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Answer: 05:00
Counting on and back
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.
Relative Time
Age 14 to 16
ShortChallenge Level 
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.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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