ratio and proportion

Question

Three 24-hour clocks show the time to be 12 noon. One of the clocks is always correct, one loses a minute every 24 hours, and one gains a minute every 24 hours. How many days will pass before all three clocks show the correct time again?

Answer 1

The clock that gains a minute a day, after 60 days will indicate that it is 1:00 p.m. when it is actually noon.

Continuing this way, in total it will take 60*24 or 1440 days before the clock is correct at noon again.

 

Similarly, the clock that loses a minute a day, after 60 days will indicate that it is 11:00 a.m. when it is actually noon.

Again continuing this way, in total it will take 60*24 ir 1440 days before the clock is correct at noon again.

 

In either case, all three clocks will indicate the right time after 1440 days.

 

Answer 2

Clocks A, B and C show 12 noon.


Clock A works OK

Clock B loses 1 minute every day therefore clock B loses 60 minutes (1 hour) every 60 days

Clock C gains 1 minute every day therefore clock C gains 60 minutes (1 hour) every 60 days



When clock B loses 24 hours it shows 12 noon again same as clock A. So 24 h * 60 days/h = 1140 days pass by.

When clock C gains 24 hours it shows 12 noon again same as clock A. so 24 h * 60 days/h = 1140 days pass by.

Answer 3

To use a common unit, change times to minutes.
Clock 1 : 1440 minutes in a day [60*24)
Clock 2: 1439 minutes in a day [loses a minute every 24 hours]
Clock 3: 1441 minutes in a day [gains a minute every 24 hours]

When will all times show 0 [that is, the minutes since starting (modulo 1440) – the actual time will be Noon}?

So, in minutes 0-1339 from the start time (t=0 is Noon), the clocks show:

Noon: Start(t=0) 1-day 2-day 3-day 4-day  . . . (n-1)-day   n-day
Clock 1:   0            0      0        0       0      . . .      0              0
Clock 2:   0        1439  1438  1437  1436   . . .      1              0
Clock 3:   0            1      2        3       4       . . .  1439           0

      n = 1440

On day 1440 all three clocks will show the time to be Noon.