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.

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20 votes

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 $60^{\circ}$?

- $6: 22$ a.m.
- $6: 27$ a.m.
- $6: 38$ a.m.
- $6: 45$ a.m.

2

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.

3

37 votes

Best answer

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: $A$

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: $A$

1

4 votes

**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.)**