Just want to poke my head up to mention that I have posted a new POTW for you. You get three for the price of one this week. The official problem is pretty straightforward, I think, so I gave you two bonus problems just for fun. Enjoy!
22/60 * 12 = 4.4 = 4:24
Let's assume S&G both adjust their watches to correct time. S turns it 25 minutes back, so now the watch is actually 35 minutes slow. G turns it 10 minutes ahead, so it is 15 minutes fast. S will miss the train by 30 (35 - 5) minutes. G has a 20 (-15 -5) minute wait.
x/4+(24-x)/2 = x. Solving, x=9.6 or 9:36
Clearly after 4 o'clock. Since it takes an hour to go from pointing directly at 4 (aka "20 minutes") to 5 ("25 minutes"), at 22 minutes it has crossed 2/5 of that time, or 24 minutes; it is therefore 4:24.
Both Simpkins and Green will aim to arrive when they think it is 7:55. Simpkins, thinking his watch is 25 minutes fast, will therefore aim to arrive when his watch says it is 8:20, at which point it will actually be 8:30 and he's 30 minutes late. Green will aim to arrive when his watch says it is 7:45, at which point it will actually be 7:50, and he'll be 5 minutes earlier than he thought he'd be (plus have 10 minutes to catch the train).
For Junior, let t be the number of minutes after midnight that it currently is. We then find that t = t/4 + (1440 - t)/2; which solves to t = 576, which means it is now 09:36.
Actually, correction: Green's watch, when it says it is 7:45, will mean that it is actually 7:40 (that is, it is actually 5 minutes fast), and he'll have a 20 minute wait for the train.
@GAZZA: I think your first interpretation of the second problem is correct. Green's watch has "gained five minutes". My assumption is that that is with respect to the "ten minutes slow" that he thought his watch was running. So he will plan to arrive when his watch says 7:45, thinking it's 7:55. It will actually be 7:50. Simpkins will indeed be a half hour late for the train.
@4: I took it to mean the other way (Gazza's second answer). Green thinks its 7:55 when his watch says 7:45 (he thinks it's slow by 10 mins), but in actuality at that time it is 7:40 (his watch is really fast by 5 mins).
You just have to remember that the hour-hand sweeps 12-minute increments per minute tick on the clock face. So it's 4:00 + 2*12 minutes = 4:24.
Simpkins arrives at 8:30 and Green arrives at 7:40. "Gained 5 minutes" means his clock reads five minutes later than it actually is.
jrosenhouse wrote (September 19, 2016):
> [...] I have posted a new POTW for you. [...]
The official problem is pretty straightforward, I think [...]
Specificly [ http://educ.jmu.edu/~rosenhjd/POTW/Fall16/POTW3F16.pdf ]:
> My accurate clock has only one hand, and hour hand.
At the precise moment that this hand points directly to the 22-minute mark,
what is [...]
Has there a definition of an "accurate clock" been given at all (e.g. in class] which would be applicable to an "hour hand pointing [sequentially] to minute marks" ?
Does it imply that the duration of the hand from having pointed at one mark until having pointed at the next following mark remains constant?
Yeah I'm pretty sure my second answer for Green is right. If you say your watch is 5 minutes fast, you usually mean that it is ahead of the actual time. I don't think the fact that he'll arrive when his watch says 7:45 is in dispute. "Slow" and "fast" must have different signs. He's arrived at 7:45 because he thinks it is fast, and uses +10. Therefore, slow must be negative, right? (And -5 in this case).
I see Eric Lund's point, though. Suppose Green sets it slow 10 minutes on purpose, so he knows to arrive when his watch says 7:45. What he doesn't know is that between the moment he set it slow and now, it has gained five minutes, so it is only 5 minutes slow. Then when his watch says 7:45 it's actually 7:50.
Well yeah, but that's introducing new parameters. There's nothing about the problem that implies they're allowed to change the time on their watch, and I would suggest that it is implied they CANNOT do this (after all, if they could, why wouldn't they have already done so? Incorrectly, sure...)
Yeah I’m pretty sure my second answer for Green is right. If you say your watch is 5 minutes fast, you usually mean that it is ahead of the actual time. I don’t think the fact that he’ll arrive when his watch says 7:45 is in dispute. “Slow” and “fast” must have different signs.
Sort of like flammable and inflammable? Or saying that yesterday's movie was cool vs. yesterday's movie was hot? :) English can be weird..
I think that's part of the point though; teaching math to students requires familiarizing them with the practice of translating somewhat opaque and confusing words into the correct mathematical representation. And while you're doing that, it's also useful to recognize that some vernacular sentences may have no 'single correct' translation. That's one of the reasons I like Jason's "any reasonable interpretation will be accepted" approach. I only wish some of my teachers had had the same attitude - I still smart at getting a zero on a QM test decades ago because I solved 'the wrong' problem. And then got told by the Prof that while he thought it was good work, if he regarded my test he'd have to re-grade a bunch of others too and he didn't want to spend the time to do that. Jerk. Grumble grumble grumble...
If you say your watch is 5 minutes fast, you usually mean that it is ahead of the actual time.
But that's not what the problem says. The problem says that Green's watch has "gained" five minutes. "Fast" and "slow" are obviously with respect to a standard reference time, but "gained" and "lost" are relative terms. For instance, if you are running a race where you are ten seconds behind after one lap but only five seconds behind after the second lap, you are said to have gained five seconds during the second lap. If you were actually leading by five seconds, the commentator would say that you were five seconds ahead.
In this case we are told within a single sentence that Green's watch was previously ten minutes slow but has gained five minutes. I would expect most people to interpret the "gained" here as being with respect to the watch's previous state, the way most people would assume that if a sentence refers to Jason, a subsequent "he" in the same sentence also refers to Jason (absent context that would suggest otherwise, such as a situation where "Jason" is known to be female). Simpkin's case is unambiguous because the problem explicitly uses the terms "fast" and "slow".
In this case we are told within a single sentence that Green’s watch was previously ten minutes slow but has gained five minutes.
Here's what it says:
"Green thinks his watch is ten minutes slow, while it has actually gained five minutes."
"Gained" is relative, but in order to know relative to what, we need to know how Green came to believe his watch is ten minutes slow. 1) Maybe he set it that way explicitly and it has since gained five minutes, so that it is now five minutes slow. 2) Or, maybe his watch was correct at some point, but when he compared it with a clock he thought was correct but which was actually ten minutes fast, he came to believe his was 10 minutes slow; and it has since gained five minutes, making it actually five minutes fast. 3) Or, maybe when he left home he correctly believed his watch was 10 minutes slow, but he's forgotten that in his travels he changed time zones, and his watch has since gained five minutes, and so when his watch says 7:45, it's actually 8:50.
I think something like the second scenario is the best interpretation because the problem says "while it has actually gained five minutes" – this implies that Green has never been correct in his belief about his watch being slow.
I'm always a little amazed by just how hard it is to come up with a perfectly pristine, totally unambiguous phrasing for problems of this sort. My source for the problem was a collection of puzzles by Henry Dudeney, and I simply copied the puzzle verbatim.
Like Eric mentioned, I will accept any solution from my students that is based on a plausible reading of the problem. If any of them devote the careful attention to this all of you have, I will be so impressed that I will not quibble if their answer is different from the one in the book.
I’m always a little amazed by just how hard it is to come up with a perfectly pristine, totally unambiguous phrasing for problems of this sort.
Its easy to think that, when you speak or write, people are getting a direct connection to what you're thinking. In my experience literally every human being (including me) thinks they are communicating more clearly than they actually are. This is the reason dangerous professions use repeat-back and formal acknowledgement, ritualized call-and-response, checklists, etc.; its to overcome the "I thought I was being perfectly clear" problem.