Proportion

$$a \times d = b \times c$$

Product of extremes = Product of means

Fourth Proportional

$$d = \frac{b \times c}{a}$$

Solving for unknown 4th term

Mean Proportional

$$b = \sqrt{a \times c}$$

Middle term in continued proportion

Third Proportional

$$c = \frac{b^2}{a}$$

Third term in continued proportion

Direct Proportion Rule

$$x_1 : x_2 :: y_1 : y_2$$

When both quantities scale together

Inverse Proportion Rule

$$x_1 : x_2 :: y_2 : y_1$$

When one scales up and other down

Conceptual Solved Examples & Case Studies

Example 1

If $7, x, 35, 45$ are in proportion, find the value of $x$.

Step-by-Step Solution:

Applying the Product Law: $\text{Product of Means} = \text{Product of Extremes}$
$$x \times 35 = 7 \times 45 \implies x = \frac{7 \times 45}{35} = \frac{45}{5} = \mathbf{9}$$
Answer: $x = \mathbf{9}$.

Example 2

Find the mean proportional between $16$ and $36$.

Step-by-Step Solution:

Let mean proportional be $b$.
Using $b^2 = a \times c$:
$$b = \sqrt{16 \times 36} = \sqrt{16} \times \sqrt{36} = 4 \times 6 = \mathbf{24}$$
Answer: The mean proportional is $\mathbf{24}$.

Example 3

6 workers can construct a boundary wall in 12 days. How many workers are needed to finish the same work in 8 days?

Step-by-Step Solution:

Decreasing days requires more workers $\implies$ Inverse Proportion.
Set up proportion: $12 : 8 :: x : 6$
$$8 \times x = 12 \times 6 \implies 8x = 72 \implies x = \frac{72}{8} = \mathbf{9\text{ workers}}$$
Answer: $\mathbf{9}$ workers are needed.

Common Misconceptions & Examiner Traps

Common Misconception

Scrambling the order of terms in a proportion.

Scientific Reality & Correction

Position order is strictly invariant: 1st × 4th = 2nd × 3rd must always hold.

Common Misconception

Using fourth proportional formulas when asked for third proportional.

Scientific Reality & Correction

Third proportional means continued proportion ($a : b :: b : c$); repeat the middle term!

Common Misconception

Setting up inverse variation as direct variation in word problems.

Scientific Reality & Correction

In inverse variation, invert the second ratio: $x_1 : x_2 :: y_2 : y_1$.

Visual Learning & Conceptual Map

WBBSE Mathematics: Proportion Balance & Geometric Compass

The Fundamental Invariant Law: Product of Extremes must equal Product of Means ($a \times d = b \times c$)

1st Term

$a$

EXTREME

2nd Term

$b$

MEAN

3rd Term

$c$

MEAN

4th Term

$d$

EXTREME

Invariant Balance: $\mathbf{(1st \times 4th) = (2nd \times 3rd) \implies a \times d = b \times c}$ | Continued Proportion: $\mathbf{b = \sqrt{ac}}$

Chapter Summary & 10 Key Takeaways

Takeaway 1

Definition: Equality of two ratios forms a proportion ($a : b :: c : d$).

Takeaway 2

Fundamental Law: $\text{Product of Extremes} = \text{Product of Means}$ ($a \times d = b \times c$).

Takeaway 3

Fourth Proportional: $d = \frac{b \times c}{a}$.

Takeaway 4

Continued Proportion: Three terms $a, b, c$ such that $a : b :: b : c \implies b^2 = ac$.

Takeaway 5

Mean Proportional: $b = \sqrt{ac}$; Third Proportional: $c = \frac{b^2}{a}$.

Takeaway 6

Direct vs. Inverse: Direct uses $x_1 : x_2 :: y_1 : y_2$; Inverse uses $x_1 : x_2 :: y_2 : y_1$.

Takeaway 7

Geometry & Shadows: Heights and shadow lengths are directly proportional under uniform sunlight.

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.

If $4, 6, 10, x$ are in proportion, find the value of $x$.

Reveal Answer & Explanation

Answer: Product of extremes = Product of means $\implies 4 \times x = 6 \times 10 \implies 4x = 60 \implies x = \mathbf{15}$.
Multiply 1st × 4th and equate to 2nd × 3rd.

Find the mean proportional between $9$ and $25$.

Reveal Answer & Explanation

Answer: $b = \sqrt{9 \times 25} = 3 \times 5 = \mathbf{15}$.
Use the formula $b = \sqrt{a \times c}$.

Find the third proportional to $4$ and $12$.

Reveal Answer & Explanation

Answer: Set up $4 : 12 :: 12 : c \implies 4c = 144 \implies c = \mathbf{36}$.
Use the formula $c = \frac{b^2}{a}$.

On a map, $2\text{ cm}$ represents $50\text{ km}$ on the ground. What ground distance is represented by $5\text{ cm}$?

Reveal Answer & Explanation

Answer: $2 : 5 :: 50 : x \implies 2x = 250 \implies x = \mathbf{125\text{ km}}$.
Map distance and actual distance are in direct proportion.

12 workers can build a wall in 20 days. How many days will 15 workers take to build the same wall?

Reveal Answer & Explanation

Answer: Inverse proportion: $20 : x :: 15 : 12 \implies 15x = 240 \implies x = \mathbf{16\text{ days}}$.
More workers take fewer days (inverse proportion).

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