Relative time
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?
Age
14 to 16
Challenge level
Problem
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.
Student Solutions
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.