The smaller angle made by the hour and the minute hands is noted when it was 1:30 PM. In which of the following intervals of time will the smaller angle between these hands again be the same?
1:46 PM and 1:47 PM
At 1:30 PM: hour hand angle = \(30\times1+0.5\times30=45^\circ\); minute hand angle = \(6\times30=180^\circ\). Smaller angle between them = 135°.
Let t minutes pass after 1:30. Hour hand = \(45+0.5t\); minute hand = \(180+6t\). The raw angular gap grows as \(135+5.5t\), and once this exceeds 180°, the smaller angle is measured as \(360-(135+5.5t)\).
Setting this equal to 135° again: \(360-(135+5.5t)=135\Rightarrow5.5t=90\Rightarrow t\approx16.36\) minutes.
Adding this to 1:30 PM gives approximately 1:46.36 PM, which falls between 1:46 PM and 1:47 PM.
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