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Reflex Angle Between Hands

Introduction

Reflex-angle questions ask for the larger angle (>180°) formed by the hour and minute hands of a clock. Knowing how to find both the smaller and reflex angles is useful because many exam questions specifically ask for the reflex angle.

Pattern: Reflex Angle Between Hands

Pattern

Key idea: compute the smaller angle first, then subtract from 360° to get the reflex angle. Use the hand-position expressions: hour = 30H + 0.5m, minute = 6m. The raw difference is Δ = |30H - (11/2)·m|.

Then:

  • Smaller angle = min(Δ, 360 - Δ).
  • Reflex angle = max(Δ, 360 - Δ) = 360 - (smaller angle).

Always compute Δ first and then decide which of Δ or 360 - Δ is > 180° (that one is the reflex angle).

Step-by-Step Example

Question

Find the reflex angle between the hour and minute hands at 5:20.

Solution

  1. Step 1: Write the raw difference formula

    Δ = |30H - (11/2)·m|

  2. Step 2: Substitute H = 5 and m = 20

    Δ = |30×5 - (11/2)×20| = |150 - 110| = 40°.

  3. Step 3: Find the reflex angle

    Reflex angle = 360 - (smaller angle). Here smaller angle = 40°, so reflex = 360 - 40 = 320°.

  4. Final Answer:

    320°

  5. Quick Check:

    Hour position = 30×5 + 0.5×20 = 150 + 10 = 160°. Minute position = 6×20 = 120°. Raw difference = |160 - 120| = 40° → smaller = 40°, reflex = 360 - 40 = 320° ✅

Quick Variations

1. If Δ > 180° then Δ is already the reflex angle (no need to subtract).

2. For times close to the hour-mark (e.g., H:00), check whether Δ or 360 - Δ exceeds 180° before answering.

3. Use the same method for any given angle question: compute Δ first, then choose smaller/reflex as required.

Trick to Always Use

  • Step 1 → Compute Δ = |30H - 11m/2| exactly (don’t round early).
  • Step 2 → If Δ ≤ 180, reflex = 360 - Δ; if Δ > 180, reflex = Δ.

Summary

Summary

  • Compute raw difference Δ = |30H - (11/2)m| first.
  • Smaller angle = min(Δ, 360 - Δ); Reflex angle = 360 - (smaller angle).
  • Always verify by calculating hour = 30H + 0.5m and minute = 6m and checking the numerical difference.

Example to remember:
At 5:20 → smaller = 40°, reflex = 320°.

Practice

(1/5)
1. Find the reflex angle between the hour and minute hands at 4:40.
easy
A. 260°
B. 220°
C. 240°
D. 140°

Solution

  1. Step 1: Write the raw-difference formula

    Δ = |30H - (11/2)·m|.

  2. Step 2: Substitute H = 4, m = 40

    Δ = |30×4 - (11/2)×40| = |120 - 220| = 100°.

  3. Step 3: Compute reflex angle

    Reflex angle = 360 - Δ = 360 - 100 = 260°.

  4. Final Answer:

    260° → Option A

  5. Quick Check:

    Smaller = 100°, reflex = 360 - 100 = 260° ✅
Hint: Find Δ first, then reflex = 360 - Δ when Δ ≤ 180°.
Common Mistakes: Confusing the smaller and reflex angles; forgetting to subtract from 360.
2. Find the reflex angle between the hour and minute hands at 2:30.
easy
A. 105°
B. 255°
C. 210°
D. 195°

Solution

  1. Step 1: Use Δ = |30H - (11/2)·m|

  2. Step 2: Substitute H = 2, m = 30

    Δ = |30×2 - (11/2)×30| = |60 - 165| = 105°.

  3. Step 3: Reflex angle

    Reflex = 360 - 105 = 255°.

  4. Final Answer:

    255° → Option B

  5. Quick Check:

    Smaller = 105°, reflex = 360 - 105 = 255° ✅
Hint: If Δ ≤ 180°, reflex = 360 - Δ.
Common Mistakes: Taking 105° as reflex instead of the smaller angle.
3. Find the reflex angle between the hour and minute hands at 9:10.
easy
A. 155°
B. 295°
C. 215°
D. 65°

Solution

  1. Step 1: Compute Δ

    Δ = |30H - (11/2)·m|.

  2. Step 2: Substitute H = 9, m = 10

    Δ = |30×9 - (11/2)×10| = |270 - 55| = 215°.

  3. Step 3: Identify reflex

    Since Δ = 215° (>180°), Δ itself is the reflex (larger) angle → 215°.

  4. Final Answer:

    215° → Option C

  5. Quick Check:

    Δ > 180°, so reflex = Δ = 215° ✅
Hint: If Δ > 180°, take Δ as the reflex angle directly.
Common Mistakes: Subtracting Δ from 360 when Δ is already >180° (gives the smaller angle).
4. Find the reflex angle between the hour and minute hands at 6:50.
medium
A. 65°
B. 265°
C. 255°
D. 275°

Solution

  1. Step 1: Compute raw difference

    Δ = |30H - (11/2)·m|.

  2. Step 2: Substitute H = 6, m = 50

    Δ = |30×6 - (11/2)×50| = |180 - 275| = 95°.

  3. Step 3: Reflex angle

    Reflex = 360 - 95 = 265°.

  4. Final Answer:

    265° → Option B

  5. Quick Check:

    Smaller = 95°, reflex = 360 - 95 = 265° ✅
Hint: Compute Δ first; when Δ ≤ 180°, reflex = 360 - Δ.
Common Mistakes: Mixing up smaller and reflex angles or arithmetic errors in Δ.
5. Find the reflex angle between the hour and minute hands at 10:25.
medium
A. 197.5°
B. 230°
C. 250°
D. 270°

Solution

  1. Step 1: Compute raw difference

    Δ = |30H - (11/2)·m|.

  2. Step 2: Substitute H = 10, m = 25

    Δ = |30×10 - (11/2)×25| = |300 - 137.5| = 162.5°.

  3. Step 3: Reflex angle

    Reflex = 360 - 162.5 = 197.5°.

  4. Final Answer:

    197.5° → Option A

  5. Quick Check:

    Smaller ≈162.5°, reflex ≈197.5° ✅
Hint: Subtract the smaller angle from 360° to get the reflex angle when Δ ≤ 180°.
Common Mistakes: Rounding too early or taking the smaller angle as reflex.

Mock Test

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