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CBSE • Class 6 • Mathematics • Ch 12

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Ratio and Proportion

In CBSE Class 6 Mathematics, "Ratio and Proportion" explores comparative math: comparing quantities by division ($a : b$), simplest form, equivalence of ratios, definition of proportion ($a : b :: c : d$), cross-product rule ($ad = bc$), and the Unitary Method for real-life rate calculations.

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⚖️ Have You Ever Wondered?

If 5 pens cost ₹50, how much will 12 pens cost, and why can you not find the ratio of 2 metres to 500 grams?

Ratios compare quantities of the SAME kind! Master proportions and the Unitary Method used in everyday commerce, cooking recipes, and speed calculations.

Why This Chapter Matters

Before You Begin (Prerequisites)

Fractions and division
Unit conversions

What You Will Learn (Core Objectives)

Express the ratio of two quantities of the same unit in simplest form $a : b$.
Determine whether two ratios form a Proportion: $a : b = c : d$.
Apply the Cross-Product Rule: $\text{Product of Extremes} = \text{Product of Means}$.
Solve practical problems using the Unitary Method (find 1 unit first, then multiply).

Chapter Roadmap & Progression

1 1. Ratio: Comparison by Division

2 2. Proportion: Equality of Two Rati...

3 3. The Unitary Method

Complete Concept Guide (100% Curriculum Coverage)

1. Ratio: Comparison by Division

A Ratio compares two quantities of the same kind and in the same units by division. Denoted by the colon symbol "$:$".

$$\text{Ratio of } a \text{ to } b = a : b = \frac{a}{b}$$

Ratios have no units (units cancel out).
Quantities must be in the same unit before forming ratio (e.g., ratio of $2\text{ m}$ to $60\text{ cm} = 200\text{ cm} : 60\text{ cm} = 10 : 3$).
Order matters: $a : b \ne b : a$.

2. Proportion: Equality of Two Ratios

If two ratios are equal, they are said to be in Proportion, denoted by "$::$":$$a : b :: c : d \iff \frac{a}{b} = \frac{c}{d}$$

$a, d$ are the Extreme terms; $b, c$ are the Middle (Mean) terms.
Fundamental Cross-Product Law:$$\text{Product of Extremes} = \text{Product of Means} \iff a \times d = b \times c$$

3. The Unitary Method

The method in which we first find the value of one unit (by division) and then find the value of the required number of units (by multiplication) is called the Unitary Method.

$\text{Value of 1 item} = \frac{\text{Total cost of given items}}{\text{Number of given items}}$
$\text{Cost of required items} = \text{Value of 1 item} \times \text{Required quantity}$

Key Formulas, Identities & Theorems

Proportion Condition

$$a : b :: c : d \iff ad = bc$$

Product of extremes equals product of means.

Unitary Rule

$$\text{Total Cost} = \left(\frac{\text{Total Cost}_1}{N_1}\right) \times N_2$$

Two-step division then multiplication.

Conceptual Solved Examples & Case Studies

Example 1

Find the ratio of 30 days to 36 hours in simplest form.

Step-by-Step Solution:

Convert to same units (hours):
$30\text{ days} = 30 \times 24 = 720\text{ hours}$.
$\text{Ratio} = \frac{720}{36} = \frac{20}{1} = \mathbf{20 : 1}$.

Example 2

If the cost of 6 cans of juice is ₹210, what will be the cost of 4 cans of juice?

Step-by-Step Solution:

Cost of 6 cans $= ₹210$.
Cost of 1 can $= \frac{210}{6} = ₹35$.
Cost of 4 cans $= 35 \times 4 = \mathbf{₹140}$.

Common Misconceptions & Examiner Traps

Common Misconception

Writing ratio without converting units (e.g. 50 paise to ₹2 as 50:2).

Scientific Reality & Correction

Units must match! ₹2 = 200 paise, so ratio is $50 : 200 = 1 : 4$.

Visual Learning & Conceptual Map

Proportion Extremes & Means

a : b :: c : d => a × d = b × c

Product of Extremes

a × d

Product of Means

b × c

Chapter Summary & 10 Key Takeaways

Takeaway 1

Ratio compares quantities of the same kind by division ($a : b$).

Takeaway 2

Proportion is equality of ratios ($a : b :: c : d \implies ad = bc$).

Takeaway 3

Unitary method finds value of 1 unit first, then multiplies for required quantity.

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.

Are 25, 30, 40, and 48 in proportion?

Reveal Answer & Explanation

Answer: Product of extremes $= 25 \times 48 = 1,200$. Product of means $= 30 \times 40 = 1,200$. Since $1,200 = 1,200$, they are in proportion.
Compare product of extremes and means.

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