Proportion — Extremes, Means & Proportionals

🎯 Learning Objective

Use product of extremes = product of means, and find third, fourth and mean proportionals.

💡 Concept

• a:b :: c:d means a/b = c/d — a, d are EXTREMES; b, c are MEANS
• Rule: product of extremes = product of means → a × d = b × c
• Fourth proportional to a, b, c → x = bc/a
• Third proportional to a, b → x = b²/a (b appears twice: a:b :: b:x)
• Mean proportional between a and b → √(ab)

🧮 Key Formulas

a:b :: c:d → ad = bc

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Fourth proportional = bc/a

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Third proportional = b²/a

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Mean proportional = √(ab)

✏️ Easy Example

Q. Find x if 4 : 6 :: 10 : x.

1. Product of extremes = product of means
2. 4x = 6 × 10 = 60
3. x = 15

Answer: 15

🇮🇳 Real-Life Example

Google Maps scale says 1 cm : 5 km. A 3 cm stretch on the map means 15 km of road — that is proportion doing the work.

📝 Exam-Level Example

Q. Find the mean proportional between 4 and 25.

1. Mean proportional = √(4 × 25)
2. = √100

Answer: 10

📝 Exam-Level Example

Q. Find the third proportional to 8 and 12.

1. 8 : 12 :: 12 : x
2. x = 12²/8 = 144/8

Answer: 18

🪄 Memory Trick

Third proportional → square the SECOND, divide by the first. Mean proportional → root of the product. Never mix the two.

⚠️ Common Mistakes

• ❌ Confusing third proportional (b²/a) with mean proportional (√ab)
• ❌ Cross-multiplying the wrong pair in a:b :: c:d
• ❌ Writing the fourth proportional as ac/b instead of bc/a

🏆 Exam Tips

• ✅ Label extremes and means before cross-multiplying — 5 seconds, zero errors
• ✅ Mean proportional answers are usually perfect squares under the root — expect clean numbers

📌 Summary

• ad = bc — the one rule of proportion
• Fourth proportional = bc/a
• Third proportional = b²/a
• Mean proportional = √(ab)