🎙 Teacher-Style Introduction

Ratio & Proportion वह chapter है जो लगभग हर competitive exam का foundation है — Time & Work, Partnership, Mixture, Speed… सब इसी पर आधारित हैं। Delhi Police exam में यह 2–3 आसान सवाल देता है, बस concept clear होना चाहिए।

📘 Important Concepts & Formulas

🔹 Ratio (अनुपात)

Comparison of two quantities.

a : b = a/b

🔹 Proportion (समानुपात)

a : b = c : d ⇒ a/b = c/d

🔹 Cross Multiplication Rule

a/b = c/d ⇒ ad = bc

🔹 Increase/Decrease ratios

• a : b → (ka) : (kb) (Multiply or divide by same number)
• अगर दो ratios को compare करना हो तो LCM method best है।

🔹 Compound Ratio

(a : b) × (c : d) = (ac : bd)

⚡ Tricks & Shortcuts

• Ratio में *units same होना ज़रूरी है* (kg–kg, ₹–₹, students–students).
• a : b = x ⇒ a = kx, b = ky (k is common multiplier).
• Direct formula for split ratio: Total = Sum of ratio parts → Each share = (Individual part / Total parts) × Total value
• DP की exam में calculation बहुत simple होती है (LCM < 30).

✔ Solved Examples / हल प्रश्न

Example 1: Ratio of 3 and 5.
Solution: 3 : 5 (simple ratio)

Example 2: If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c
LCM of 3 & 4 = 12
a : b = 8 : 12
b : c = 12 : 15
Final = 8 : 12 : 15

Example 3: Divide ₹600 in ratio 2 : 3.
Total parts = 5
Share1 = (2/5)×600 = 240
Share2 = 360

Example 4: a : b = 4 : 6. Simplify.
Divide by 2 → 2 : 3

Example 5: If a/b = 3/7, find a when b = 28.
a = (3/7)×28 = 12

Example 6: a : b = 5 : 9. If sum = 70, find numbers.
Total parts = 14
a = 5/14 × 70 = 25
b = 45

Example 7: Simplify compound ratio (2:3)×(4:5).
Result = (8 : 15)

Example 8: Ratio of 45 to 60.
45 : 60 → divide by 15 → 3 : 4

✔ MCQs (Delhi Police Pattern)

1. Simplify: 18 : 24 → 3 : 4
2. Ratio of 50 to 20 = 5 : 2
3. Divide 300 in ratio 1 : 2 → 100, 200
4. If a : b = 3 : 5, find a when b = 40 → 24
5. a : b = 4 : 6, b : c = 3 : 5. Find a : b : c → 4 : 6 : 10
6. Simplify: 72 : 90 → 4 : 5
7. Sum = 84, ratio = 1 : 3 → 21, 63
8. a/b = 2/7, b=35 → a= 10
9. Compound ratio of 3:4 and 2:5 = 6 : 20 = 3 : 10
10. Find ratio of 0.4 to 2 → 1 : 5

💡 Motivation: Ratio is the “engine” of arithmetic — इसे strong कर लिया तो बाकी सब आसान लगने लगेगा! 🚓🔥