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Opposite Directions

Opposite Directions problems ask for the times when the hour and minute hands point in exactly opposite directions (form a straight line but pointing away from each other, 180° apart). These problems are a special case of the specific angle problems with θ = 180°.

10Worksheets
200+Practice Questions
IntermediateDifficulty
2-3 hoursHours to Master

What You'll Learn

Introduction & Overview Why This Matters How to Solve Pro Tips & Tricks
Shortcut Methods Common Mistakes Practice Worksheets Exam Importance

Introduction to Opposite Directions

Opposite Directions problems ask for the times when the hour and minute hands point in exactly opposite directions (form a straight line but pointing away from each other, 180° apart). These problems are a special case of the specific angle problems with θ = 180°.

Prerequisites

Angle formula: |30H - 5.5M| = 180 Solving linear equations Understanding of 11 opposite positions in 12 hours Time formatting
Why This Matters: Opposite Directions problems appear in 1-2 questions in SSC CGL and Banking PO exams. They test understanding of collinear opposite positions.

How to Solve Opposite Directions Problems

1

Step 1: Set up the equation |30H - 5.5M| = 180

2

Step 2: Solve for M in terms of H: M = (30H ± 180)/5.5

3

Step 3: For each hour H (0 to 11, where 0=12), calculate both solutions

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Step 4: Keep only solutions where 0 ≤ M < 60

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Step 5: For H=0 (12:00), 30×0 - 180 = -180 → M negative (discard)

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Step 6: Format valid solutions as H:MM

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Step 7: Answer with the time(s) when hands are opposite

Pro Strategy: For opposite directions (180°), the formula gives only one valid solution per hour (the other solution is negative or ≥60). The hands are opposite 11 times in 12 hours (excluding the 6:00 position counted once).

Example Problem

Example: At what time between 4:00 and 5:00 are the hands opposite? Solution: Step 1: H = 4, |120 - 5.5M| = 180 Step 2: Case 1: 120 - 5.5M = 180 → -5.5M = 60 → M = -10.9 (invalid) Step 3: Case 2: 120 - 5.5M = -180 → -5.5M = -300 → M = 300/5.5 = 54.545 minutes Step 4: 4:54:33 Answer: Approximately 4:54:33

Pro Tips & Tricks

  • Opposite times: 12:32:44, 1:38:11, 2:43:38, 3:49:05, 4:54:33, 6:00, 7:05:27, 8:10:55, 9:16:22, 10:21:49, 11:27:16
  • Formula: M = (30H + 180)/5.5 for H = 0 to 10 (one valid solution each)
  • At H=6, M = (180 + 180)/5.5 = 360/5.5 = 65.45 (rolls to 7:05:27), but 6:00 is also opposite
  • Actually 6:00 is the only exact hour where hands are opposite
  • The hands are opposite 11 times in 12 hours (including 6:00)
  • Between H and H+1, there is exactly one opposite position (except between 5-6 and 6-7 where 6:00 is the boundary)

Shortcut Methods to Solve Faster

M = (30H + 180)/5.5 for H = 0 to 5 gives times before 6:00
M = (30H - 180)/5.5 for H = 6 to 11 gives times after 6:00
At 6:00, hands are exactly opposite
Between 5:00 and 7:00, the opposite time is 6:00

Common Mistakes to Avoid

Forgetting that 6:00 is an opposite position
Using H=11 gives M=60 which rolls to next hour
Confusing opposite (180°) with overlap (0°)
Not checking both ± cases for the equation

Exam Importance

Opposite Directions is an important topic for various competitive exams. Here's how frequently it appears:

SSC CGL
1-2 questions
BANKING PO
1-2 questions
RAILWAYS RRB
1-2 questions
CAT
0-1 questions
INSURANCE
1-2 questions

Ready to Master Opposite Directions?

Start with Worksheet 1 and work your way up to expert level! Each worksheet includes:

20 practice questions
Detailed solutions
Step-by-step explanations
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