At what angle the hands of a clock are inclined at 15 minutes past 5?

### Concept & Angle Between Clock Hands To calculate the exact angle of inclination between the hands at $H:M$, apply the fundamental formula: $$\theta = \left| 30H - \frac{11}{2}M \right|$$ ### Step-by-Step Solution 1. **Given:** The time is 5:15 ("15 minutes past 5"). $H = 5$, $M = 15$. 2. Substitute these values into the formula: $$\theta = \left| 30(5) - \frac{11}{2}(15) \right|$$ 3. Calculate the hour position relative to 12 o'clock: $30 \times 5 = 150^\circ$ 4. Calculate the minute displacement term: $5.5 \times 15 = 82.5^\circ$ 5. Find the absolute difference between the two positions: $$\theta = |150^\circ - 82.5^\circ| = 67.5^\circ$$ 6. Convert the decimal to a mixed fraction to match the options: $67.5^\circ = 67\frac{1}{2}^\circ$. ### Exam Strategy & Shortcut Mentally visualize the clock: The hour hand at 5 o'clock is at $150^\circ$. In 15 minutes, it moves forward by $15 \times 0.5^\circ = 7.5^\circ$. So it is at $157.5^\circ$. The minute hand at 15 is exactly on the 3, which is $90^\circ$. Subtract the two: $157.5^\circ - 90^\circ = 67.5^\circ$. ### Common Pitfall Students often struggle with multiplying by 5.5. Using the fraction $11/2$ makes calculation safer ($15 \times 11 = 165$, and $165 / 2 = 82.5$). ### Final Answer Therefore, the correct answer is **$67\frac{1}{2}^\circ$**.