⭐ Sabse Important Points · एक नज़र में
- Ratio a:b: Represents the fraction a/b; both a and b must be in the same unit.
- Proportion a:b = c:d: Means extremes property: a×d = b×c.
- Simplification: Divide both terms by their GCD to get the simplest form.
- Direct Proportion: If a/b = constant, then a and b are directly proportional (e.g., more items, more cost).
- Equivalent Ratios: Multiply or divide both terms by the same non-zero number to get equivalent ratios.
- Inverse Proportion: If a×b = constant, then a and b are inversely proportional (e.g., more workers, less time).
Introduction · परिचय
Ratio and Proportion is a fundamental topic in arithmetic that deals with comparing quantities and finding equivalent relationships. It is frequently tested in Agniveer Army GD exams through direct problems, word problems, and its applications in partnership, mixtures, and time-speed-distance. This short revision note covers the core concepts, formulas, and problem-solving techniques.
Ratio: Concept and Simplification
A ratio compares two quantities of the same kind by division. It is denoted as a:b, where a and b are positive numbers. The ratio a:b means a/b. Ratios are simplified by dividing both terms by their greatest common divisor (GCD). Ratios can be scaled up or down by multiplying or dividing both terms by the same non-zero number. A ratio has no units.
📐 Formula / Rule Box · सूत्र याद रखो
Ratio as fraction
a:b = a/b
Simplification
a:b = (a/GCD(a,b)) : (b/GCD(a,b))
⭐ Key Points · याद रखने वाली बातें
- Ratio a: b: Represents the fraction a/b; both a and b must be in the same unit.
- Simplification: Divide both terms by their GCD to get the simplest form.
- Equivalent Ratios: Multiply or divide both terms by the same non-zero number to get equivalent ratios.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Simplify the ratio 24:36.
✅ Solution: 2:3
💡 Explanation: GCD of 24 and 36 is 12. Divide both by 12: 24/12 = 2, 36/12 = 3. So ratio is 2:3.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: If a:b = 3:4 and b:c = 5:6, find a:c.
✅ Solution: 5:8
💡 Explanation: Make b common. LCM of 4 and 5 is 20. a:b = 3:4 = 15:20; b:c = 5:6 = 20:24. So a:b:c = 15:20:24, hence a:c = 15:24 = 5:8.
Proportion: Concept and Types
A proportion is an equation stating that two ratios are equal. It is written as a:b = c:d or a/b = c/d. The terms a and d are called extremes, and b and c are called means. In a proportion, product of extremes = product of means (a×d = b×c). There are two main types: direct proportion (both quantities increase or decrease together) and inverse proportion (one increases while the other decreases).
📐 Formula / Rule Box · सूत्र याद रखो
Proportion
a:b = c:d ⇒ a×d = b×c
Direct proportion
a/b = constant
Inverse proportion
a×b = constant
| Direct Proportion | Inverse Proportion |
|---|---|
| Both quantities increase or decrease together | One increases while the other decreases |
| Ratio a/b is constant | Product a×b is constant |
| Example: More items → More cost | Example: More workers → Less time |
⭐ Key Points · याद रखने वाली बातें
- Proportion a: b = c:d: Means extremes property: a×d = b×c.
- Direct Proportion: If a/b = constant, then a and b are directly proportional (e.g., more items, more cost).
- Inverse Proportion: If a×b = constant, then a and b are inversely proportional (e.g., more workers, less time).
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find x if 4:5 = 12:x.
✅ Solution: x = 15
💡 Explanation: In proportion, product of extremes = product of means: 4×x = 5×12 → 4x = 60 → x = 15.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: If 8 workers can build a wall in 12 days, how many days will 6 workers take?
✅ Solution: 16 days
💡 Explanation: Workers and days are inversely proportional: 8×12 = 6×d → 96 = 6d → d = 16 days.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Simplify the ratio 45:60.
👀 Answer dekho
✅ Answer: 3:4
💡 Explanation: GCD of 45 and 60 is 15. Divide both by 15: 45/15=3, 60/15=4. So ratio is 3:4.
Q2If a:b = 2:3 and b:c = 4:5, find a:b:c.
👀 Answer dekho
✅ Answer: 8:12:15
💡 Explanation: Make b common. LCM of 3 and 4 is 12. a:b = 2:3 = 8:12; b:c = 4:5 = 12:15. So a:b:c = 8:12:15.
Q3If 5 pens cost ₹40, what is the cost of 8 pens?
👀 Answer dekho
✅ Answer: ₹64
💡 Explanation: Direct proportion: cost per pen = 40/5 = ₹8. So 8 pens cost 8×8 = ₹64.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting to convert quantities to the same unit before forming a ratio (e.g., mixing metres and centimetres).
- In inverse proportion problems, mistakenly using division instead of multiplication (or vice versa).
- Not simplifying ratios to their lowest terms, leading to incorrect answers in multiple-choice questions.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Ratio a: b: Compares two quantities of the same kind; simplify by dividing by GCD.
- Proportion a: b = c:d: Product of extremes equals product of means (a×d = b×c).
- Direct vs Inverse: Direct proportion → ratio constant; Inverse proportion → product constant.