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# KCC's Quizzes AQQ230 about a clock and angle puzzle

In an analog clock, the angle formed by the hour and the minute hands varies constantly.

Questions:

1. Which angle θ is obtained when it is 5:20 ?
2. Determine when exactly at around 4, the angle will be exactly 90°?

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[edited by: emassa at 1:54 PM (GMT -4) on 29 Mar 2023]
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• 1. θ = 40°

Solution:

θ from 12 for each number = 360° / 12 = 30°

Hour θ movement for each minute = 30° / 60 = 0.5°

For 20 minutes,

number = 4

θ = 4 (30°) = 120°

For 5,

θ = 5 (30°) = 150°

Hour θ movement = 20 (0.5°) = 10°

Total = 150° + 10° = 160°

θ difference = 160° - 120° = 40°

2. 4:05 (minute is 5.45 to be exact)

Solution:

θ from 12 for each number = 360° / 12 = 30°

Hour θ movement for each minute = 30° / 60 = 0.5°

θ for each minute = 360° / 60 = 6°

For 4,

θ = 4 (30°) = 120°

Equation for 90°:

90° = [120° + x (0.5°)] - x (6°)

90° = [120° + 0.5°x] - 6°x

6°x - 0.5°x = 120° - 90°

5.5°x = 30°

x = 5.45

Time would be 4:05 (5.45 to be exact)

• Thanks Mark for your so fast feedback! Rendez-vous next week to compare your answers with the official ones!

• For the second question, it is obvious there will be 2 possible solutions: the 90° angle can be made with the minute hand before or after the hour hand! But with the way the question has been asked (i.e. "around 4"), there is one solution better than the other...

• Each number on clock makes 30°.
The hour hand also moves, so for each minute, it moves 30/60 = 0.5 deg. for 20 minutes it moves 10 deg.

At 5.20 , θ is 40°.
At around  4:5:30,  θ is 90°.

• 1. Theta=40°

2. There are two solutions

2a: Between 4:05 and 4:06

2b: between 4:38 and 4:39

Given h=hours and m=minutes, you can write the follwing equations for the angle alpha the minute hand forms with the "vertical" and the angle beta the hour hand forms with the vertical

alpha=6*m

beta= 30*h + 0.5m

• Thanks Rajesh for your prompt reply! We will confirm your answers next week!

• Thanks Gaetano! Confirmation next week!

• 1. Which angle θ is obtained when it is 5:20 ?

Long hand: 20 past is 120 degrees from 12:00
Short hand: is 150 degrees + 1/3 of the way between 5 and 6 (30/3 degrees) = 160 degrees
Angle between them is 40 degrees

2. Determine when exactly at around 4, the angle will be exactly 90°?

I started iterating it...

4:00pm
Long hand: 0
Short Hand: 120
Difference: 120

4:10pm
Long hand: 60
Short Hand: 120 + 30/6 = 5
Difference: 65

4:05pm
Long hand: 5/60*360 = 30
Short Hand: 120 + 30*5/60 = 2.5
Difference: 92.5 Too high - add time

4:06pm
Long hand: 6/60*360 = 36
Short Hand: 120 + 30*6/60 = 3
Difference: 87 Too low - subtract time

At this point we know it is between 4:05 and 4:06.  I iterated some more but this successive approximation was a little painful, so I thought a bit more and came up with the equation I was essentially using.

l=long hand angle
s=short hand angle
m=minutes past

s-l = 90
l=6*m
s=120 +(30*m/60)
So
120 + (0.5*m) - 6*m = 90
-5.5*m = -30
m = 30/5.5 = 5.45 minutes past 4 or 4:05:27

• Wonderful Martin!

• OK guys, time to publish the official answer!

Solution:

1. The angle θ between the 2 hands is simply the difference between the angles covered by the hour hand and the minute hand. In a full journey, the hour hand will cover the full rotation (360°) in 12 hours and the minute hand will cover the full rotation (360°) every 60 minutes.

If the time is given as HH:MM, let’s then call HH=h and MM=m.

The angle covered by the hour hand is (h+m/60)360°/12 = 30h +m/2

The angle covered by the minute hand is m.360/60 = 6m

Therefore the angle difference θ is 30h +m/2 – 6m = 30h – 11m/2

When it is 5:20 (h=5 and m=20), the angle θ is 30*5 – 11*20/2 = 40°

2. We can re-use the above established equation: θ = 30h – 11m/2

θ = 90 = 30h – 11m/2 ; hence 30h=11m/2 +90 or h = 11m/60 +90/30 or m=(h-3)*60 / 11

Around 4, m=5.45 minutes or 5 minutes and 27 seconds

In fact, one can observe -90 degree can also be considered since the angle between the 2 hands can be measured in 2 ways.

The above relation becomes : -90 = 30h - 11m/2 and with h=4, we obtain m=38.1818.

Thus at 4:38:11 the angle between the 2 hands will be also 90°

Big applause for our 4 first winners:

1. Hal KURKOWSKI, Retired Executive Director, L-Maxim MSS Business Unit, Dallas, USA

2. Jeffrey EVANKO, Key Account Manager, ADI, Cleveland (OH), USA

3. Mark CEE, Snr Systems Integration Engineer, ADI, Calabarzon, Philippines

4. Rajesh KUMAR, IC layout designer, ADI, Jharkhand, India

And now, be ready for the next challenge (AQQ231)!