Clock

Clocks-based problems in aptitude test your grasp of angles, time calculations, and relative motion of the hour and minute hands.

You’re typically asked to:

• Find angles between hands at a certain time.
• Find the time when hands meet/form a certain angle.
• Detect when hands are opposite or perpendicular.
• Solve mirror-image time problems.

Think of it as a circular relative-speed problem, where both hands move continuously but at different rates.

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2. Key Formulas & Shortcuts

A. Degrees Covered by Hands

• Minute hand: $6^\circ$ per minute
• Hour hand: $0.5^\circ$ per minute or $30^\circ$ per hour

B. Angle Between Two Hands

$$\theta = \left| 30H - \frac{11}{2}M \right|$$

Where $H$ = hour, $M$ = minutes

C. When Hands Overlap

Hands overlap at:

$$\text{Time} = \frac{H \times 60}{11} \text{ minutes past } H \text{ o'clock}$$

E.g., between 2 and 3 → $\frac{2 \times 60}{11} = 10\frac{10}{11} \text{ minutes past 2}$

D. When Hands are at Right Angle (90°)

They are at right angles twice in every hour:

$$\text{Time} = \frac{H \times 60 \pm 15}{11}$$

E. When Hands are Opposite (180°)

$$\text{Time} = \frac{H \times 60 \pm 30}{11}$$

F. Mirror Image Time

To find real time from mirror image:

$$\text{Real Time} = 12:00 - \text{Mirror Time}$$

E.g., Mirror shows 2:10 → Real time = 9:50

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3. Conceptual Tips & Common Mistakes

• Use absolute value for angle problems (since angle is non-negative).
• If a time like 2:50 is given, hour hand is not exactly at 2 — it has moved ahead!
• Hands overlap 11 times in 12 hours (not 12!).
• When converting minutes, always ensure decimal to fraction properly.
• Don’t forget that both hands move continuously, not stepwise.

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4. Visual Explanation

At 3:00 → angle between hands = 90°
At 6:00 → angle = 180°
At 12:00 → overlap
At 1:05 → hands are almost aligned again
This illustrates that relative motion creates periodic patterns.

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5. Solved Examples

Example 1: Angle Between Hands

Q: Find the angle between the hands at 2:30.

$$\theta = |30H - \frac{11}{2}M| = |30 \times 2 - \frac{11}{2} \times 30| = |60 - 165| = 105^\circ$$

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Example 2: When Hands Overlap

Q: At what time between 1 and 2 do the hands overlap?

$$\text{Time} = \frac{1 \times 60}{11} = \frac{60}{11} = 5 \frac{5}{11} \text{ minutes past 1}$$

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Example 3: Hands at Right Angle

Q: At what times between 3 and 4 are the hands at right angles?

$$\text{Time} = \frac{3 \times 60 \pm 15}{11} = \frac{180 \pm 15}{11} = \frac{165}{11}, \frac{195}{11} = 15, 17\frac{8}{11}$$

So at 3:15 and 3:17:49, hands are perpendicular.

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Example 4: Hands in Straight Line

Q: At what time between 5 and 6 are hands 180° apart?

$$\text{Time} = \frac{5 \times 60 \pm 30}{11} = \frac{300 \pm 30}{11} = \frac{270}{11}, \frac{330}{11} = 24\frac{6}{11}, 30$$

Ans: 5:24:33 and 5:30

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Example 5: Mirror Image

Q: Mirror shows 1:40. What is the actual time?

$$\text{Real time} = 12:00 - 1:40 = 10:20$$

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Example 6: Hour Hand Position

Q: Where is the hour hand at 3:20?

• Every hour = 30°
• Extra $\frac{20}{60} = \frac{1}{3}$ hr → $\frac{1}{3} \times 30 = 10^\circ$

Ans: At 3:20, hour hand is at $90 + 10 = 100^\circ$