Hands Together, Opposite & at Right Angles

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title

Hands Together, Opposite & at Right Angles

  • Clock
  • Hands Together, Opposite & at Right Angles
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Learning Objective

Find when the hands coincide, oppose or form right angles, and count these events in 12/24 hours.

🎯 Learning Objective

Find when the hands coincide, oppose or form right angles, and count these events in 12/24 hours.

💡 Concept

  • Relative speed = 6 − 0.5 = 5.5° per minute — the minute hand keeps gaining on the hour hand
  • Hands coincide every 720/11 = 65 5/11 minutes
  • Between H and H+1 o'clock, the hands meet at M = 60H/11 minutes past H
  • In 12 hours: together 11 times, right angles 22 times, opposite 11 times (in 24 h: 22, 44, 22)
  • For any required angle A, solve |30H − 5.5M| = A

🧮 Key Formulas

Hands coincide every 720/11 = 65 5/11 min

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Meet between H and H+1 at M = 60H/11 min past H

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12 h: together 11, right angles 22, opposite 11

✏️ Easy Example

Q. At what time between 1 and 2 o'clock are the hands of a clock together?

  1. At 1:00 the gap is 30°
  2. Time = 30 ÷ 5.5 = 60/11
  3. = 5 5/11 min past 1

Answer: 5 5/11 minutes past 1

🇮🇳 Real-Life Example

The match starts at 2 pm sharp; the clock's hands overlap mid-over at 10 10/11 minutes past 2 — that is 60×2/11. Try the formula for any hour of play.

📝 Exam-Level Example

Q. At what time between 3 and 4 o'clock are the hands of a clock at right angles?

  1. |90 − 5.5M| = 90
  2. 5.5M = 180 → M = 360/11
  3. = 32 8/11 min past 3

Answer: 32 8/11 minutes past 3

📝 Exam-Level Example

Q. How many times do the hands of a clock form a right angle in 24 hours?

  1. Normally 2 per hour, but 2 moments merge in every 12 hours → 22
  2. 24 hours → 2 × 22 = 44

Answer: 44 times

🪄 Memory Trick

Answers in this chapter love elevenths — 5 5/11, 32 8/11, 65 5/11. In MCQs, the option carrying a /11 fraction is usually the right lane.

⚠️ Common Mistakes

  • ❌ Saying the hands meet 12 times in 12 hours — only 11, the 12 o'clock meeting is shared
  • ❌ Counting 24 right angles in 12 hours — two moments merge, so 22
  • ❌ Missing the second right-angle position within an hour

🏆 Exam Tips

  • ✅ Gap to cover ÷ 5.5 = minutes needed — works for any target angle
  • ✅ If the hands meet every 65 minutes instead of 65 5/11, the clock is running fast — the classic gain/loss question

📌 Summary

  • Relative speed 5.5°/min
  • Meet at 60H/11 min past H; every 65 5/11 min
  • 12 h: 11 together, 22 right angles, 11 opposite
  • Clock answers love /11 fractions
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