Poor geppetto. After solving his problems with Pinocchio, he returns to his clock shop to find that his clocks are off schedule. His clocks are all supposed to go off at the same time. However, the train whistle clocks are going off every 9 minutes, the bell clocks are all going off every 3 minutes, the cuckoos are going off every 6 minutes and the fire alarms every 4. If all the clocks have just gone off at the same time, how many times will one or more clocks go off before they all go off at the same time again?
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TWCs go off every 9 mins
BCs go off every 3 mins
Cs go off every 6 mins
FAs go off every 4 mins
You can see that all of them go off at the same time at every 36 minutes, since 36 is the least number that can be perfectly divided by all of 9 , 3 , 6 and 4
So
TWCs go off 4 times in 36 mins
BCs go off 12 times in 36 mins
Cs go off 6 times in 36 mins
FAs go off 9 times in 36 mins
BCs go off every 3 mins
Cs go off every 6 mins
FAs go off every 4 mins
You can see that all of them go off at the same time at every 36 minutes, since 36 is the least number that can be perfectly divided by all of 9 , 3 , 6 and 4
So
TWCs go off 4 times in 36 mins
BCs go off 12 times in 36 mins
Cs go off 6 times in 36 mins
FAs go off 9 times in 36 mins
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Do this logically via the smallest amount of time.
3 mins pass, nothing else hits. 3 MINUTES
(fure alarms hit)
3 mins pass, the cuckoos hit. 6 MINUTES
(fire alarms hit)
3 mins pass, the whistles go off. 9 MINUTES
3 mins pass, the cuckoos and fire alarms hit. 12 MINUTES
3 mins pass, nothing else hits. 15 MINUTES
(fire alarms hit)
3 mins pass, whistles hit, cuckoos hit. 18 MINUTES
(fire alarms hit)
3 mins pass, noting else hits. 21 MINUTES
3 mins pass, cuckoos hit, fire alarms hit. 24 MINUTES
3 mins pass, and whistles hit. 27 MINUTES
(fire alarms hit)
3 mins pass, cuckoos hit. 30 MINUTES
(fire alarms hit)
3 mins pass, nothing else hits. 33 MINUTES
3 mins pass, cuckoos hit, fire alarms hit, whistles hit. DONE AT 36 MINUTES
Now you could've realized that 3, 4, 6, and 9 all go into the lowest possible number of 36, and then made sure to count the multiples up until that point [3 goes into 36 TWELVE times, 4 goes into 36 NINE times, 6 goes into 36 SIX times, 9 goes into 36 FOUR times.... 12 + 9 + 6 + 4 = 31]
But this way really shows you all the alarms that hit.
3 mins pass, nothing else hits. 3 MINUTES
(fure alarms hit)
3 mins pass, the cuckoos hit. 6 MINUTES
(fire alarms hit)
3 mins pass, the whistles go off. 9 MINUTES
3 mins pass, the cuckoos and fire alarms hit. 12 MINUTES
3 mins pass, nothing else hits. 15 MINUTES
(fire alarms hit)
3 mins pass, whistles hit, cuckoos hit. 18 MINUTES
(fire alarms hit)
3 mins pass, noting else hits. 21 MINUTES
3 mins pass, cuckoos hit, fire alarms hit. 24 MINUTES
3 mins pass, and whistles hit. 27 MINUTES
(fire alarms hit)
3 mins pass, cuckoos hit. 30 MINUTES
(fire alarms hit)
3 mins pass, nothing else hits. 33 MINUTES
3 mins pass, cuckoos hit, fire alarms hit, whistles hit. DONE AT 36 MINUTES
Now you could've realized that 3, 4, 6, and 9 all go into the lowest possible number of 36, and then made sure to count the multiples up until that point [3 goes into 36 TWELVE times, 4 goes into 36 NINE times, 6 goes into 36 SIX times, 9 goes into 36 FOUR times.... 12 + 9 + 6 + 4 = 31]
But this way really shows you all the alarms that hit.
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The key is the 9x4 because it's the first available factor of 9 that evenly divides by 4.
So, in 36 minutes, the Whistle clock will have gone off 4 times, the bell clock 12, the cuckoo 6 and the fire alarm 9 and they all will be in sync after 36 minutes.
So, in 36 minutes, the Whistle clock will have gone off 4 times, the bell clock 12, the cuckoo 6 and the fire alarm 9 and they all will be in sync after 36 minutes.