Ratio and Proportion

Overview

Ratio and proportion form the backbone of quantitative reasoning in primary mathematics and appear consistently in JKTET Paper I. This topic tests a candidate's ability to compare quantities, solve real-world problems involving sharing, mixing, and scaling, and apply the unitary method—a fundamental technique that underpins much of arithmetic problem-solving.

For the JKTET, you must be comfortable with expressing ratios in simplest form, recognizing equivalent ratios, setting up and solving proportions, and using the unitary method to find unknown quantities. These concepts connect directly to topics like percentage, profit-loss, and mensuration, making them essential building blocks. Exam questions typically involve word problems requiring you to find a missing term, divide quantities in a given ratio, or apply direct and inverse proportion reasoning.

Mastering this topic means internalizing the relationship between ratios as comparisons and proportions as equalities of ratios—and knowing when to apply each technique efficiently under exam conditions.

Key Concepts

• **Ratio** is a comparison of two quantities of the same kind expressed as a:b or a/b. The quantities must share the same unit before forming a ratio.

• **Antecedent and Consequent**: In the ratio a:b, 'a' is the antecedent (first term) and 'b' is the consequent (second term).

• **Simplest Form**: A ratio is in simplest form when the HCF of both terms is 1. For example, 12:18 simplifies to 2:3.

• **Equivalent Ratios**: Ratios that represent the same comparison. Multiplying or dividing both terms by the same non-zero number gives equivalent ratios (2:3 = 4:6 = 6:9).

• **Proportion** is an equality of two ratios. If a:b = c:d, we say a, b, c, d are in proportion, written as a:b :: c:d.

• **Extremes and Means**: In a:b :: c:d, 'a' and 'd' are extremes; 'b' and 'c' are means. The product of extremes equals the product of means (a × d = b × c).

• **Unitary Method**: A technique where we first find the value of one unit, then multiply to find the value of the required number of units.

• **Direct Proportion**: Two quantities are in direct proportion if an increase in one causes a proportional increase in the other (more items cost more money).

• **Inverse Proportion**: Two quantities are in inverse proportion if an increase in one causes a proportional decrease in the other (more workers take fewer days to complete work).

Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Ratio of a to b | a:b = a/b (both in same units) | | Simplifying ratio | Divide both terms by their HCF | | Proportion test | a:b :: c:d if and only if a × d = b × c | | Mean proportional | If a:x :: x:b, then x = √(a × b) | | Third proportional | If a:b :: b:c, then c = b²/a | | Dividing Q in ratio a:b | Parts are Q × a/(a+b) and Q × b/(a+b) | | Unitary method (direct) | If M items cost ₹P, then 1 item costs ₹P/M | | Unitary method (inverse) | If M workers take D days, then 1 worker takes M × D days | | Continued proportion | a:b:c means a/b = b/c |

Worked Examples

**Example 1: Simplifying and Comparing Ratios**

*Problem*: Express 45:60 in simplest form and check if it equals 3:4.

*Solution*:

• HCF of 45 and 60 = 15
• 45 ÷ 15 = 3; 60 ÷ 15 = 4
• Simplest form = 3:4
• Yes, 45:60 equals 3:4 ✓

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**Example 2: Finding Missing Term in Proportion**

*Problem*: If 5:8 :: x:24, find x.

*Solution*:

• Using product of extremes = product of means
• 5 × 24 = 8 × x
• 120 = 8x
• x = 120 ÷ 8 = 15

*Answer*: x = 15

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**Example 3: Dividing a Quantity in a Given Ratio**

*Problem*: Divide ₹630 between Amir and Bilal in the ratio 4:5.

*Solution*:

• Sum of ratio parts = 4 + 5 = 9
• Amir's share = 630 × 4/9 = 2520/9 = ₹280
• Bilal's share = 630 × 5/9 = 3150/9 = ₹350

*Verification*: 280 + 350 = 630 ✓; Ratio 280:350 = 4:5 ✓

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**Example 4: Unitary Method (Direct Proportion)**

*Problem*: If 12 notebooks cost ₹180, what is the cost of 20 notebooks?

*Solution*:

• Cost of 1 notebook = 180 ÷ 12 = ₹15
• Cost of 20 notebooks = 15 × 20 = ₹300

*Answer*: ₹300

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**Example 5: Unitary Method (Inverse Proportion)**

*Problem*: If 6 workers can build a wall in 10 days, how many days will 15 workers take?

*Solution*:

• Total work = 6 × 10 = 60 worker-days
• Days for 15 workers = 60 ÷ 15 = 4 days

*Answer*: 4 days

Common Mistakes

• **Comparing quantities with different units** → Always convert to the same unit before forming a ratio. Convert 50 cm to 0.5 m (or 2 m to 200 cm) before comparing with 2 m.

• **Writing ratio in wrong order** → "Ratio of boys to girls" means boys:girls, not girls:boys. Read the question carefully for the sequence.

• **Forgetting to simplify** → Exam options often include unsimplified ratios as distractors. Always reduce to lowest terms unless asked otherwise.

• **Confusing direct and inverse proportion** → More workers means less time (inverse). More items means more cost (direct). Ask yourself: "If one increases, does the other increase or decrease?"

• **Adding instead of using the formula for division** → When dividing ₹100 in ratio 2:3, students sometimes compute ₹2 and ₹3. Correct approach: 100 × 2/5 = ₹40 and 100 × 3/5 = ₹60.

• **Applying extremes-means incorrectly** → In a:b :: c:d, the cross-multiplication is a×d = b×c, not a×c = b×d.

Quick Reference

• Ratio compares; proportion equates two ratios.
• a:b :: c:d ⟹ a × d = b × c (product rule).
• To divide Q in ratio a:b: first part = Q × a/(a+b).
• Unitary method: find value of 1, then scale up.
• Direct proportion: both quantities move in same direction.
• Inverse proportion: one up means the other down.
• Always check units match before forming any ratio.