Relative Time
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?
Albert Einstein is experimenting with two unusual clocks which both have 24-hour displays. One clock goes at twice the normal speed. The other clock goes backwards, but at the normal speed. Both clocks show the correct time at 13:00.
At what time do the displays on the clocks next agree?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Answer: 05:00
Counting on and back
Fast-forward | Backwards |
---|---|
13:00 | 13:00 |
15:00 | 12:00 |
17:00 | 11:00 |
+6 | $-$3 |
23:00 | 08:00 |
01:00 | 07:00 |
+4 | $-$2 |
05:00 | 05:00 |
Counting the difference between the clocks
Every hour, one clock goes forwards by two hours and the other goes back by one, so the difference between them grows by 3 hours. Eventually, after 8 hours, they will be 24 hours apart, or in other words they show the same time again. 8 hours after 13:00 is 21:00, at which time the clocks will both be showing 05:00.
Using algebra
After $x$ hours, the first clock will have gone forward $2x$ hours and the second clock will have gone back $x$ hours. So the next time they agree is when they have run through a total of 24 hours together, i.e. when $2x + x = 24$, that is when $x = 8$.
At 21:00 (13:00 + 8 hours) both clocks will be showing 05:00.