Question 45

# At what time between 9PM and 10PM minute hand and hour hand will be opposite to each other?

Solution

The hour hand and the minute hands will be opposite to each other when the angle between them= $$180^{\circ\ }$$.

At 9PM, angle between the two hands= 270$$^{\circ\ }$$

The minute hand covers 360$$^{\circ\ }$$ in 60 minutes. So, in 1 minute, it covers $$\ \frac{\ 360^{\circ\ }}{60}$$= 6$$^{\circ\ }$$

Similarly for the hour hand, it covers 360$$^{\circ\ }$$ in 12 hours or 720 minutes.

.'. The hour hand covers $$\ \frac{\ 360^{\circ\ }}{720}$$= 0.5$$^{\circ\ }$$.

Using the concept of relative speed, since both the hands move in the same directions, the effective speed= (6-0.5)$$^{\circ\ }$$ =5.5 or $$\ \frac{\ 11}{2}$$ degrees per minute.

Effective distance to cover= $$270^{\circ}-180^{\circ\ }=\ 90^{\circ\ }$$

Therefore time taken= $$\ \frac{\ 90}{\ \frac{\ 11}{2}}=\ \ \frac{\ 180}{11}=\ 16\ \frac{\ 4}{11}$$ minutes after 9 PM.

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