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At what time between $6$ a. m. and $7$ a. m. will the minute hand and hour hand of a clock make an angle closest to $60°$?

1. $6: 22$ a.m.
2. $6: 27$ a.m.
3. $6: 38$ a.m.
4. $6: 45$ a.m.
edited | 2k views

At $6$ a.m. the hour hand and minute hand are separated by $180$ degree. Now,

Speed of hour hand = $360$ degree/12 hour (clock is $12$ $hrs$ as am/pm is given) = $30$ degrees /hr = $0.5$ degree per minute

Speed of minute hand = $360$ degree per $60$ minutes = $6$ degrees per minute.

So, we want the relative distance between minute and hour hand to be $60$ degree as per question which would mean a relative distance traversal of $180-60 = 120$ degrees. This happens after $x$ minutes such that

$$6x - 0.5x = 120 \implies x = \frac{120}{5.5} = 21.81$$

So, closest time is $6:22$ a.m.
edited
+2
Nice Explanation !
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Can You Plz explain why you have taken 6z-0.5x=120 ?

I am not able to get it ?

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@Manis (and in case anyone else has the same doubt)

Distance = Speed * Time

Here, Distance = 120 degrees

You could say that the relative speed of the hour and minute hand is 6 - 0.5 = 5.5 degrees/minute

The time, which we need to calculate, is x minutes

So, we get 120 = (6 - 0.5) * x = 5.5x

hour hand moves 0.5° in 1 minute
minute hand moves 6° in 1 minute

in $x$ time minute and hour hand have moved. Now, from the figure it is deducible that:

\large \begin{align*} {\color{Blue} 6x} +60&= 180+{\color{Red} 0.5x}\\ 5.5x &= 120\\ x &= \frac{120}{5.5} \\ &= 21.8181\\ &\approx 22 \end{align*}

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nice explanation @amarvashisth

Minute hand rotates with the speed:   360°/(60min) = 6° per minute.

Hour Hand rotates with the speed:   360°/((24*60)mins) = 0.25° per minute.

At 6 o' clock, minute hand is vertical up & hour hand is vertical down.

Let us assume, minute hand is at 0°, this means hour hand is at 180°.

So, angle between these two hands after t minutes from 6 a.m is either (180° + 25t) - 6t or 6t - (180° + 25t)

Angle given in the question is 60°.

(180° + 25t) - 6t = 60

-> t = 120/5.75

-> t= 20.86min i.e. 6:20am

OR

6t - (180° + 25t) = 60°

-> t= 41.73min i.e 6:41am

6:22am is most closer to the actual answer .

Therefore, Option A is correct

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you have taken 24 hour clock.
there was also formula

2/11(30*x + A) or 2/11(30*x - A)

here x is starting time (here x=6)

A is angle

now put the value

2/11(30 * 6 + 60) or 2/11(30 * 6 - 60)

480/11 or 240/11

43.63 or 21.81

So  closest time is 6:22 a.m.
+1 vote
option a
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55 spaces in clock are covered in 60 min... So 20 spaces are covered in 20/55  *  20  = 21.81 = close to 22 min.. therefore answer is 22 min. past 6..

So option (A)

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