Combining Ratios — LCM Method

🎯 Learning Objective

Merge a:b and b:c into a:b:c by equalising the common term.

💡 Concept

• To combine A:B and B:C, make the B value SAME in both ratios
• Take LCM of the two B values, scale each ratio up to it
• A:B = 2:3 and B:C = 4:5 → LCM(3,4) = 12 → 8:12 and 12:15 → A:B:C = 8:12:15
• Longer chains (A:B, B:C, C:D) — repeat the same step pair by pair
• Ratio-change questions: (a x + k)/(b x + k) = new ratio → solve for x

🧮 Key Formulas

A:B:C = (A×LCM/B₁) : LCM : (C×LCM/B₂)

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New ratio: (ax + k)/(bx + k) = p/q

✏️ Easy Example

Q. If A:B = 2:3 and B:C = 4:5, find A:B:C.

1. LCM of 3 and 4 = 12
2. A:B = 8:12 and B:C = 12:15
3. A:B:C = 8:12:15

Answer: 8:12:15

🇮🇳 Real-Life Example

You know Amit earns 2/3 of Vijay, and Vijay earns 4/5 of Raju — equalise Vijay in both ratios and all three salaries line up for comparison.

📝 Exam-Level Example

Q. If A:B = 3:4 and B:C = 5:6, divide ₹5,900 among A, B and C.

1. LCM of 4 and 5 = 20 → A:B:C = 15:20:24
2. Total parts = 59; one part = 5900/59 = 100
3. A = 1500, B = 2000, C = 2400

Answer: A = ₹1,500, B = ₹2,000, C = ₹2,400

📝 Exam-Level Example

Q. Two numbers are in the ratio 3:5. If 9 is added to each, the ratio becomes 3:4. Find the numbers.

1. (3x + 9)/(5x + 9) = 3/4
2. 12x + 36 = 15x + 27
3. 3x = 9 → x = 3
4. Numbers = 9 and 15

Answer: 9 and 15

🪄 Memory Trick

Middle term ka LCM — that is the whole method. Scale both ratios to it and read off a:b:c.

⚠️ Common Mistakes

• ❌ Joining A:B and B:C directly as A:C without equalising B
• ❌ Scaling only one ratio to the LCM and forgetting the other
• ❌ Simplifying the combined ratio before dividing the amount (allowed, but then recompute parts)

🏆 Exam Tips

• ✅ Write the two ratios one below the other — LCM step becomes visual
• ✅ For A:B, B:C, C:D chains, combine two at a time from the left

📌 Summary

• Common term equal → ratios merge
• Use LCM of the middle values
• 2:3 & 4:5 → 8:12:15 (the model example)
• Add-to-both questions → assume 3x, 5x and solve