Ratio and Proportion

Overview

Ratio and proportion form the backbone of quantitative reasoning at the primary level and appear consistently in JTET Paper I mathematics. These concepts connect arithmetic operations to real-world comparisons—sharing sweets among children, mixing ingredients, or scaling maps. Mastery here builds the foundation for percentage, profit-loss, and advanced problem-solving in later classes.

For JTET, expect direct questions on simplifying ratios, finding missing terms in proportions, and applying the unitary method to everyday situations. The pedagogy component may also ask how to introduce these concepts using concrete materials. Focus on the relationship between ratio, proportion, and the unitary method as three interconnected tools rather than isolated topics.

Key Concepts

• **Ratio** compares two quantities of the same kind using division. The ratio of a to b is written as a : b or a/b. Both quantities must be in the same unit before forming a ratio.

• **Terms of a ratio**: In a : b, 'a' is the antecedent (first term) and 'b' is the consequent (second term). The ratio has no unit—it is a pure number.

• **Simplest form**: A ratio is in simplest form when the HCF of both terms is 1. Example: 12 : 18 simplifies to 2 : 3 by dividing both by 6.

• **Proportion** states that two ratios are equal. If a : b = c : d, we write a : b :: c : d. Here a and d are called extremes, b and c are called means.

• **Fundamental property of proportion**: Product of extremes = Product of means. That is, a × d = b × c. This rule helps find an unknown term.

• **Unitary method** finds the value of one unit first, then uses it to find the value of any number of units. It applies the principle: if more items cost more (direct) or fewer items take more time (inverse), adjust accordingly.

• **Direct proportion**: When one quantity increases, the other increases in the same ratio. Example: more notebooks cost more money.

• **Inverse proportion** (awareness level for primary): When one quantity increases, the other decreases. Example: more workers finish work in less time.

Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Ratio of a to b | a : b = a/b (same units required) | | Simplifying ratio | Divide both terms by their HCF | | Proportion test | a : b :: c : d ⇔ a × d = b × c | | Finding fourth proportional | If a : b :: c : x, then x = (b × c)/a | | Finding third proportional | If a : b :: b : x, then x = b²/a | | Mean proportional of a and c | x = √(a × c) where a : x :: x : c | | Unitary method (direct) | Value of 1 unit = Total value ÷ Number of units | | Scaling a ratio | Multiply or divide both terms by the same non-zero number |

**Must-remember facts:**

• Order matters: 3 : 5 ≠ 5 : 3
• Ratios compare like quantities; you cannot form a ratio of 5 kg to 3 hours
• A ratio remains unchanged if both terms are multiplied or divided by the same number
• In a : b : c (three-term ratio), compare pairwise: a : b and b : c

Worked Examples

**Example 1: Simplifying a ratio**

*Problem*: Express 45 : 60 in simplest form.

*Solution*:

• 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.
• Divide both terms: 45 ÷ 15 = 3, 60 ÷ 15 = 4.
• **Answer: 3 : 4**

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**Example 2: Finding a missing term in proportion**

*Problem*: If 4 : 7 :: 12 : x, find x.

*Solution*:

• Apply product of extremes = product of means: 4 × x = 7 × 12
• 4x = 84
• x = 84 ÷ 4 = 21
• **Answer: x = 21**

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**Example 3: Unitary method**

*Problem*: If 8 pens cost ₹96, what is the cost of 13 pens?

*Solution*:

• Cost of 1 pen = 96 ÷ 8 = ₹12
• Cost of 13 pens = 12 × 13 = ₹156
• **Answer: ₹156**

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**Example 4: Dividing a quantity in a given ratio**

*Problem*: Divide ₹350 between A and B in the ratio 3 : 4.

*Solution*:

• Total parts = 3 + 4 = 7
• Value of 1 part = 350 ÷ 7 = ₹50
• A's share = 3 × 50 = ₹150
• B's share = 4 × 50 = ₹200
• **Answer: A gets ₹150, B gets ₹200**

Common Mistakes

• **Ignoring units before forming ratio** → Always convert to the same unit first. To compare 2 m and 50 cm, write 200 cm : 50 cm = 4 : 1.

• **Reversing the order carelessly** → Students write "ratio of boys to girls" as girls : boys. Read the question carefully; order matters.

• **Not simplifying to lowest terms** → Exam options often list the simplest form. Always reduce using HCF.

• **Applying cross-multiplication incorrectly** → Remember: extremes multiply together, means multiply together. Mixing positions gives wrong answers.

• **Forgetting to find the value of one unit in unitary method** → Jumping directly to multiplication without first dividing leads to errors. Always write the one-unit step explicitly.

• **Adding instead of multiplying when scaling** → To convert 2 : 3 to an equivalent ratio with first term 6, multiply both terms by 3 (not add 4). Result: 6 : 9.

Quick Reference

• Ratio a : b means a/b; both terms must share the same unit.
• Simplify by dividing both terms by their HCF.
• Proportion: a × d = b × c (extremes × extremes = means × means).
• Unitary method: Find value of 1 unit first, then scale.
• To divide amount Q in ratio m : n: each part = Q/(m + n), then multiply.
• Order is sacred—read "A to B" carefully before writing the ratio.