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I'm stuck with the following question which is taken from SAT-like exam:

How many $ 5 \times 5 $ ceramic tiles are needed to cover rectangular area whose size is $ 10^4 $ units squared (you can't cut the tiles).

The answer that I've got is $400$ (${10^4} \over {5*5}$), however, it is wrong. I don't understand where's the problem. I would appreciate any help.

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The question is not well-worded. Your answer of $400$ is reasonable under certain natural assumptions. But if the rectangle has one side equal to $1$ unit, or $10\sqrt{2}$ units, then $400$ won't do the job. If the last suggested answer had been "It depends," that choice would have been right. – André Nicolas Dec 12 '12 at 16:52
@AndréNicolas I pasted here the question "as it is". I'm quite sure that the question suggests that the sides are divisible by 5, because you can't cut the tiles and they must completely fill the whole rectangular form. Let's think logically - if the sides hadn't been divisible by 5, then you would have to cut the tiles (as they are 5×5). But we aren't allowed to cut them, so the sides must be divisible by 5. However, then the only answer is 400. So I'm certain that this is a mistake or just a very bad-worded question. – artyh Dec 12 '12 at 17:04
By "cover" one can mean that possibly some of the tiles stick out beyond the rectangle. That in fact is the technical meaning of cover. (But that technical meaning cannot be expected to be known to people taking a SAT-level test.) By the way, I am replying "blind" since the link, if there was one, doesn't work. – André Nicolas Dec 12 '12 at 17:05
@AndréNicolas Well, it could be. However there is no such answer as "depends on shape", so it could be that the question is poorly-worded or there is some mistake in the book. Anyway, thank you for your suggestions. – artyh Dec 12 '12 at 17:08

If you mean you got $400$ by dividing $10^4$ by $5*5$ it looks good to me. The only problem I see is if the question meant $(10^4$ units$)^2$, but using "rectangular" seems to suggest it was $10^4$ units$^2$

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Yes, I meant $10^4/25=400$. And yes, it is $units^2$ (like $meters^2$ => area). In this question, there are 4 possible answers: 400, 500, 600, none of them. In the answers list, they say it is "none of them". So it means that the answer is something other than 400. That's why I'm wondering what it could be. – artyh Dec 12 '12 at 16:23
My point on $(10^4$ units$)^2$ was that it could represent a square $10^4$ on a side, when you would need many more squares. But I agree with you that the answer should be $400$ – Ross Millikan Dec 12 '12 at 16:32

It depends. If the sides of the ractangle are divisible by $5$, then $400$ will suffice. If the rectangle is $1\times 10000$, you need $2000$ tiles (well, I could get along with about $1500$, but those won't be parallel to the axis). If the rectangle is $0.001\times 1000000$, the number gets even bigger. Thusthe correct answer in my opionion would have been "It depends on the exact shape of the rectangle". Which is not among 400, 500, 600, so it's indded "none of them".

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I thought about it as well, however it is said that the tile is $ 5 \times 5$, so it means that tile's area is $25$ units$^2$, and according to you, if it is $2000$ the total area would be $25 * 2000$ which is not $10^4$ units$^2$. And they should fit the rectangular shape fully (because you can't cut the tiles). – artyh Dec 12 '12 at 16:42
The hint that you cannot cut tiles should give it away: As soon as the shape of the rectangle would require you to cut tiles for a wasteless cover, you know that a cover without cutting is wasteful, hence may require more tiles than the mere quotient of areas suggests. – Hagen von Eitzen Dec 12 '12 at 18:47

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