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Hands Opposite Each Other

Introduction

Problems asking when the hour and minute hands are exactly opposite (180° apart) are common in reasoning tests. This pattern is important because opposite positions link directly to straight-line concepts and help count occurrences and solve timing puzzles.

Pattern: Hands Opposite Each Other

Pattern

The hands are opposite when their angle difference is 180°. Using hour = 30H + 0.5m and minute = 6m, set: |(30 × H) - (11/2 × m)| = 180.

Solving gives two linear forms and the minute formulas:

  • (11/2)·m = 30H + 180 → m = (60/11)·(H + 6)
  • (11/2)·m = 30H - 180 → m = (60/11)·(H - 6)

Accept only minute values with 0 ≤ m < 60 for the specified hour. Negative or ≥60 results lie in adjacent hours and must be interpreted accordingly.

Step-by-Step Example

Question

When are the hands opposite each other between 1 and 2 o’clock?

Solution

  1. Step 1: Write the opposite condition

    Use |30H - (11/2)m| = 180.

  2. Step 2: Plug H = 1 into both linear forms

    (a) (11/2)m = 30×1 + 180 = 210 → m = 420/11 = 38 2/11 minutes.
    (b) (11/2)m = 30×1 - 180 = -150 → m = -300/11 = -27 3/11 minutes (discard for 1-2).

  3. Step 3: Interpret valid root within the hour

    The only valid time between 1:00 and 2:00 is 1:38 2/11. The negative root refers to a time in the previous hour.

  4. Final Answer:

    1:38 2/11

  5. Quick Check:

    Hour position = 30×1 + 0.5×38.181... ≈ 30 + 19.091 = 49.091°. Minute = 6×38.181... ≈ 229.091°. Difference = 180° ✅

Quick Variations

1. For hour H, compute m = (60/11)(H ± 6) and accept values in 0-59.999… for that hour.

2. If m = 60 exactly for (H + 6), the opposite occurs exactly at the next hour (e.g., some cases give 6:00 or 12:00).

3. Negative m from (H - 6) means the opposite happened in the previous hour.

Trick to Always Use

  • Step 1: Use m = (60/11)(H + 6) and m = (60/11)(H - 6).
  • Step 2: Keep only 0 ≤ m < 60 for the hour; interpret negative or ≥60 as adjacent-hour events.
  • Step 3: Verify by computing hour = 30H + 0.5m and minute = 6m and checking the 180° difference.

Summary

Summary

  • Opposite condition: |30H - (11/2)m| = 180.
  • Minute formulas: m = (60/11)(H ± 6); accept only 0 ≤ m < 60 for that hour.
  • Opposites occur 11 times in 12 hours (same frequency as coincidences).
  • Always verify by substitution - negative or ≥60 roots indicate times in adjacent hours.

Example to remember:
Between 1 and 2 → 1:38 2/11 (opposite)

Practice

(1/5)
1. When are the hour and minute hands opposite each other between 2 and 3 o’clock?
easy
A. 2:43 7/11
B. 2:21 9/11
C. 2:27 3/11
D. 2:49 1/11

Solution

  1. Step 1: Use opposite condition

    The hands are opposite when |30H - (11/2)m| = 180. This gives m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 2

    Compute m = (60/11)(2 + 6) = (60/11)×8 = 480/11 = 43 7/11 minutes. The other root (H - 6) is negative and discarded for 2-3.

  3. Step 3: Write the time

    The hands are opposite at 2:43 7/11.

  4. Final Answer:

    2:43 7/11 → Option A

  5. Quick Check:

    Hour ≈ 60 + 0.5×43.636 = 81.818°; minute ≈ 6×43.636 = 261.818°; difference = 180° ✅
Hint: Use m = (60/11)(H + 6) for the later opposite in the hour; discard negative roots.
Common Mistakes: Accepting negative minute values inside the same hour.
2. When are the hands opposite each other between 4 and 5 o’clock?
easy
A. 4:54 6/11
B. 4:49 1/11
C. 4:38 2/11
D. 4:21 9/11

Solution

  1. Step 1: Use the formula

    m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 4

    m = (60/11)(4 + 6) = (60/11)×10 = 600/11 = 54 6/11 minutes. The (H - 6) root is negative and discarded for 4-5.

  3. Step 3: Write the time

    The hands are opposite at 4:54 6/11.

  4. Final Answer:

    4:54 6/11 → Option A

  5. Quick Check:

    Hour ≈ 120 + 0.5×54.545 = 147.273°; minute ≈ 327.273°; difference mod 360 = 180° ✅
Hint: Compute (H + 6) first for opposites; if result ≤ 60 it is the later opposite in that hour.
Common Mistakes: Confusing which algebraic root maps to the current hour.
3. At what time between 9 and 10 o’clock will the hands be opposite each other?
easy
A. 9:27 3/11
B. 9:16 4/11
C. 9:43 7/11
D. 9:32 8/11

Solution

  1. Step 1: Use opposite formula

    m = (60/11)(H ± 6).

  2. Step 2: Substitute H = 9

    m = (60/11)(9 - 6) = (60/11)×3 = 180/11 = 16 4/11 minutes. The (H + 6) root exceeds 60 and is invalid for 9-10.

  3. Step 3: Write the time

    The hands are opposite at 9:16 4/11.

  4. Final Answer:

    9:16 4/11 → Option B

  5. Quick Check:

    Hour ≈ 270 + 0.5×16.364 = 278.182°; minute ≈ 98.182°; difference = 180° ✅
Hint: If (H - 6) gives a small positive value, that gives the earlier opposite inside the hour.
Common Mistakes: Not checking which root stays within 0-60 minutes.
4. When are the hands opposite each other between 11 and 12 o’clock?
medium
A. 11:21 9/11
B. 11:38 2/11
C. 11:27 3/11
D. 11:05 5/11

Solution

  1. Step 1: Use m = (60/11)(H ± 6)

  2. Step 2: Substitute H = 11

    m = (60/11)(11 - 6) = (60/11)×5 = 300/11 = 27 3/11 minutes. The (H + 6) root > 60 and is invalid for 11-12.

  3. Step 3: Write the time

    The hands are opposite at 11:27 3/11.

  4. Final Answer:

    11:27 3/11 → Option C

  5. Quick Check:

    Hour ≈ 330 + 0.5×27.273 = 343.636°; minute ≈ 163.636°; difference = 180° ✅
Hint: Try (H - 6) first for the earlier opposite time; verify 0 ≤ m < 60.
Common Mistakes: Selecting the root that lies outside the hour range.
5. How many times do the hands of a clock stand opposite each other in 24 hours?
medium
A. 11
B. 22
C. 24
D. 23

Solution

  1. Step 1: Opposites in 12 hours

    The hour and minute hands are opposite each other 11 times in 12 hours.

  2. Step 2: Extend to 24 hours

    Since the same cycle repeats twice in a full day (24 hours), total opposites = 11 × 2 = 22 times.

  3. Step 3: Verify logic

    Each 12-hour cycle (AM or PM) produces 11 opposite positions, not 12.

  4. Final Answer:

    22 → Option B

  5. Quick Check:

    11 opposites in 12 hours × 2 = 22 in 24 hours ✅
Hint: Opposites occur 11 times in 12 hours → multiply by 2 for 24 hours = 22 times.
Common Mistakes: Mistaking the count as 24 or 44 by assuming one per hour.

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