Ratio and Proportion

Overview

Ratio and proportion form the backbone of quantitative reasoning in HP TET Mathematics. This topic connects arithmetic to real-world problem solving—comparing quantities, scaling recipes, distributing amounts, and calculating costs. Questions typically test your ability to simplify ratios, find missing terms in proportions, and apply the unitary method to word problems.

For HP TET, expect 2–4 questions from this area, often presented as practical scenarios involving money distribution, time-work, or distance-speed relationships. Mastery here also strengthens your performance in percentage, profit-loss, and mensuration topics. The key is building a strong conceptual foundation and practising quick mental calculations.

Key Concepts

• **Ratio** is a comparison of two quantities of the same kind expressed as a:b or a/b. The first term (a) is called the antecedent, the second (b) is the consequent.

• **Ratios have no units** since they compare like quantities. A ratio of 3:5 means "for every 3 units of the first, there are 5 units of the second."

• **Proportion** states that two ratios are equal. If a:b = c:d, we write a:b :: c:d (read as "a is to b as c is to d"). Here a and d are extremes, b and c are means.

• **In a proportion, product of means = product of extremes.** This gives us b × c = a × d, which is the fundamental tool for finding unknown terms.

• **Unitary method** involves finding the value of one unit first, then multiplying to get the required quantity. It works on direct and inverse variation principles.

• **Direct proportion**: When one quantity increases, the other increases proportionally (more items cost more money).

• **Inverse proportion**: When one quantity increases, the other decreases proportionally (more workers take less time to complete a task).

• **Continued proportion**: Three quantities a, b, c are in continued proportion if a:b = b:c. Here b is called the mean proportional, and b² = a × c.

Formulas / Key Facts

| Concept | Formula/Fact | |---------|--------------| | Ratio simplification | Divide both terms by their HCF | | Proportion rule | If a:b :: c:d, then a × d = b × c | | Mean proportional | Between a and c is √(a × c) | | Third proportional | To a and b is b²/a | | Fourth proportional | To a, b, c is (b × c)/a | | Componendo | If a/b = c/d, then (a+b)/b = (c+d)/d | | Dividendo | If a/b = c/d, then (a−b)/b = (c−d)/d | | Duplicate ratio | Of a:b is a²:b² | | Sub-duplicate ratio | Of a:b is √a:√b | | Triplicate ratio | Of a:b is a³:b³ |

**Unitary Method Steps:** 1. Find value of 1 unit from given information 2. Multiply by required number of units

Worked Examples

**Example 1: Simplifying Ratios**

*Problem:* Express the ratio 84:126 in simplest form.

*Solution:*

• Find HCF of 84 and 126
• 84 = 2² × 3 × 7
• 126 = 2 × 3² × 7
• HCF = 2 × 3 × 7 = 42
• Divide both: 84 ÷ 42 = 2, 126 ÷ 42 = 3
• **Simplest ratio = 2:3**

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

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

*Solution:*

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

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**Example 3: 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 4: Unitary Method (Inverse Proportion)**

*Problem:* If 6 workers complete a task in 15 days, how many days will 10 workers take?

*Solution:*

• Total work = 6 × 15 = 90 worker-days
• Days for 10 workers = 90 ÷ 10 = 9 days
• **Answer: 9 days**

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**Example 5: Distribution in a Given Ratio**

*Problem:* Divide ₹630 among A, B, C in the ratio 2:3:4.

*Solution:*

• Total parts = 2 + 3 + 4 = 9
• Value of 1 part = 630 ÷ 9 = ₹70
• A's share = 2 × 70 = ₹140
• B's share = 3 × 70 = ₹210
• C's share = 4 × 70 = ₹280
• **Answer: A = ₹140, B = ₹210, C = ₹280**

Common Mistakes

• **Comparing unlike quantities** → Ratios must compare same units. Convert 2 hours and 45 minutes to the same unit (both to minutes: 120:45) before simplifying.

• **Forgetting to simplify completely** → Students stop at 6:9 instead of 2:3. Always check if both terms share a common factor.

• **Confusing direct and inverse proportion** → More workers means less time (inverse), not more time. Ask: "If one quantity increases, does the other increase or decrease?"

• **Wrong cross-multiplication order** → In a:b :: c:d, the equation is a × d = b × c, not a × c = b × d. Remember: extremes multiply, means multiply.

• **Ignoring units in word problems** → When the question gives km and metres, or rupees and paise, convert to the same unit before setting up the ratio.

• **Adding ratios incorrectly** → To combine ratios like A:B = 2:3 and B:C = 4:5, first make B common (8:12 and 12:15), giving A:B:C = 8:12:15.

Quick Reference

• **Ratio = comparison, Proportion = equality of two ratios**

• **Product of extremes = Product of means** (the golden rule for proportions)

• **Unitary method: Find value of 1 → Multiply by required quantity**

• **Direct proportion: both increase/decrease together; Inverse: one up, other down**

• **To distribute in ratio a:b:c, total parts = a+b+c, each share = (part × total) ÷ total parts**

• **Mean proportional of a and c = √(ac)**