How long will it take Marie to saw another board into 3 pieces?

Question

So this is supposed to be really simple, and it's taken from the following picture:

book image

Text-only:

It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into $3$ pieces?

I don't understand what's wrong with this question. I think the student answered the question wrong, yet my friend insists the student got the question right.

I feel like I'm missing something critical here. What am I getting wrong here?

This is simultaneously wonderful and sad. Wonderful for the student who was level-headed enough to answer this question correctly, and sad that this teacher's mistake *could* be representative of the quality of elementary school math education. — Jared, May 03 '13 at 03:38
I took five minutes per _cut_. One cut yields two pieces. Two cuts yield three pieces. And so on..... — Michael Hardy, May 03 '13 at 04:40
The question is asking about the time required to make number of pieces, while the grader has calculated the time required to make number of cuts — user13267, May 03 '13 at 05:18
I think the issue is that the language and image are incongruent. The question should have been "10 minutes to saw 2 pieces from a board" (2 cuts), then the teachers answer would be correct. As it is stated, it implies sawing a board in half (1 cut). — Vijay, May 03 '13 at 05:23
The language and the image are in perfect agreement- the image shows two pieces resulting from a single cut. — JayL, May 03 '13 at 08:01
By the way, originally this was posted as "Teacher Math Fail" at zerooutoffice: http://zerooutoffive.blogspot.com/2010/10/teacher-math-fail.html — azimut, May 06 '13 at 20:21
Perhaps a trivial point, but the illustration does not show a *board*, which is a piece of timber whose larger dimensions are at least 6 times the smallest one. It would also be practically impossible to saw a thin stave of wood unsupported at one end and supported on the other side at a distance at least half a saw's length from the cut. The miseducation thus extends from arithmetic (or logic) to carpentry. — John Bentin, May 12 '13 at 20:22
This reminds me of a similar problem I posted to some students.. "One electric water boiler can heat water from room temperature to boiling in two minutes. How many minutes does it take with three identical electric water boilers?" I got all kinds of answers... — Peter Franek, Jan 02 '16 at 07:25
The way to see this is that you're making twice as many cuts for 3 pieces. — Marcos, Jan 15 '16 at 00:08
For what it's worth, I'd added this example to the blog http://innumeracy.net/cutting-a-board/ with attribution to this site and member. This was a great question, and a fine example of innumeracy, in my opinion. — JTP - Apologise to Monica, Feb 17 '16 at 18:11
I still look back at this question, to appreciate how a third grader boggled so many people, with this applied reasoning. haha. — MonK, Jul 12 '16 at 14:18
Our [SMP](https://en.wikipedia.org/wiki/School_Mathematics_Project) textbook included a question asking how long an orchestra of twice as many people takes to play the same symphony (in the usual [Rule of Three](https://en.wikipedia.org/wiki/Cross-multiplication#Rule_of_Three) phraseology) — a good way to keep people on their toes. — PJTraill, Oct 25 '17 at 19:41
The language is a big part of the problem. The problem is part of a broader area of misunderstanding, what I call "fences and fence-posts". Examples: 1 When there are N posts in a line connected in series by fencing, there are N - 1 fences, not N. 2 When we write 2^10, there are 10 of "2' but only 9 of "*" multiplications, so it is wrong to say "2^10 is 2 multiplied 10 times". 3 When you make a sandwich with another piece of bread so as to have three slices, it is a double-decker, not a triple-decker as many restaurants claim, as we are counting the decks of tuna, not slices of bread! — Dr. Michael W. Ecker, Jan 13 '21 at 03:34
Honestly, I don't believe that it really needs 10 minutes to cut the thin piece of wood in the figure ... ;-) — Dr. Wolfgang Hintze, Feb 25 '21 at 07:18

score 885 · Accepted Answer · edited Oct 03 '16 at 23:25

885

Haha! The student probably has a more reasonable interpretation of the question.

Of course, cutting one thing into two pieces requires only one cut! Cutting something into three pieces requires two cuts!

------------------------------- 0 cuts/1 piece/0 minutes
---------------|--------------- 1 cut/2 pieces/10 minutes
---------|-----------|--------- 2 cuts/3 pieces/20 minutes

This is a variation of the "fence post" problem: how many posts do you need to build a 100 foot long fence with 10 foot sections between the posts?

Answer: 11 You have to draw the problem to get it...See below, and count the posts!

|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----| 
0-----10----20----30----40----50----60----70----80----90---100

edited Oct 03 '16 at 23:25

suomynonA

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answered May 03 '13 at 03:33

Twiceler

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Wow, I feel completely stupid. That was so obvious, yet I didn't figure it out. I guess it isn't too late to start at square one – yuritsuki May 03 '13 at 03:36
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In all fairness, neither did whoever graded this problem. This question made me smile :) – Gyu Eun Lee May 03 '13 at 03:37
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This is an example of a [red herring](http://en.wikipedia.org/wiki/Red_herring) in a word problem. The number of pieces (2) distracts from the actual variable - the number of cuts (1). I think this is a very common (and important) technique to teach children, because it happens so often in solving real world problems – xdumaine May 03 '13 at 17:20
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This is a perfect example of SAT-style math, especially relating to the fencepost problem. – apnorton May 03 '13 at 19:33
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Agreed and not only that but the question does say "a board" into "two pieces." A board = one board. Going from one board to two boards takes one cut. I can't see how this could be interpreted any other way. They don't talk about sawing equal length pieces or cutting two equal pieces from a very long board needing 3 cuts (you would have 3 pieces by then). anyway, my wife and I home school our kids because of stupid crap like this. – Eric M May 03 '13 at 20:02
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@Retrosaur, the problem is in the question. It's framed in the same way as 'if it takes 5 minutes to make one widget then it must take 15 minutes to make three'. The sawing a thin pole scenario doesn't match this, despite 'board' being a noun similar to 'widget' and 'saw' being a verb similar to 'make'. The kid might grow up to be an economics major ;) – conjectures May 04 '13 at 07:57
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And yet it only takes ten poles if you build the fence around something. – Marcks Thomas May 07 '13 at 15:29
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This is why we start counting at zero in C. :-) – Jens May 08 '13 at 11:13
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Well, it depends on the topology of the board. The teacher is right if the board has the topology of, say, a ring or a torus. ;) – Heidar May 24 '13 at 23:11
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This is also similar to why reading pages 255-260 means you've read 6 pages. – JoeCool Oct 28 '13 at 20:30
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**The only two hard problems in computer science:** cache invalidation, naming things, and off-by-one errors. – Wildcard Aug 23 '17 at 21:56
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"See below, and count the posts!" 27 below this one, being the top answer of 28. – MD XF Dec 29 '17 at 03:03
The question clearly stated ANOTHER board. So yes 2 cuts in another board for 3 pieces. So times 2. The student is correct. Student should be teacher :) – mick Nov 24 '19 at 16:23

score 424 · Answer 2 · edited Dec 02 '14 at 08:53

424

Well, the information is incomplete, so they're both right and wrong. Since the question is for $3^{rd}$ graders the correct answer should be $20$ minutes ($2$ cuts $\times$ $10$ min), though the teacher is right if you do cut it like this (first red, then green):

long vertical cut, then a short horizontal cut

The problem is that the question doesn't say anything about how you have to cut it, so the blue cut would have been good enough too. That cut should only have taken a few seconds.

edited Dec 02 '14 at 08:53

yuritsuki

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answered May 03 '13 at 05:51

AndSoYouCode

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However if you look at the drawing of the piece of wood being cut, this approach would be extremely difficult. – Mark Adler May 03 '13 at 06:42
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@Mark Yes, looking at the drawing of the piece of wood, I think the answer would rather be close to an hour... – Axel May 03 '13 at 07:48
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The text is indeed incomplete, and this square is exactly what came to my mind. But then I noticed the picture next to the question. – ugoren May 03 '13 at 08:01
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The question is hypothetical, and doesn't give the information about board size. So I think we might interpret 'as fast' to be that it means Marie can cut any board into 2 pieces in 10 minutes regardless of its size and length of the cut. – tia May 03 '13 at 08:32
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@MarkAdler Hehe, I didn't see the picture at first. But it still depends on how thin you cut the first piece. And I must say that the picture doesn't look like anything I'd call a "Board". Are we sure the picture is of the object Marie is cutting? – AndSoYouCode May 03 '13 at 11:03
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The picture shows something close to a 2x4 and a saw cutting the shortest width. I couldn't imagine someone ripping a 2x4 with a handsaw like your red line shows. I see your point but based on the text and picture, I don't think these types of cuts on a new board would be expected for 3rd grader. In addition the teacher has 10=2 but I suspect that should be 10=1 for one cut needed for two pieces. It took 10 minutes to get 2 pieces from 1 cut. And you only need two cuts for 3 pieces. And so I think it's reasonable to assume all cuts are the same effort then. – Eric M May 03 '13 at 20:12
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If this kind of arbitrary cut is allowed, then what stops me from hacking off a tiny corner piece in ten seconds and calling that two boards? – Kaz May 04 '13 at 00:29
-1 (kidding) for making it seem like the teacher could be correct. =) – Kevin Fegan May 04 '13 at 01:28
You should all know that in these kind of tests, picture is not drawn to scale. That image is actually an image of a board drawn with altered scale to fool students that trued using rulers to get the answer. – Lie Ryan May 04 '13 at 05:53
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it says ANOTHER board, it's not the same one! – lel May 04 '13 at 09:35
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-1. This is makes absolutely no sense. Who said the red cut has to be in the middle? – user2345215 Mar 10 '14 at 14:34
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This answer, my friends, demonstrates what we call "being a smartass". – Zhuli Apr 09 '14 at 12:05
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I'd like to point out that, as an armchair philosopher, I would lean back and point out that particulates of the board are leeching into its surroundings at every passing moment, and that any grain of sawdust from the first cut could itself be considered a new board. Therefore, it takes zero time to get a third board. The board is the Sorites paradox. – Parthian Shot Oct 22 '15 at 00:22
Surely if you want to pay attention to "how you have to cut it" then you should consider how the temperature might change after 10 minutes, maybe that will have an effect... And what about the energy you have left in your muscles after cutting a board for 10 minutes, it will probably take you longer the second time. And how fast will you get tired? are you a fit person? What did you eat for breakfast?? – John Cataldo Mar 03 '18 at 18:17
Sorry, but They Said “ another “ board. So no , this is not more logical. Assuming the other boards are similar ( logical ) the kid gave the best answer. – mick Jun 11 '18 at 00:05
No , the teacher asked about a new (!) board. So we start over, make 2 cuts and have 3 pieces. Student wins. – mick Nov 24 '19 at 16:24

amWhy · Answer 3 · 2020-04-11T23:23:19.387

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The student was correct:

Sawing a board into two pieces requires exactly one cut to be made. Sawing the board into three pieces requires exactly two cuts...

Hence, if it took $\bf 10$ minutes to make one cut, then cutting a board twice, at the same pace, would take $\;2 \times 10 = \bf 20$ minutes.

The instructor should receive tutoring from the student, I'm afraid.

edited Apr 11 '20 at 23:23

answered May 03 '13 at 03:35

amWhy

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@amWhy Diagonally across the image is a watermark that says zerooutoffive.com, where the image was at some point submitted. – Karl Kroningfeld May 03 '13 at 09:43
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Retrosaur: Oh! I was just wondering if this came from actual work submitted by a student... Sometimes we see samples of work or questions/solutions where the educator went awry...that's all. ;-) $\;\;$ I didn't mean to question its authenticity. Just wondering if this actually happened to a student. – amWhy May 03 '13 at 13:11

score 211 · Answer 4 · 2018-03-06T03:48:40.300

211

You can actually do it in ten minutes but your saw must look like this:

|     |
|     |
|     |
|     |      <- cutting edges
|     |
|     |
+--+--+
   |         <- handle
   |

:-)

edited Mar 06 '18 at 03:48

answered May 04 '13 at 08:22

80

Well of course... but why 2 blades, why not 10 or 100. Obvously it could not be done in 10 minutes. Adding a second blade doubles the friction, so to complete the cut with 2 blades in the same time she'd have to increase the force applied, and/or the rate she moved the blade. But, if she could increase the force and rate, she could make a single cut with 1 blade in perhaps 5 minutes, or even 1 minute, invalidating the relationship of time, and any two cuts. So, it's clear she must work with the same force and rate. Therefore, with 2 blades it would take 20 minutes. Still 10 minutes per cut. – Kevin Fegan May 04 '13 at 20:37
Well, we could just have a blade of negligible thickness, and then it would take negligible energy and time to do. – PyRulez May 06 '13 at 20:29
@KevinFegan - Force and/or rate might not be the limiting factor - could be setting up tools or something like that. – psr May 06 '13 at 20:59
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@PyRulez: Nope. What's involved here is surface energy. To create new surface area, you need to supply this energy to the material by doing work on it. As the blade thins out, the closer the work you expend becomes to the minimum surface energy. That's asymptotic behavior with asymptote most definitely *not* at zero. – Kuba hasn't forgotten Monica May 06 '13 at 22:53
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Use a circular saw: 1 minute. – naught101 May 07 '13 at 03:03
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And, bods, for the love of `$DEITY`, please don't vote this up so far that it exceeds the correct answers. It was just a bit of humor (hence the community wiki) and I never expected it to be quite this popular - if it starts to look like it's threatening the correct answers, I'll have to delete it, and the world will be a drabber place :-) – May 07 '13 at 05:51
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This answer is cute, but in my experience, cut rate is limited by the power input to the blade. Assuming the student tried as hard as possible for the first cut, cutting with two blades at once would not speed the operation. – Phil Jul 29 '13 at 17:32
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@KevinFegan Not only does it increase the friction, but it increases the weight of the saw. Therefore requiring more energy, and therefore time, to saw the board. – Keltari Aug 28 '13 at 03:07
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Come on, guys, even in *Physics* class you can assume 0 friction, 0 air resistance, infinitesimal widths etc. etc. – Patrick M Sep 25 '13 at 04:33

Keltari · Answer 5 · 2018-10-12T11:04:02.187

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Another correct answer would be 10 minutes. One could infer from:

If she works just as fast

that "work" is the complete amount of time to do the job.

edited Oct 12 '18 at 11:04

answered May 03 '13 at 12:30

Keltari

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I once took a test that had the question "What is the last thing you should do before handing in a test?" and I answered (incorrectly, sadly), choice (c) which was "Staple a $20 bill to the test" which is the *last* thing I would ever consider doing. I tried to explain my reasoning to the teacher but they just laughed and said I was an existential philosopher... – Michael May 03 '13 at 20:59
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@Michael That is why mathematics puzzles as english words seldom often have ambiguous answers. The english language is rich with ways to say one thing and mean another. – Jonathan May 03 '13 at 21:42
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@Jonathan or, apparently, to say two opposite things at the same time…! (“seldom often”?) – Aant May 04 '13 at 17:33
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@AAnt Whoops! I was going to say "words seldom have only one meaning", but when I changed the comment to refer to the puzzles, I left in an unintended "seldom". Well spotted. :) – Jonathan May 07 '13 at 07:15
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@Michael: (c) was not one of the answers, it was a directive. – Michael Apr 22 '14 at 16:17

score 121 · Answer 6 · answered May 20 '13 at 13:46

121

The topologists among us may perhaps enjoy the following defense of the teacher's answer: if the board is in the shape of a ring, it will take two cuts to get two pieces, and three cuts to get 3 pieces.

answered May 20 '13 at 13:46

Mikhail Katz

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Of course, if you look at the board itself, it clearly *isn't* in the shape of a ring... – Joe Z. Nov 28 '13 at 09:46
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Actually an examination of the figure suggests that the question may originally have been phrased in terms of "cutting off" such and such number of pieces (with the other end being off the board). This formulation would make the teacher's answer correct. One can speculate that the phrasing was carelessly changed by someone somewhere along the way. – Mikhail Katz Nov 28 '13 at 19:16
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(Also, if the board is in the shape of a ring, you do indeed have to "saw it into one piece" first, as some people were jokingly pointing out.) – Joe Z. Jan 17 '14 at 08:20
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But if your saw was long enough, you could just make one cut to cut it in half – MCMastery Oct 12 '17 at 00:51
I'm late to the party, but you can saw the ring to make 2 pieces with 1 cut if you saw through the plane part. – Sally G Jun 18 '19 at 10:59

score 88 · Answer 7 · edited Mar 23 '19 at 02:46

88

Let P : pieces

Let m : minutes

Let C : cuts

Let t : time per slice = 10

$$C(m) = m/t , \{m| m < \text{Life}(\text{Marie})\, \{C < \text{length}(\text{board})\}$$

$$P(C(m)) = \text{floor}(C(m)) +1 , \{m| m < \text{Life}(\text{Marie})\}$$

You're right that clearly isn't a simple grade 3 problem, but the answer is still 20. $$P(C(20)) = \text{floor}(C(20)) + 1 = \text{floor}(20/10)+1 = \text{floor}(2) + 1 = 2 + 1 = 3\ \text{pieces} $$

edited Mar 23 '19 at 02:46

NetherGranite

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answered May 03 '13 at 18:27

awiebe

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Notation overflow! – user13107 May 05 '13 at 16:39
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I don't understand... How can you have an inequality `C < length(board)` where the LHS is an amount of cuts and the RHS is the length of the board? They're completely different units. Isn't this a case of comparing apples and pears? – Alderath Jan 10 '14 at 15:40
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A ==|==|== B The length of the board is the distance from A to B, the length of C is a real number, the set of C has an integral length. The granularity of the length of the board is infinite, but the number of times Marie is capable of subdividing it is finite. The length of a cut occupies some arbitrary amount of space which is of the same unit in which you are measuring the board. – awiebe Jan 11 '14 at 03:23
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Couldn't you find a way to throw some tensor calculus in there? – richard1941 Jan 29 '17 at 03:41

Niels B. · Answer 8 · 2013-05-03T22:43:51.710

The student answered the question the most correct way possible. First it is stated that Marie spends 10 minutes on sawing a board into two pieces. And then the student must answer how long it will take to saw another board into three pieces.

So we are not talking about chopping off pieces from an undefined source. We are talking about splitting a board.

However, it's poorly phrased because it's not explained how the board must be cut. It can be cut in infinite ways. Also, we don't know if the two boards are identical, so we must rely on assumptions here.

David Aldridge · Answer 9 · 2013-05-04T07:15:26.230

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The teacher would be correct if the question was "... to cut two pieces from the end of a board ...", implying more strongly that the pieces were being cut so as to leave another remaining piece.

I don't think that a reasonable person would interpret the question in that way, though.

edited May 04 '13 at 07:15

answered May 03 '13 at 08:28

David Aldridge

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Kaz · Answer 10 · 2013-05-03T22:14:00.097

One part of what the teacher suggests is possible. Four pieces can be obtained in twenty minutes, because this takes only two cuts: cut it in two, then parallel the pieces and cut again, such that the saw goes through both at the same time. (The assumption is that the extra energy doesn't take more time, just more effort per stroke: not realistic, but let's go with it).

The mistake is interpolating between the two possibilities. If two pieces takes ten minutes, and four can be had in twenty, it does not follow that three pieces can be had in fifteen. However, six pieces can be had in thirty minutes which averages out to three in fifteen.

Suppose two workers are put on the job, and suppose it is somehow possible for them to divide a cut between themselves by attacking it from opposite sides without hindering each other, so they can meet in the middle in five minutes and complete the cut. They can execute this at the beginning to make one board into two. Then they double up the board, and each makes a ten minute double cut through both boards: six pieces in fifteen minutes, so basically three pieces per worker per fifteen minutes.

So if we think about just a one-off job carried out by a single person with a saw, then the student is right. However, if we were talking about productivity over multiple pieces, and possibly with multiple workers, then the teacher would also be right; the problem is, nothing of the sort is suggested in the way the question is posed.

Everybody has the wrong idea. It took ten minutes to make two pieces. That is five minutes per piece. So three pieces take fifteen minutes. That is how the teacher seems to be thinking (but not me). — richard1941, Aug 07 '19 at 06:47
@richard1941 I'll bet pretty much everybody understands that interpretation. The issue is that it is physically absurd. — electronpusher, Feb 13 '22 at 18:35

score 52 · Answer 11 · edited May 05 '22 at 10:31

The student is absolutely correct (as Twiceler has correctly shown).

The time taken to cut a board into $2$ pieces (that is $1$ cut) : $10$ minutes
Therefore, The time taken to cut a board into $3$ pieces (that is $2$ cuts) : $20$ minutes

The question may have different weird interpretations as I am happy commented:<br

Time taken to cut it into one piece = $0$ minutes
So Time taken to cut it into $3$ pieces = $0 \times 3$ minutes = $0$ minutes.

So $0$ can be an answer. but it is illogical just like the teacher's answer
and as Keltari said

Another correct answer would be 10 minutes. One could infer, "If she works just as fast," that "work" is the complete amount of time to do the job. -Keltari

This is logical but you can be sure that this is not what the question meant, but the student has chosen the most relevant one. The teacher's interpretation is mathematically incorrect.

The teacher may have put the question for the students to have an idea of Arithmetic Progression and may have thought that the students will just answer the question without thinking hard. In many a schools, at low grades children are thought that real numbers consists of all the numbers. Only later in higher grades do they learn that complex numbers also exist. (I learned just like that.) So the question was put as a question on A.P. thinking that the students may not be capable of solving the answer the correct way.

Or as Jared rightly commented:

This is simultaneously wonderful and sad. Wonderful for the student who was level-headed enough to answer this question correctly, and sad that this teacher's mistake could be representative of the quality of elementary school math education. – Jared

Whatever may be the reason, there is no doubt that the student has been accurate in answering the question properly and that the teacher's answer is illogical.

You are *probably* right that the question is not looking for the answer I gave. However, you most certainly cannot be *sure*. You are making a decision based on *assumptions*. The age of the test takers and the wording of the question are points to consider. However, there is *nothing* to prove or disprove that my answer is, or is not, what they are looking for. — Keltari, Sep 15 '13 at 22:35

score 45 · Answer 12 · answered Sep 10 '13 at 20:38

45

Teaching children, you have to be fair to them:

Think as they do.
Third graders are budding topologists, just very far from graduation.
Children are honest and direct in their assessments - it would not likely occur to them to "cut into 3 pieces of equal length" because that was not in the question. Nether would they likely think of any of the alternative cuts offered here in the various answers --- precisely because:
People (especially children) tend to be very visual. DUH there is a picture of the saw cutting the board. The board IS a board, not a paper cut out, not a piece of rectangular plywood; and the way the saw is positioned very strongly implies the next cut would be made in a similar fashion. Honestly, how many of you looked at that picture and almost unconsciously imagined moving the saw to the right (or maybe to the left) of the current cut ? I did - and I bet the children would too ... because:
Children are hands on.

When I read the problem, I thought the test grader just muffed it misreading the scoring sheet. Wow - I guess I'm childish :-P

answered Sep 10 '13 at 20:38

Howard Pautz

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While this is a nice insight on teaching practices, this doesn't actually answer the question :C – yuritsuki Sep 11 '13 at 02:48
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@Retrosaur thx. Oh sorry, I tried to address the underlying cause of the problem. You say you "don't understand what's wrong with this question. " Take it at face value, as a child would. I bet every kid would get the answer correct! You ask, "What am I getting wrong here?" My answer is: The grader or teacher is being too much an adult, thinking too much into it. :)) And when I read "I feel like I'm missing something critical here," I was prompted to give the answer above. The missing link is knowing when to throttle back the intellect, before it overpowers playfulness. Is the answer clearer? – Howard Pautz Sep 11 '13 at 18:33
And a follow up @renegadeballoon aka Retrosaur - by the looks of it months later, very few other people 'actually answered the question' too :C (Just proves math people are as incorrigible as engineers *ouch* :-P ) – Howard Pautz Aug 07 '14 at 02:16
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Didn't intend to come of that way, just that it didn't directly answer the question. But still very informative and thorough, which is why I gave it a +1; it'd do you some good to add just a small bit to the bottom explaining the actual answer – yuritsuki Aug 07 '14 at 02:19

score 38 · Answer 13 · answered May 06 '13 at 04:18

38

There is a similar problem that needs an argument quite analogous to what the student seems to have used:

A clock takes 12 seconds to strike 4 o'clock, how long will it take to strike 8 o'clock?

The interpretation is that the time is spent between the strikes, so the answer is 28 seconds instead of 24.

answered May 06 '13 at 04:18

ronno

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I'd argue it would take at least another four hours. – Marcks Thomas May 08 '13 at 13:47
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@Marcks That would be my argument too. Except, by the time 4 o'clock has been struck, we are now already 12 seconds into the first hour. So it would be 3 hours 59 minutes and 48 seconds. – daviewales May 18 '13 at 13:07
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@daviewales Plus the 28 seconds to chime. 4 hours 16 seconds. – Paul Fleming May 19 '13 at 13:17
I was taking the time from the end of the chime at 4, until the start of the the chime at 8. – daviewales May 20 '13 at 00:52
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Are we assuming the clock being measured is accurate? Does the time we use to measure the clock fall on a Daylight Savings day? When you say "how long" are you referring to time or distance the hands traveled? :D – Keltari Aug 28 '13 at 03:23
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@Keltari Since Daylight Savings switches at 2:00, which is not between 4 o'clock and 8 o'clock, it's irrelevant to the question. – Joe Z. Jan 17 '14 at 08:18

prog_24 · Answer 14 · 2013-05-14T16:24:36.263

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The answer will have to be 20. If it takes 'Marie' 10 minutes to cut the board into two pieces then that means it has taken her 10 minutes to make that chop.

Three pieces would require two chops therefore the teacher is wrong:

2 * 10 = 20

edited May 14 '13 at 16:24

answered May 03 '13 at 10:31

prog_24

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The student is actually right. The teacher is wrong... – Michal B. May 03 '13 at 12:22
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why this answer has down vote. it he correct also like student. – Mowgli May 03 '13 at 14:20
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I'm assuming it was down voted because it said the student was wrong then they weren't. – CramerTV May 03 '13 at 18:14
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Since all of their logic is correct, I assume it was just a mistake in typing *'student'* instead of *'teacher'*. – Kevin Fegan May 04 '13 at 01:40
Absolutely, it was a typo in the haste of things. The logic was simple enough – prog_24 Dec 21 '14 at 22:31

score 30 · Answer 15 · answered May 03 '13 at 15:07

30

Considering they show a drawing of the piece of wood on the problem itself, the assumption is the cut would be made in the same way, hence the student was right to start with.

answered May 03 '13 at 15:07

Jeffrey Kinzer

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score 27 · Answer 16 · edited Sep 12 '17 at 21:19

27

I would think of it this way: how long would it take to cut it in to $1$ piece... $0$ minutes because it is already in one piece. The model is: time = cuts $\cdot$ $10$. As $1$ cut $= 10$ minutes.

edited Sep 12 '17 at 21:19

Troutinator_104

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answered May 03 '13 at 10:18

someSortOfProgrammer

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How log to cut it into zero pieces? negative ten minutes? – richard1941 Jan 29 '17 at 03:47
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@richard1941 how long ago was the board cut from the tree? – Ekadh Singh - Reinstate Monica Apr 05 '20 at 16:21

score 25 · Answer 17 · edited May 10 '17 at 00:33

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Sawing once takes $10$ minutes and obtains $2$ pieces. So, since we obtain $3$ pieces when we saw twice, it takes $2 \cdot 10 = 20$ minutes.

edited May 10 '17 at 00:33

Troutinator_104

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answered May 06 '14 at 06:54

zscoder

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score 23 · Answer 18 · edited Nov 09 '15 at 06:34

Quite a random question. The answer would depend on where and how the cuts are made.

e.g. Let's say the $2$ blocks are identical (assumption on my part) each of $10$cm $\times$ $5$cm $\times$ $1$cm Let's say first block is cut lengthwise, i.e. into 2 blocks of 10cmx5cmx0.5cm. This means it took 10 mts to cut through and area of $10\times5 = 50cm^2$

Now let's look at the second block. Cut along width to create 2 blocks each of 5cmx5cmx1cm. Then you take one of these and cut off along the thickness to create two pieces 5cmx2.5cmx1cm. In this case you have just gone through an area of 5x1+5x1 = $10cm^2$ so it should only take 2mts.

Of course, if this was a question in a $3^{rd}$ grade exam, none of the above is relevant. The way the question is written, the answer could be 20 (if you are cutting pieces off a long piece of wood as the diagram indicates) or 15 (if you cut a block into half and then use the second cut to cut one of the halves into half).

+++++++++++ | +++++++++++ | +++++++++++   
+++++++++++ | +++++++++++ | +++++++++++   
+++++++++++ | +++++++++++ | +++++++++++   
+++++++++++ | +++++++++++ | +++++++++++   
+++++++++++ | +++++++++++ | +++++++++++

Answer = 20

++++++++++ | +++++++++++++++++  
++++++++++ | +++++++++++++++++   
-----------| +++++++++++++++++   
++++++++++ | +++++++++++++++++   
++++++++++ | +++++++++++++++++

Answer = 15

That being said, it is a terribly poorly framed question.

Given the length of your second cut there, I'd say it would take 25 minutes, not 15. — Joe Z., May 22 '13 at 18:28

Timon · Answer 19 · 2013-05-22T06:05:06.763

Well, i have something else in my mind.I know it is not that practical still i want to share my views about the question.

Assuming the board is of 1 meter long and we have to cut it into 3 pieces of equal length i.e. finally each of the pieces should be length of 1/3 meter long (see the picture given in the question). So we have to make two cuts at length 1/3 and at length 2/3. Now note that after cutting it first time(which will take 10 minutes), we will have two piece. One is of 1/3 meter long and the other portion is of 2/3 meter long.Now we have to make a cut to the last portion(which is of 2/3 meter long) and make it in two pieces.

Now the interesting part comes. If we assume the board has a uniform resistance against the saw, then after losing its one third end, board will lose its resistance uniformly (assuming resistance depends on length proportionally). In that case, it will take another $10*(2/3)=6.67$ minutes to get another two pieces.

hence we need total $16.67$ minutes.

I know it is not practical, still...

Since you are cutting across the width of the board, the resistance to the saw is a function of the width of the board, not the length of the board. — Kevin Fegan, Sep 04 '13 at 00:47

score 21 · Answer 20 · answered Jan 04 '15 at 12:01

21

The problem is If she works just as fast, that means that if she cuts with the same speed...

$$ \vec{v} = \frac{\triangle \vec{x}}{\triangle \vec{t}} $$

If Marie saw a board into 2 pieces in 10 minutes, that means 1 cut in 10 minutes ($ \vec{v} = \frac{1}{10} $ cuts per minute ).

So to perform 2 cuts to obtain 3 pieces, we have:

$$ \triangle \vec{t} = \frac{\triangle \vec{x}}{\vec{v}} = \frac{2}{\left (\frac{1}{10} \right )} = 2\times 10 = 20 $$

answered Jan 04 '15 at 12:01

saclyr

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In the absence of any indication of how she made the cut, I think you have to assume that *just as fast* is to be measured in cuts per unit time rather than speed, or even velocity. (By the way, I've heard of *the arrow of time*, but shouldn't $t$ be a scalar?) – Martin Rattigan Feb 28 '17 at 04:59
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The cutting speed is not a distance speed (perhaps proportional to one). Speed is not a vector anyway, it is a scalar. And you have a time vector? In the denominator? It's rare to see someone know how to use a language like Latex so well with so little respect for the conventions of the symbols. You can define vector division and such any way you like and I will happily accept it, but to offer it as if it should be understood is an abuse of notation. https://en.m.wikipedia.org/wiki/Abuse_of_notation?wprov=sfla1 – electronpusher Apr 22 '17 at 02:36

score 21 · Answer 21 · answered Feb 28 '17 at 04:02

21

Ten minutes. $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $

answered Feb 28 '17 at 04:02

Martin Rattigan

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Cryptic. $\_\_\_$ – pjs36 Feb 28 '17 at 04:28
@pjs36. The dotted line is meant to be one sawcut = 10 mins. – Martin Rattigan Feb 28 '17 at 04:42
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@Leucippus. Why does it not answer the question? There's nothing in the question posed that says the cut must be a straight line (or are we meant to teach children to think in straight lines?). – Martin Rattigan Feb 28 '17 at 05:14
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+1 for creativity. But we must be sure the path length of your cut is the same as that of the (presumably linear) reference cut taking 10 minutes. – electronpusher Apr 22 '17 at 02:40
True, but if the time is taken to be proportional to the length of the cut there is no definite answer, as noted by AndSoYouCode above. In the cut shown both the start and end points and can be as close as you please to the bottom left hand corner and the loop as close as you please to zero (saw permitting), so the length of the cut can be any positive number. – Martin Rattigan Apr 24 '17 at 10:20

score 20 · Answer 22 · answered Jan 04 '15 at 11:16

Both the teacher and the student are right. It depends on how you look at the problem. We have the rate: 10 minutes to saw 2 pieces.

First way to understand the problem (from the student's perspective): We have a board. We make one cut and get 2 pieces (the sawed off part and the remaining part). From this point of view, every cut yields 2 pieces. So, first cut (takes 10 minutes) and we get 2 pieces. The second cut (takes another 10 minutes - same rate) and we get 3 pieces Total time = 10 + 10 = 20 minutes

Second way to understand the problem (from the teacher's perspective): We have a board. We make one cut and get 1 piece (the sawed off part. The remaining piece of the board is not counted) From this point of view, every cut yields only 1 piece. So, the rate is 10/2 = 5 minutes/1 piece So, first cut (5 minutes), we get 1st piece Second cut (5 minutes), we get 2nd piece Third cut (5 minutes), we get 3rd piece Total time = 5 + 5 + 5 = 15 minutes

Seraphim · Answer 23 · 2015-02-28T20:28:05.283

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Teachers method: 10 minutes per 2 pieces, hence 5 minutes per 1 piece 3 pieces implies 5(3) = 15.

Common sense: 10 minutes for 1 cut of a board (which makes 2 pieces) Therefore 3 pieces requires 2 cuts hence 2(10)=20

Too much missing info,I just asked my boss this question and he said that the answer would make sense if you were slicing a square board in half and that took 10 minutes, then slicing one half in half would make it 3 pieces in half the time as the original cut since it is half the size... 10 + 10/2 = 15

edited Feb 28 '15 at 20:28

answered Feb 28 '15 at 20:18

Seraphim

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This shows how weak an appeal to "common sense" is. No way should the listener have to know what the speaker has assumed to be true or assumed to be false in arguing that something is "common sense". – Rosie F Mar 25 '18 at 08:45

score 18 · Answer 24 · answered Sep 20 '13 at 22:16

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Perhaps if it was "can marie saw the board into 3 pieces in 10 minutes?" then it would be correct. Maybe it was a misprint.

answered Sep 20 '13 at 22:16

Don Larynx

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score 15 · Answer 25 · edited May 05 '22 at 10:31

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Simple question, simple answer:

Think about it, how many cuts do you need to make 2 pieces?

======|======

One cut.

How many cuts do you need to make 3 pieces?

====|====|====

Two cuts.

So $10$ minutes for the single cut means $2\times10=20$ minutes for the double cut.

I'm surprised at your teacher. I think you should seriously hunt them down and duke it out verbally.

edited May 05 '22 at 10:31

Cheese Cake

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answered Nov 04 '15 at 15:24

Daniel

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All the answers there seems to point already on this? – nicael Nov 08 '15 at 19:54
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I needed to vent. – Daniel Nov 09 '15 at 01:05

score 14 · Answer 26 · answered Feb 08 '15 at 03:15

Well, it took her ten minutes to cut the board into two pieces. To cut something into two pieces, you have to slice it only one time. Also, the number of pieces needs to be one more than the number of cuts if it's cut evenly enough. This means to get three pieces, you need two cuts, so $10\times2=20$. Remember that she works at the same speed.

score 13 · Answer 27 · answered Feb 08 '15 at 03:58

The student gave the correct answer since it takes 2 cuts to make 3 pieces-each cut using up 10 minutes!

That's hilarious,the reason the student got the answer "right" and the teacher got it "wrong" is because the student actually thought through the question from first principles-realizing 2 cuts,each eating up 10 minutes,would be needed-and the teacher,used to thinking in "tricks", assumed an equal amount of time would be needed for each piece rather then each cut!

I'm rather ashamed to say I thought that was the answer,too-until I tossed out all my preconceptions and looked at it from scratch using only what I was given-as the student clearly and correctly did!And this is really the essence of correct mathematical thinking,which the student has and the teacher has obviously lost laboring with inferior educational methodology: The correct approach to any mathematical problem is to begin from first principles knowing only what is given. The power of mathematics is to produce a method of solution to a problem where none existed before.Sadly,most of our school mathematics-especially in America-is designed to produce purely practical thinking with spoon fed algorithms-and original thinking is not only discouraged, but indirectly punished.

score 9 · Answer 28 · answered May 27 '16 at 09:58

Well this was a incredibly difficult question considering the teacher got it wrong!

Indeed it seems like this question is slightly ambiguous, however the student's answer is definitely correct!

Answer:

$1$ cut from the saw take 10 minutes, as $1$ cut from a saw should cut a board into $2$ pieces.

In order to cut another board into $3$ pieces however, $2$ cuts are needed.

As $1$ cut takes $10$ minutes, $2$ cuts will take $10 \times 2 =20$ minutes!

$\mathcal Therefore,$ the student is correct with an answer of 20 minutes!

Dan Kowalczyk · Answer 29 · 2018-02-07T19:49:42.993

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Assume the board is flexible. Fold the board in half and cut. It takes 10 minutes.

edited Feb 07 '18 at 19:49

answered Jan 24 '18 at 08:10

Dan Kowalczyk

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Rosie F · Answer 30 · 2022-05-05T10:39:44.580

Something is wrong, certainly. Several answers have debated the question of whether it was the student or the teacher who was wrong. I see another possibility: the question is wrong.

The student's been taught, in a maths lesson, about proportions or "rule of three" or something similar. So it's natural for the student to assume that the question's purpose is to test the student's ability to use that technique. It also makes the student assume that the question is sound, and that the student is not responsible for looking for errors in it and dealing with them (if there are any). So even if the student spots the out-by-one error, they might assume that that is merely an error in the way the question was presented, and then go on to answer the proportions question that they think was intended.

"$15$ minutes" and "$20$ minutes" are both reasonable responses to a question which was unsound in what the circumstances seem to be. The correct thing would've been to present the underlying proportions question in a different and clearer way.

And what way do you believe would have been best appropriate in presenting the question, considering how this question was for third graders? — Mr Pie, Mar 25 '18 at 09:06
Some way that doesn't involve an out-by-one error. So, not in terms of cutting a long thing into pieces, or of making a fence. — Rosie F, Mar 25 '18 at 10:01
Downvoter, please would you care to say why you downvoted? Note that I wrote "I see another possibility" and did not mean to imply that this must've been the truth. — Rosie F, Jan 08 '20 at 12:07

Captain Morgan · Answer 31 · 2018-05-14T09:28:48.207

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What's critical is the action at hand. The rate should not be of making pieces, but cuts. This is what's shifting the rate. A cut in 10 minutes mean 2 cuts in 20 minutes, which is necessary for 3 pieces.

edited May 14 '18 at 09:28

answered May 11 '18 at 21:03

Captain Morgan

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score 1 · Answer 32 · answered Jan 05 '20 at 16:13

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You can cut the board in less than a second because assuming you can cut out 2 extremely small pieces of the board and if she works at the same rate this should take less than a second.

answered Jan 05 '20 at 16:13

Ekadh Singh - Reinstate Monica

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How long will it take Marie to saw another board into 3 pieces?

32 Answers32

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