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Given Angle Between Hands

Introduction

In many clock problems, instead of asking for the time difference or overlap, the question gives an angle such as 45°, 90°, or 135° and asks at what time the hour and minute hands form that angle. Understanding this pattern is crucial because it combines relative speed and angular distance concepts of clock hands.

Pattern: Given Angle Between Hands

Pattern

Key formula: To find the time when the hands make an angle θ (where θ can be 0°, 45°, 90°, 135°, or 180°):

Angle = |30H - (11/2)M|

Rearranging for M (minutes past H o’clock):

  • M = (60/11) × (H ± θ/30)

The “±” sign gives two possible times - one when the hands are apart by θ and another when they are on the opposite sides.

Step-by-Step Example

Question

Find the times between 2 and 3 o’clock when the hands are 45° apart.

Solution

  1. Step 1: Write the formula

    M = (60/11) × (H ± θ/30)

  2. Step 2: Substitute H = 2, θ = 45°

    M = (60/11) × (2 ± 45/30) = (60/11) × (2 ± 1.5)

  3. Step 3: Find both possible values

    For + → (60/11) × 3.5 = 210/11 = 19 1/11 minutes For - → (60/11) × 0.5 = 30/11 = 2 8/11 minutes

  4. Step 4: Write both times

    The hands are 45° apart at 2:02 8/11 and 2:19 1/11.

  5. Final Answer:

    2:02 8/11 and 2:19 1/11

  6. Quick Check:

    Substituting back gives |30×2 - (11/2)×(2.73)| ≈ 45° ✅

Quick Variations

1. Use θ = 90° for right angles and θ = 180° for opposite directions.

2. For acute and obtuse angles, compute both “+” and “-” results separately.

3. If M ≥ 60, subtract 60 and move to the next hour.

Trick to Always Use

  • Step 1 → Use M = (60/11) × (H ± θ/30)
  • Step 2 → Evaluate both + and - to get two valid times.
  • Step 3 → If M > 60, convert to the next hour by subtracting 60.

Summary

Summary

  • Formula: M = (60/11) × (H ± θ/30)
  • Two times occur for each given angle (except at special boundaries).
  • Always check for M within 0-60 range for a valid time.
  • Reflex or larger angles can also be found by using (360 - θ).

Example to remember:
At 2 o’clock, hands are 45° apart at 2:02 8/11 and 2:19 1/11.

Practice

(1/5)
1. Find the times between 3 and 4 o’clock when the hands are 45° apart.
easy
A. 3:08 2/11 and 3:24 6/11
B. 3:10 10/11 and 3:49 1/11
C. 3:12 2/11 and 3:48 2/11
D. 3:15 and 3:45

Solution

  1. Step 1: Write formula

    M = (60/11) × (H ± θ/30) where θ = 45°.

  2. Step 2: Substitute H = 3, θ = 45°

    M = (60/11) × (3 ± 1.5) = (60/11) × {4.5, 1.5}.

  3. Step 3: Calculate minutes

    For 4.5 → m = 270/11 = 24 6/11 minutes → 3:24 6/11.
    For 1.5 → m = 90/11 = 8 2/11 minutes → 3:08 2/11.

  4. Final Answer:

    3:08 2/11 and 3:24 6/11 → Option A

  5. Quick Check:

    Substitute into |30H - 11m/2| gives ≈45° for both times ✅
Hint: Use H ± θ/30 inside (60/11) to get both roots.
Common Mistakes: Forgetting the smaller root (use ± both ways).
2. Find the time between 4 and 5 o’clock when the hands are 135° apart.
easy
A. 4:14 7/11
B. 4:46 4/11
C. 4:47 3/11
D. 4:13 1/11

Solution

  1. Step 1: Use formula

    M = (60/11)(H ± θ/30) with θ = 135° → θ/30 = 4.5.

  2. Step 2: Substitute H = 4

    M = (60/11)(4 ± 4.5) → (60/11)×8.5 = 510/11 = 46 4/11 minutes; the other root is negative (discard).

  3. Final Answer:

    4:46 4/11 → Option B

  4. Quick Check:

    |30×4 - (11/2)×46.3636| ≈ 135° ✅
Hint: If one root is negative, the valid time lies in the other root (or adjacent hour).
Common Mistakes: Including negative minute roots as times within the same hour.
3. Find the times between 6 and 7 o’clock when the hands are 90° apart.
easy
A. 6:16 4/11 and 6:49 1/11
B. 6:14 6/11 and 6:47 3/11
C. 6:20 and 6:50
D. 6:10 10/11 and 6:49 1/11

Solution

  1. Step 1: Formula

    M = (60/11)(H ± θ/30) with θ = 90° → θ/30 = 3.

  2. Step 2: Substitute H = 6

    M = (60/11)(6 ± 3) → (60/11)×9 = 540/11 = 49 1/11, and (60/11)×3 = 180/11 = 16 4/11.

  3. Final Answer:

    6:16 4/11 and 6:49 1/11 → Option A

  4. Quick Check:

    Both times give |30H - 11m/2| ≈ 90° ✅
Hint: For right angles use ±3 inside the (60/11) factor.
Common Mistakes: Dropping one of the ± roots.
4. Find the time between 8 and 9 o’clock when the hands are 135° apart.
medium
A. 8:19 1/11
B. 8:14 7/11
C. 8:47 3/11
D. 8:38 2/11

Solution

  1. Step 1: Formula

    M = (60/11)(H ± θ/30) with θ = 135° → θ/30 = 4.5.

  2. Step 2: Substitute H = 8

    m = (60/11)(8 - 4.5) = (60/11)×3.5 = 210/11 = 19 1/11 minutes. The (H + 4.5) root gives >60 (belongs to next hour).

  3. Final Answer:

    8:19 1/11 → Option A

  4. Quick Check:

    |30×8 - (11/2)×19.0909| ≈ 135° ✅
Hint: If one root exceeds 60, the valid time in the hour is the other root.
Common Mistakes: Not shifting >60 roots to the next hour when interpreting answers.
5. Find the times between 9 and 10 o’clock when the hands are 45° apart.
medium
A. 9:08 2/11 and 9:41 9/11
B. 9:10 10/11 and 9:49 1/11
C. 9:12 2/11 and 9:47 3/11
D. 9:40 10/11 and 9:57 3/11

Solution

  1. Step 1: Use formula

    M = (60/11)(H ± θ/30) with θ = 45° → θ/30 = 1.5.

  2. Step 2: Substitute H = 9

    m = (60/11)(9 ± 1.5) → (60/11)×10.5 = 630/11 = 57 3/11, and (60/11)×7.5 = 450/11 = 40 10/11 minutes.

  3. Final Answer:

    9:40 10/11 and 9:57 3/11 → Option D

  4. Quick Check:

    Both values satisfy |30H - 11m/2| ≈ 45° ✅
Hint: Evaluate both ± roots; normalize if one is near the hour boundary.
Common Mistakes: Swapping minute roots or using wrong hour when m > 60.

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