The angle between the minute hand and the hour hand of a clock when the time is 8.30, is

### Concept & Angle Between Clock Hands The fastest way to determine the angle $\theta$ between the hands of a clock at any given time $H:M$ is to use the clock angle formula: $$\theta = \left| 30H - \frac{11}{2}M \right|$$ ### Step-by-Step Solution 1. **Given:** Time is 8:30. Therefore, $H = 8$ and $M = 30$. 2. Substitute the values into the formula: $$\theta = \left| 30(8) - 5.5(30) \right|$$ 3. Calculate the position of the hour hand without the minute hand offset: $$30 \times 8 = 240°$$ 4. Calculate the offset and minute hand position term: $$5.5 \times 30 = \frac{11}{2} \times 30 = 11 \times 15 = 165°$$ 5. Subtract the two values to find the absolute difference: $$\theta = |240° - 165°| = 75°$$ ### Exam Strategy & Shortcut Use logical deduction: At 8:30, the minute hand is perfectly on the 6. The hour hand is exactly halfway between the 8 and the 9. The space between 6 and 8 is 2 hours ($2 \times 30° = 60°$). The hour hand is halfway between 8 and 9, which adds another $15°$ ($30° / 2$). Total angle = $60° + 15° = 75°$. This visual method often avoids math errors. ### Common Pitfall A common mistake is assuming the hour hand is exactly on the 8 at 8:30, which leads to an incorrect angle calculation of $240° - 180° = 60°$. Always account for the hour hand moving forward as the minutes pass. ### Final Answer Therefore, the correct answer is **75°**.