SOLUTION: 3 clocks, hour hands are missing, have minute hands running fast. Clock P, Q, R gain 2, 6, 12 mins per hour respectively. They start at midday - pointing to 12. Find the n

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Question 1098484: 3 clocks, hour hands are missing, have minute hands running fast.
Clock P, Q, R gain 2, 6, 12 mins per hour respectively.
They start at midday - pointing to 12.
Find the number of hours later when all 3 hands show the same number of minutes.
Found 3 solutions by Alan3354, ikleyn, ankor@dixie-net.com:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
3 clocks, hour hands are missing, have minute hands running fast.
Clock P, Q, R gain 2, 6, 12 mins per hour respectively.
They start at midday - pointing to 12.
Find the number of hours later when all 3 hands show the same number of minutes.
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They make 1 revolution in 58, 54 and 48 minutes.
The smallest number evenly divisible by the 3 times is
29*27*16 = 12528 minutes
= 208 hours 48 minutes
= 8 days, 16 hours, 48 minutes
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The 3 minute hands will again coincide at 12.
Corrected after I realized I missed a factor of 3.
============
There might be another conjunction sooner at some other position.
I'm thinking about that.
==========
Remember, a stopped clock is right twice each day.
======================
Interesting problem.
The only coincident position is the 12 o'clock position, which is not obvious at first.
----
30 hours is obviously correct. Actually, every 30 hours.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
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My previous solution was  WRONG.   So, I took it out.

Below find the correct solution   // (I came to it after seeing the tutor's  @ankor@dixie-net.com  solution,  which is absolutely correct).
============== 


1.  The readings difference between clocks Q and P has the rate of 4 min per hour.

    It means that their readings will coincide every 15 hours, when their minute hands are at the same positions  "verically up".


    It happens every 15 hours, and there is no other time moments, when their positions coincide.



2.  The readings difference between clocks R and P has the rate of 10 min per hour.

    It means that their readings will coincide every 6 hours, when their minute hands are at the same positions  "verically up".


    It happens every 6 hours, and there is no other time moments, when their positions coincide.



3.  The readings difference between clocks Q and R has the rate of 6 min per hour.

    It means that their readings will coincide every 10 hours, when their minute hands are at the same positions  "verically up".


    It happens every 10 hours, and there is no other time moments, when their positions coincide.



4.  It implies that the next and closest time moment when the readings of all 3 clock coincide comes in  LCM (15,6,10) which is 30 hours.

    LCM is "Leatt Common Multiple).


Answer.   In  30  hours.

Again:  the answer by the tutor  @ankor@dixie-net.com  solution is absolutely correct.



Answer by ankor@dixie-net.com(22740)   (Show Source): You can put this solution on YOUR website!
I came up with 30 hrs all three hands will be straight up at 12
I reasoned thus
P hand: After 30 hrs, it will be 30*2 = 60 min or exactly 1 hour
Q hand: After 30 hrs, it will be 30*6 = 180 min or exactly 3 hrs ahead
R hand: After 30 hrs, it will be 30*12 = 360 min or exactly 6 hrs ahead
:
I may be full of the well-known waste product here, but there it is.
Let me know: ankor@att.net, I have made many mistakes in my life and am used to it.
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