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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 $60^{\circ}$?

1. $6: 22$ a.m.
2. $6: 27$ a.m.
3. $6: 38$ a.m.
4. $6: 45$ a.m.

Use: $0.182(30h\pm A)$

$0.182(30*6 \pm 60°)$

=> $0.182(120 \ or \ 240)$

=> $21.84 \ or \ 43.68$

So, 6 hours and 21.84 minutes; or 6 hours and 43.68 minutes

=> 6:22 or 6:44

Option A matches the closest.

Some basic points related to the clock:

• Hour hand makes an angle of $30^{\circ}$ in one hour.
• The minute hand makes an angle of $30^{\circ}$ in five minutes.
• Hour hand covers a distance of $5$ minutes in one hour.
• $1$ minute contains $6^{\circ}.$
• In each hour, the minute hand covers $55$ minutes distance more than the hour hand.
• When both hand make the angle od $90^{\circ}$, then both lie $15$ minutes from each other.

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 $12\;\text{hrs}$ 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.

Correct Answer: $A$
by

Nice Explanation !

Can You Plz explain why you have taken 6z-0.5x=120 ?

I am not able to get it ?

@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
This explanation reminds me of those relative distance between trains problems. If both trains are moving in the same direction, we subtract (speed x time) values in the manner given above.

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:

$\color{blue}{6x} + 60 = 180 + 0.5x$

$\implies5.5x = 120$

$\implies x = \frac{120}{5.5} = 21.8181 \approx 22.$

nice explanation @amarvashisth
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.

Correct Option: (A)

TECHNIQUE 1:-
For Hour Hand
1 hour = 5 places covered= 5 * (360 / 60) degree covered= 30 degree covered
=> 1 min = (30/60) degree covered = 0.5 degree covered

For Min. Hand
1 hour = 60 places covered= 60 * (360 / 60) degree covered = 360 degree covered
=> 1 min = (360 / 60) = 6 degree covered

Let, at x min. after 6:00 AM, the angle between hour and min hand be 120 degree.

At x min after 6
Angle covered by hour hand = (x * 0.5) degree
Angle covered by min hand = (x * 6) degree

Therefore, a/q
{(0.5 * x) degree + 180 degree} - {(6 * x) degree} = 60 degree ,(180 degree is added because initial angle between hour and min hand is 180 degree)
On solving the above equation, we get x = 21.81 min = 22 min(approx.)

TECHNIQUE 2:-
For Hour Hand
1 hour = 5 places covered= 5 * (360 / 60) degree covered= 30 degree covered
=> 1 min = (30/60) degree covered = 0.5 degree covered

For Min. Hand
1 hour = 60 places covered= 60 * (360 / 60) degree covered = 360 degree covered
=> 1 min = (360 / 60) = 6 degree covered

Therefore, at every 1 min, the difference between the angles covered by hour hand and min. hand is (6 - 0.5) degree = 5.5 degree
In other words, the min. hand and hour hand move in a manner such that 5.5 degree angular difference is covered every min.

Now, a/q
Initially, angle between hour and minute hand = 180 degree (At 6:00 AM, hour and min. hand are diametrically opposite)

Finally, say x min. after 6:00 AM, angle between hour and min hand = 60 degree

Therefore, angular difference covered in this interval = (180 - 60) degree = 120 degree

Also, angular difference covered in x min if 5.5 degree is covered every min is given by (5.5 * x) degree

Therefore, (5.5. * x) degree = 120 degree
=> x = 21.81 in = 22 min (approx.)

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