Water clocks puzzle

[jack]

Puzzle: I have two identical cylindrical lemonade bottles I punched small holes into the sides of, just above the bottom¹. I find that when I fill them each with water the first empties in 225 seconds and the second more quickly in 75 seconds. Without any other form of measurement, how do I use them to time 25 seconds?

Q. That's just one of those hourglass/numerology puzzles. If one hourglass empties in time p minutes, and the second in time q, then calculate p^-1 mod q, and you know pp^-1=1 (mod q) so pp^-1=1+nq. Run glass Q n times and glass P p^-1 times; when the first has finished you have exactly one minute before the second does.
A. If you think that's easy, try it.

Q. It's a trick, isn't it.
A. Probably.

Q. Tilt one of the bottles...
A. If you like. They *are* cylindrical, so you *can* halve the amount of water by tilting them until the water only just covers the base, if you think it'll help.

Q. I could use a...
A. No other equipment is necessary. You can use additional vessels of unknown size if you want to.

Q. How quickly did you solve it?
A. Actually, I made the puzzle up myself last night.

Q. Are you sure your solution works?
A. I may be mistaken. In which case I apologise, and promise to feel very embarassed.

Q. Do I need to use much maths for this?
A. Some. Nothing not taught at A-level IIRC.

Q. Can I find a different solution?
A. I don't think so, but try; it might be better.

[-1] That's not a footnote, that means 'inverse of p'
[1] Really.

Comments

[fanf]

Fill both bottles, and let them empty through their holes. When the fast one empties, block the hole on the other one while you re-fill the first. Let it empty again, and when it finishes block the hole on the other again. It will be 1/3 full. Pour the contents of the slow bottle into the fast bottle. You now have a 25s timer.

[cartesiandaemon.livejournal.com]

I believe the change in volume is not linear in time.

This is the maths I was doing: a drip is constant, but if it's ejected under pressure I think it isn't. And I hope I haven't messed up the maths.

If I'm right I should have chosen the numbers so your solution didn't work, but I didn't think to.

[simont]

I believe the change in volume is not linear in time.

In that case the volume remaining in the slow bottle won't be 1/3 of its total volume; but it will still be the correct volume to empty out of the slow bottle in 75 seconds. Therefore, assuming that the fast bottle empties at an instantaneous rate three times that of the slow bottle when they're equally full (which seems reasonable in spite of the nonlinearity you mention), it should still be the right amount to empty out of the fast bottle in 25 seconds.

[cartesiandaemon.livejournal.com]

Damn. You're right, that does work. I think my solution would require less setup time, but, even if I got my equations right, it's probably less general.