What is the acute angle (in degrees) between the hour hand and the minute hand of a standard analog clock at $3:40$?

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### 1. Strategy & Theoretical Principles The angle $\theta$ between the hour hand and minute hand at $H$ hours and $M$ minutes is given by the formula: $$\theta = |30 H - 5.5 M|$$ where $30^\circ$ is the speed of the hour hand per hour and $5.5^\circ$ is the relative speed between the hands per minute. ### 2. Analytical Formulation & Calculation For $H = 3$ and $M = 40$: $$\theta = |30(3) - 5.5(40)| = |90 - 220| = |-130| = 130^\circ$$ Since $130^\circ \le 180^\circ$, the acute/minor angle between the hands is $130^\circ$. ### 3. Distractor Forensics - Candidate taking reflex angle gets $360 - 130 = 230^\circ$. - Candidate ignoring hour hand movement from the 3 mark gets $|30(3) - 30(8)| = 150^\circ$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Remember the hour hand advances by $0.5^\circ$ every minute: at 40 minutes, it has moved $20^\circ$ past the 3 o'clock mark. - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*.

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