Ratio and Proportion

Overview

Ratio and proportion form the backbone of comparative mathematics at the primary level. These concepts help children understand relationships between quantities—how many times one quantity contains another, and how two such relationships can be equal. For CG TET Paper I, this topic carries significant weight as it connects directly to real-life situations that primary teachers must help students navigate.

In the exam, expect questions testing your conceptual clarity on forming ratios, simplifying them, checking proportions, and applying the unitary method to solve word problems. The questions often blend ratio-proportion with topics like fractions, percentages, and simple arithmetic, making it essential to master the foundational ideas thoroughly.

Teachers must understand not just the procedures but also why these methods work—this pedagogical understanding helps in both the content questions and the pedagogy section where you may be asked how to teach these concepts effectively to young learners.

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Key Concepts

• **Ratio** is a comparison of two quantities of the same kind by division. If there are 4 boys and 5 girls, the ratio of boys to girls is 4:5 (read as "4 is to 5").

• **Order matters in ratios.** The ratio 3:4 is different from 4:3. The first quantity mentioned always comes first in the ratio.

• **Ratios have no units.** When comparing 20 kg to 30 kg, the ratio is 2:3, not "2 kg : 3 kg."

• **Equivalent ratios** are formed by multiplying or dividing both terms by the same non-zero number. Example: 2:3 = 4:6 = 6:9.

• **Proportion** states that two ratios are equal. If a:b = c:d, we say a, b, c, d are in proportion, written as a:b :: c:d.

• **In a proportion a:b :: c:d, the product of extremes equals the product of means.** That is, a × d = b × c. This is the cross-multiplication rule.

• **Unitary method** finds the value of one unit first, then uses it to find the value of the required number of units. It is the practical application of proportion.

• **Direct proportion**: When one quantity increases, the other increases proportionally (more goods → more cost). **Inverse proportion**: When one increases, the other decreases (more workers → less time).

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Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Ratio of a to b | a : b = a/b | | Simplest form | Divide both terms by their HCF | | Proportion test | a:b :: c:d if a × d = b × c | | Mean proportional | If a:x :: x:b, then x = √(a × b) | | Unitary method | Value of 1 unit = Total value ÷ Number of units | | Direct proportion | If x₁/y₁ = x₂/y₂ | | Inverse proportion | If x₁ × y₁ = x₂ × y₂ |

**Must-remember facts:**

• To compare ratios, convert them to fractions with the same denominator or to decimals.
• A ratio remains unchanged if both terms are multiplied or divided by the same number.
• In continued proportion a:b :: b:c, the value b² = a × c.

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Worked Examples

**Example 1: Simplifying a Ratio**

*Reduce 45:60 to its simplest form.*

Step 1: Find HCF of 45 and 60.

• Factors of 45 = 1, 3, 5, 9, 15, 45
• Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• HCF = 15

Step 2: Divide both terms by 15.

• 45 ÷ 15 = 3
• 60 ÷ 15 = 4

**Answer: 3:4**

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**Example 2: Checking Proportion**

*Are 8, 12, 10, 15 in proportion?*

Step 1: Write as ratios → 8:12 and 10:15

Step 2: Apply cross-multiplication test.

• Product of extremes = 8 × 15 = 120
• Product of means = 12 × 10 = 120

Since both products are equal, **yes, they are in proportion.**

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**Example 3: Unitary Method (Direct Proportion)**

*If 5 notebooks cost ₹75, what is the cost of 8 notebooks?*

Step 1: Find cost of 1 notebook.

• Cost of 1 notebook = 75 ÷ 5 = ₹15

Step 2: Find cost of 8 notebooks.

• Cost of 8 notebooks = 15 × 8 = ₹120

**Answer: ₹120**

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**Example 4: Inverse Proportion**

*If 6 workers can complete a task in 12 days, how many days will 9 workers take?*

Step 1: Recognise inverse proportion (more workers → fewer days).

Step 2: Use the rule x₁ × y₁ = x₂ × y₂

• 6 × 12 = 9 × y₂
• 72 = 9 × y₂
• y₂ = 72 ÷ 9 = 8

**Answer: 8 days**

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Writing ratio of 5 girls to 3 boys as 3:5 | Always maintain the order as stated in the question—5:3 here. | | Leaving ratio as 6:9 without simplifying | Always reduce to simplest form by dividing by HCF → 2:3. | | Adding units to ratios (e.g., "2 m : 3 m") | Ratios are pure numbers; drop the units after ensuring both quantities have the same unit. | | Using direct proportion for all problems | Check whether quantities move in the same direction (direct) or opposite directions (inverse). | | Cross-multiplying incorrectly: a × c instead of a × d | Remember: extremes are first and last terms; means are middle two. Product of extremes = product of means. |

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Quick Reference

• **Ratio a:b** means a/b; always express in simplest form.
• **Proportion check**: Cross-multiply → a × d must equal b × c.
• **Unitary method**: Find value of 1 unit first, then scale up or down.
• **Direct proportion**: Both quantities increase or decrease together.
• **Inverse proportion**: One increases while the other decreases.
• **Equivalent ratios**: Multiply or divide both terms by the same number.