At 3.40, the hour hand and the minute hand of a clock form an angle of

### Concept & Angle Between Clock Hands The angle $\theta$ between the hour and minute hands at $H$ hours and $M$ minutes is given by the standard clock angle formula: $$\theta = \left| 30H - 5.5M \right|$$ This formula works perfectly by calculating the absolute difference between the positions of the two hands from the 12 o'clock mark. ### Step-by-Step Solution 1. **Given:** Time is 3:40. Here, $H = 3$ and $M = 40$. 2. Apply the formula: $$\theta = \left| 30(3) - 5.5(40) \right|$$ 3. Calculate the hour term: $30 \times 3 = 90°$ 4. Calculate the minute term: $5.5 \times 40 = \frac{11}{2} \times 40 = 11 \times 20 = 220°$ 5. Find the absolute difference: $$\theta = |90° - 220°| = |-130°| = 130°$$ ### Exam Strategy & Shortcut If you prefer visual logic over formulas: At 40 minutes, the minute hand is exactly on the 8 ($8 \times 30° = 240°$). At 3:40, the hour hand has moved past the 3 by $\frac{40}{60}$ (or $\frac{2}{3}$) of the way to 4. Position of 3 is $90°$. $\frac{2}{3}$ of $30° = 20°$. Hour hand position = $90° + 20° = 110°$. Angle = $|240° - 110°| = 130°$. ### Common Pitfall Making arithmetic errors when multiplying 5.5 by the minutes. Always convert 5.5 to $11/2$ to simplify the multiplication ($11/2 \times 40 = 11 \times 20 = 220$). ### Final Answer Therefore, the correct answer is **130°**.