CUET/General Test•9mos
yoloppolo

Ye CUET ka question hai, can someone provide a good explaination?

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Comments (4)

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basicallybhavesh
IIM Ranchi (IR 2)
MOD•9mos

Use relative speed (like in Time–Speed problems).

  • Minute hand speed = 360° in 60 min = 6°/min
  • Hour hand speed = 360° in 12 hr = 360° in 720 min = 0.5°/min
  • So, relative speed (minute w.r.t. hour) = 6 − 0.5 = 5.5°/min

Let t = minutes after 12:00.

Then the angle “gained” by minute hand over hour hand = 5.5t degrees.

For a right angle, the hands must be 90° apart (or the other right angle, 270° apart).

Case 1: 5.5t = 90 + 360k

Multiply by 2 to avoid decimals:

11t = 180 + 720k

t = (180 + 720k) / 11

Now count how many values lie in 0 ≤ t < 720 (i.e., within 12 hours):

  • k = 0 gives t = 180/11 ≈ 16.36
  • k = 10 gives t = (180 + 7200)/11 = 7380/11 ≈ 670.91
  • k = 11 would exceed 720 minutes

So k = 0 to 10 → 11 times

Case 2: 5.5t = 270 + 360k

11t = 540 + 720k

t = (540 + 720k) / 11

Within 0 ≤ t < 720:

  • k = 0 to 10 works (k=10 gives ≈ 703.64)

So again 11 times

Therefore, in 12 hours: 11 + 11 = 22 times

In 24 hours (double): 22 × 2 = 44 times

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M
minimax31
•9mos

An even more intuitive explanation:

The minute hand moves faster than the hour hand. In every 12 hours, the minute hand laps the hour hand 11 times. This is because when the hour hand completes a full circle, the the minute hand completes 12 full circles of the clock. When, for example, the hour hand goes from 00:00 to 01:00, the minute hand takes over the hour hand once. A second time between 01:00 to 02:00, and third time between 02:00 to 03:00.... and the eleventh time between 11:00 to 12:00.

And each time the minute hand moves relative to the hour hand one full circle, it makes a right angle twice: at 90 and at 270. So, twice in one hour. Hence, every 12 hours the right angle is formed 11 x 2 = 22 times. Consequently, every 24 hours, the right hand is formed 22 x 2 = 44 times.

Now this is exactly the workings behind the workings done above!

Hope that helps!

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basicallybhavesh
IIM Ranchi (IR 2)
MOD•9mos

Ooh, very clean -- thank you so much!

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M
minimax31
•9mos

I actually made a minor mistake. The 11th time the hands crosses is between 10:00 and 11:00. And the next crossing is exactly at 12:00 which is essentially the same position as the starting one!

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