Question
Find the angle between hour-hand and minute-hand in a clock at : quarter to six
✓
Answer
At 5:45, the minute-hand is at mark 9 and hour- hand has crossed $(\frac{3}{4})$th of the angle between 5 and 6.
Angle between two consecutive marks
$=360^{\circ} / 12=30^{\circ}$
Angle traced by hour-hand in 45 minutes
$\frac{3}{4}\left(30^{\circ}\right)=(22.5)^{\circ}=\left(22 \frac{1}{2}\right)^{\circ}$
Angle between marks 5 and 9
$=4 \times 30^{\circ}=120^{\circ}$
$\therefore$ Angle between two hands of the clock at quarter to
$\begin{aligned}
\operatorname{six} & =120^{\circ}-\left(22 \frac{1}{2}\right)^0 \\
& =\left(97 \frac{1}{2}\right)^{\circ} \\
& =97^{\circ}+\frac{1}{2}^{\circ} \\
& =97^{\circ} 30^{\prime}
\end{aligned}$
Angle between two consecutive marks
$=360^{\circ} / 12=30^{\circ}$
Angle traced by hour-hand in 45 minutes
$\frac{3}{4}\left(30^{\circ}\right)=(22.5)^{\circ}=\left(22 \frac{1}{2}\right)^{\circ}$
Angle between marks 5 and 9
$=4 \times 30^{\circ}=120^{\circ}$
$\therefore$ Angle between two hands of the clock at quarter to
$\begin{aligned}
\operatorname{six} & =120^{\circ}-\left(22 \frac{1}{2}\right)^0 \\
& =\left(97 \frac{1}{2}\right)^{\circ} \\
& =97^{\circ}+\frac{1}{2}^{\circ} \\
& =97^{\circ} 30^{\prime}
\end{aligned}$
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