Exact angle between clock hands at 1:40 — Compute the smaller angle between the hour and minute hands at 1:40 am.

Introduction / Context:
Use the standard absolute-angle method and then select the smaller of the two possible angles (θ and 360° − θ). Many options deliberately list reflex angles or nearby round numbers to trap careless calculations.

Given Data / Assumptions:

• Time: 1:40.
• Hour-hand angle = 30 × H + 0.5 × M.
• Minute-hand angle = 6 × M.
• Smaller angle is requested.

Concept / Approach:
Compute both hand angles from 12 O’clock, take the absolute difference, and if the difference > 180°, replace it with 360° − difference to obtain the smaller angle.

Step-by-Step Solution:
Hour-hand angle: 30 × 1 + 0.5 × 40 = 30 + 20 = 50°.Minute-hand angle: 6 × 40 = 240°.Raw difference: |50 − 240| = 190°.Smaller angle: 360° − 190° = 170°.

Verification / Alternative check:
Visualize: at 1:40, the minute hand is past 7 (≈ 240°), and the hour hand is past 1 (≈ 50°); the obtuse smaller gap is clearly < 180°, around 170°.

Why Other Options Are Wrong:
180°, 190°, 175° do not match the computed 170°. Therefore, “None of these” is correct.

Common Pitfalls:
Reporting the absolute difference (190°) instead of the smaller angle, or rounding to a nearby offered distractor like 175°.

Final Answer:
None of these (correct smaller angle = 170°)