1. Introduction to Clocks

A clock is a circular structure divided into 12 equal parts, representing hours, with 360° around the face. Time and angles between clock hands are calculated using geometric principles.

2. Key Concepts

• Hour Hand: Moves 30° per hour or 0.5° per minute.
• Minute Hand: Moves 6° per minute.
• Angle Between Hands: The angle between the hour and minute hands is crucial in solving clock problems.

3. Formula to Calculate Angle Between Hands

The angle between the hour and minute hands can be calculated as:

$$\theta = |(30H - 5.5M)|$$

• H: Hour on the clock (1 to 12).
• M: Minutes past the hour (0 to 59).

Explanation:

• Hour hand moves \(30°\) per hour → \(30H\).
• Minute hand moves \(6°\) per minute, relative motion of hour hand → \(5.5M\).

The absolute value ensures the angle is always positive.

4. Types of Angles in Clock Problems

• Smaller Angle: Calculated using the formula directly.
• Larger Angle: Found by subtracting the smaller angle from \(360°\).

Example: If the angle between hands is \(120°\), the larger angle is \(360° - 120° = 240°\).

5. Example: Finding Angle Between Hands

Problem: Calculate the angle at 3:15.

Solution:

• \(H = 3\), \(M = 15\)
• Apply the formula:

$$\theta = |(30H - 5.5M)|$$

$$\theta = |(30 × 3 - 5.5 × 15)|$$

$$\theta = |(90 - 82.5)| = 7.5°$$

• Angle: 7.5°.

6. Finding Time From a Given Angle

To find the time when a specific angle is given:

• Use the formula to set the angle \(\theta\) and solve for \(M\) (minutes).
• Reorganize the formula: \(M = \frac{(30H \pm \theta)}{5.5}\).

Example: Find the time when the angle between hands is \(90°\) at 3:00.

Solution:

• Angle formula rearranged for \(M\):

$$M = \frac{(90 ± 90)}{5.5}$$

$$M = 16.36 ≈ 16 \text{ minutes}$$

• Time: 3:16.

7. Coincidence of Hands

Hands coincide (overlap) when their angle is \(0°\).

Formula:

$$M = \frac{60H}{11}$$

• At \(H = 1\): \(M = \frac{60 × 1}{11} ≈ 5.45\) (5 minutes and 27 seconds).

8. Hands Perpendicular to Each Other

The hands are perpendicular when the angle is \(90°\).

• Use the angle formula and set \(\theta = 90°\).
• Solve for \(M\) to find the time.

9. Example: Hands Coincide Between 2 and 3

Problem: Find when the hands overlap between 2 and 3.

• \(H = 2\).
• Apply coincidence formula:

$$M = \frac{60H}{11}$$

$$M = \frac{60 × 2}{11} ≈ 10.9 \text{ minutes}$$

• Time: 2:11.

10. Summary of Key Formulas

• Angle Between Hands: $$\theta = |(30H - 5.5M)|$$
• Time for Coincidence: $$M = \frac{60H}{11}$$
• Perpendicular Hands: Solve for \(M\) using \(\theta = 90°\).