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Straight Line / Coinciding Hands

Introduction

Straight-line clock questions ask when the hour and minute hands are exactly together (coinciding) or exactly opposite (180° apart). These are core clock patterns - knowing them makes many other angle questions (like overlaps, opposites, and frequency counts) straightforward.

Pattern: Straight Line / Coinciding Hands

Pattern

Key formulas - derived from positions of hour and minute hands (hour = 30H + 0.5m; minute = 6m):

  • Hands coincide (0°): solve 30H - (11/2)m = 0 → m = (60/11)·H
  • Hands opposite (180°): solve |30H - (11/2)m| = 180 → m = (60/11)·(H ± 6)

Only minute values with 0 ≤ m < 60 are valid for the given hour. If m is negative or ≥ 60, the event lies in an adjacent hour.

Step-by-Step Example

Question

When are the hands of a clock coincident between 5 and 6 o’clock? Also find when they are opposite between 5 and 6.

Solution

  1. Step 1: Use the coincidence formula

    For coincidence: m = (60/11)·H. Put H = 5.

  2. Step 2: Compute m for coincidence

    m = (60/11) × 5 = 300/11 = 27 3/11 minutes. So the hands coincide at 5:27 3/11.

  3. Step 3: Use the opposite-hand formula

    For opposite: m = (60/11)·(H ± 6). Put H = 5.

  4. Step 4: Compute m for opposite positions

    m₁ = (60/11)·(5 + 6) = (60/11)·11 = 60 → corresponds to 6:00 (edge of interval).
    m₂ = (60/11)·(5 - 6) = (60/11)·(-1) = -60/11 = -5 5/11 (falls in previous hour). Therefore, there is no strictly interior 5-6 time where hands are opposite; one is at 6:00 exactly and the other falls outside 5-6.

  5. Final Answer:

    Coincide → 5:27 3/11.
    Opposite → 6:00 (edge) - no interior opposite time in 5-6.

  6. Quick Check:

    Hour position at 5:27 3/11 = 30×5 + 0.5×(300/11) = 150 + 150/11 = 163 7/11°. Minute position = 6×(300/11) = 1800/11 = 163 7/11° → same → coincide ✅

Quick Variations

1. To find the next coincidence after H, increase H by 1 and compute m = (60/11)·(H+1) - coincidences occur roughly every 65 5/11 minutes. 2. For opposites, if (H + 6) yields m = 60 exactly, the opposite falls at the next hour (e.g., some opposites occur exactly on the hour). 3. If a computed m is negative, the event belongs to the previous hour; if ≥60 it belongs to the next hour.

Trick to Always Use

  • Step 1 → For coincidence use m = (60/11)·H. For opposite use m = (60/11)·(H ± 6).
  • Step 2 → Immediately check 0 ≤ m < 60. If not, move to adjacent hour.
  • Step 3 → Verify by computing hour-position = 30H + 0.5m and minute-position = 6m and comparing.

Summary

Summary

  • Use m = (60/11)·H to find when hands coincide (0° apart) for hour H.
  • Use m = (60/11)·(H ± 6) to find opposite positions (180° apart).
  • Accept only minute values between 0 and 60; negative or ≥60 results lie in adjacent hours.
  • Hands coincide 11 times in 12 hours (roughly every 65 5/11 minutes); opposites occur 11 times in 12 hours as well.

Example to remember:
Coincide at 5:27 3/11; opposite at 6:00 (edge of interval).

Practice

(1/5)
1. When do the hour and minute hands coincide between 3 and 4 o’clock?
easy
A. 3:16 4/11
B. 3:21 9/11
C. 3:27 3/11
D. 3:10 10/11

Solution

  1. Step 1: Use the coincidence formula

    For coincidence use m = (60/11)·H, where H is the hour.

  2. Step 2: Substitute H = 3

    m = (60/11) × 3 = 180/11 = 16 4/11 minutes.

  3. Step 3: Write the time

    So the hands coincide at 3:16 4/11.

  4. Final Answer:

    3:16 4/11 → Option A

  5. Quick Check:

    Hour pos = 30×3 + 0.5×(16 4/11) ≈ 90 + 8.182 = 98.182°; minute = 6×16.364 ≈ 98.182° → coincide ✅
Hint: Use m = (60/11)·H to find coincidence quickly.
Common Mistakes: Forgetting to convert the fractional minutes (elevenths) correctly.
2. When are the hands opposite (180° apart) between 1 and 2 o’clock?
easy
A. 1:21 9/11
B. 1:38 2/11
C. 1:27 3/11
D. 1:10 10/11

Solution

  1. Step 1: Use opposite-hand formula

    For opposite positions use m = (60/11)·(H ± 6).

  2. Step 2: Substitute H = 1

    m₁ = (60/11)(1 + 6) = 420/11 = 38 2/11 minutes; m₂ = (60/11)(1 - 6) = (60/11)(-5) (negative, discard for 1-2).

  3. Step 3: Choose valid time

    Valid opposite time in 1-2 is 1:38 2/11.

  4. Final Answer:

    1:38 2/11 → Option B

  5. Quick Check:

    Hour pos ≈ 30 + 0.5×38.182 = 30 + 19.091 = 49.091°; minute ≈ 229.091°; difference = 180° ✅
Hint: Compute (H + 6) first for opposites; check the (H - 6) root for previous hour.
Common Mistakes: Accepting negative m as valid inside the same hour.
3. At what time after 12:00 will the hands next coincide?
easy
A. 12:05 5/11
B. 12:10 10/11
C. 1:05 5/11
D. 12:16 4/11

Solution

  1. Step 1: Use coincidence formula

    m = (60/11)·H gives coincidence times for hour H.

  2. Step 2: Find next hour's coincidence

    For H = 1, m = (60/11) × 1 = 60/11 = 5 5/11 minutes → time = 1:05 5/11.

  3. Step 3: Interpret

    After 12:00 the next coincidence happens when H = 1 at 1:05 5/11.

  4. Final Answer:

    1:05 5/11 → Option C

  5. Quick Check:

    Hour pos ≈ 30 + 0.5×5.455 = 30 + 2.727 = 32.727°; minute = 6×5.455 = 32.727° → coincide ✅
Hint: The first coincidence after 12:00 is at 1:05 5/11 (use H=1).
Common Mistakes: Looking for a coincidence inside the 12th hour instead of the next hour.
4. How many times do the hands coincide in a 12-hour period?
medium
A. 11
B. 12
C. 10
D. 13

Solution

  1. Step 1: Understand spacing of coincidences

    Hands coincide roughly every 65 5/11 minutes, not exactly once each hour.

  2. Step 2: Count within 12 hours

    Because coincidences repeat every ≈65.454 minutes, they occur 11 times in 12 hours (the 12th-hour overlap falls at the next cycle boundary).

  3. Step 3: Conclude

    Total coincidences in 12 hours = 11.

  4. Final Answer:

    11 → Option A

  5. Quick Check:

    Check early sequence: 12:00, ~1:05 5/11, ~2:10 10/11 … after 11 occurrences the next is just past 12 hours ✅
Hint: Remember: coincidences occur 11 times per 12 hours (not 12).
Common Mistakes: Assuming one coincidence per hour (gives 12 incorrectly).
5. Between 7 and 8 o’clock, when are the hands exactly opposite (180° apart)?
medium
A. 7:21 9/11
B. 7:27 3/11
C. 7:38 2/11
D. 7:05 5/11

Solution

  1. Step 1: Use opposite-hand formula

    m = (60/11)·(H ± 6).

  2. Step 2: Substitute H = 7

    m₁ = (60/11)(7 - 6) = (60/11)×1 = 5 5/11 minutes. m₂ = (60/11)(7 + 6) = (60/11)×13 = 780/11 = 70 10/11 (invalid for 7-8).

  3. Step 3: Choose valid time

    The valid opposite time in 7-8 is 7:05 5/11.

  4. Final Answer:

    7:05 5/11 → Option D

  5. Quick Check:

    Hour pos ≈ 210 + 0.5×5.455 = 210 + 2.727 = 212.727°; minute ≈ 32.727°; difference = 180° ✅
Hint: For opposites, try (H - 6) first; it often gives the earlier valid time in the hour.
Common Mistakes: Accepting >60-minute root as valid for the same hour.

Mock Test

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