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Question Number 144025 by ZiYangLee last updated on 20/Jun/21

$$\mathrm{Between}12\mathrm{p}.\mathrm{m}.\mathrm{today}\mathrm{and}12\mathrm{p}.\mathrm{m}.$$$$\mathrm{tomorrow},\mathrm{how}\mathrm{many}\mathrm{times}\mathrm{do}\mathrm{the}$$$$\mathrm{hour}\mathrm{hand}\mathrm{and}\mathrm{the}\mathrm{minute}\mathrm{hand}\mathrm{on}\mathrm{a}$$$$\mathrm{clock}\mathrm{form}\mathrm{an}\mathrm{angle}\mathrm{of}120°?$$

Answered by mr W last updated on 21/Jun/21

$$attimeh:mthepositionofminute$$$$handism\times 6°,thepositionofhour$$$$handis(h+\frac{m}{60})\times 30°.$$$$(0⩽h⩽11,0⩽m<60)$$$$$$$$m\times 6=(h+\frac{m}{60})\times 30+120$$$$\Rightarrow 11m=60(h+4)$$$$0⩽m=\frac{60(h+4)}{11}<60$$$$\Rightarrow 0⩽h⩽6\Rightarrow 7times$$$$$$$$m\times 6=(h+\frac{m}{60})\times 30+240$$$$\Rightarrow 11m=60(h+8)$$$$0⩽\frac{60(h+8)}{11}<60$$$$\Rightarrow 0⩽h⩽2\Rightarrow 3times$$$$$$$$m\times 6=(h+\frac{m}{60})\times 30-120$$$$\Rightarrow 11m=60(h-4):$$$$0⩽\frac{60(h-4)}{11}<60$$$$\Rightarrow 4⩽h⩽11\Rightarrow 8times$$$$$$$$m\times 6=(h+\frac{m}{60})\times 30-240$$$$\Rightarrow 11m=60(h-8):$$$$0⩽\frac{60(h-8)}{11}<60$$$$\Rightarrow 8⩽h⩽11\Rightarrow 4times$$$$$$$$totally7+3+8+4=22timesin12hours$$$$or44timesinaday.$$

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