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A goat is tied to the corner of a shed 12 feet long and 10 feet wide. If the rope is 15 feet long, over how many square feet can the goat graze ?

I know that this question has already been asked a number of time, but no matter what I do I cannot find the same answer as the one provided in the textbook. I proceed like in this thread so I have :

$\frac{3}{4}15²\pi + \frac{1}{4}3²\pi + \frac{1}{4}5²\pi = \frac{1}{4}709\pi$

However the answer given by the textbook is $177\frac{1}{4}\pi$.

Am I missing something here or is the textbook wrong ?

Emperor_Udan
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    I believe that your answer is the correct one: $177.25\pi$. Perhaps a typo put the fraction in front when it belonged in back of the $177$. – poetasis Sep 29 '19 at 12:51
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    Do you mean that $177\frac{1}{4}\pi$ is not the same thing as $\frac{1}{4}177\pi$ ? – Emperor_Udan Sep 29 '19 at 13:05
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    No, the former is additive while the latter is multiplicative. – poetasis Sep 29 '19 at 13:51
  • That is probably what happened: the textbook used mixed-fractions for some reason... – Shon Verch Sep 29 '19 at 14:11
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    +Shon Verch Yes this is it, the textbook is from the 80's – Emperor_Udan Sep 29 '19 at 14:16
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    To me (schooled in 1960's UK), $177\frac14$ means unambiguously $177+\frac14$, just like $1\frac12$ means $1+\frac12$. – TonyK Sep 30 '19 at 01:16
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    While this may or may not be strictly necessary in this case, I would expect an explicit multiplication symbol ($\frac{1}{4}.709\pi$) or brackets ($\frac{1}{4}(709\pi)$) to denote multiplication of 2 numbers instead of just putting them next to one another. In this case, I would put the second number in the numerator instead ($\frac{709}{4}\pi$). – NotThatGuy Sep 30 '19 at 08:46
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    @TonyK One person isn't enough to determine if a notation is ambiguous or not. $177\frac{1}{4}$ means $\frac{177}{4}$ to me. Since it "unambiguously" means two different things to two different people, it is ambiguous. – Eric Duminil Sep 30 '19 at 09:57
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    @EricDuminil: I agree entirely! – TonyK Sep 30 '19 at 11:14
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    @EricDuminil I don't think it was supposed to mean "this is therefore unambiguous to anyone", but instead "I cannot imagine understanding this in a different way." Which holds for me as well, BTW (schooled in the Czech Repbulic in the 1980s). – Angew is no longer proud of SO Sep 30 '19 at 12:44
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    @EricDuminil I am frankly dubious that if you or anyone were buying $177\frac 1 4$ of some product you'd expect $\frac {177} 4$ of that product. – StephenG - Help Ukraine Sep 30 '19 at 13:07
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    @StephenG: Okay, I suppose it's mostly cultural. I'll try to find products with $1\frac{1}{2}$ of anything, in France or Germany (as opposed to 1.5). I don't remember ever seeing any over here, though. – Eric Duminil Sep 30 '19 at 14:34
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    @EricDuminil Yes, seems likely cultrual. It's more common (IMO) to see e.g. $2.50$ per whatever now than $2\frac 1 2$. I suspect it might also be a shift away from fraction in many settings. May also be to do with the use of metric systems and (supposedly :-)) higher standard of education now. That said I'd still be surprised if e.g. a French person seeing $3\frac 1 2$ expected to pay $1.50$ and not $3.50$ if you see what I mean. – StephenG - Help Ukraine Sep 30 '19 at 15:23
  • @StephenG: On products, I'd expect a slightly different typography between $3$ and $\frac{1}{2}$ to indicate that they're not to be multiplied. For example, the fraction could be written as superscript (which means that the product could cost $\sqrt{3}$ :D), smaller or with some single quotes inbetween. – Eric Duminil Sep 30 '19 at 15:45
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    While it is clearly ambiguous as to what the notation means -- indeed, the above discussion, with different people thinking different things, verifies this! -- when a textbook answer differs so drastically from yours, you should try to think how can their answer be right? what different assumptions can be made?. Given that you've seen their answer is something to do with $177$ and yours is $709/4$, you should really be looking for some relation: namely, $709/4 = 177 + 1/4$. Questioning why the book gave the answer they did is often very helpful for unpacking their notation/assumptions :) – Sam OT Sep 30 '19 at 20:06
  • @EricDuminil Besides the fact TonyK says "To me", I think that "unambiguous" should be understood to mean within the linguistic context. "Gift" might be ambiguous to you as to whether it means "present" or "poison", but within an English-specific context, it is not ambiguous. To my knowledge, using the notation given in the question to indicate mixed fractions is quite standard in the English speaking world. – Acccumulation Sep 30 '19 at 22:09
  • @SamT, looking at this question, and the two possible answers, I did just that. Then I calculated 709/4 = 177.25, and immediately started wondering how the textbook had managed to get the answer wrong twice: first by a factor of 4, and then by dropping the 0.25. – ilkkachu Oct 01 '19 at 01:43
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    @StephenG, nobody buys "177 1/4" of anything. At most, they might buy 177.25 kg (or liters, or whatever) of something, but it's more likely that they'd just buy 180 of whatever units that stuff is sold in. – ilkkachu Oct 01 '19 at 01:47
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    @ilkkachu You're arguing about the specific example when the rest of us are debating the general idea. – StephenG - Help Ukraine Oct 01 '19 at 02:05
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    @StephenG, which one? That first was an anecdote about how an ambiguous expression can be interpreted in different ways by different people. The point of the second was that $177 \frac{1}{4}$ seems significantly different from, say, $2 \frac{1}{2}$ in that it actually makes sense in an everyday context to buy two and a half of something, and so it makes sense to use that ambiguous everyday notation for 2.5. But it doesn't make sense for 177.25, let alone $177.25 \cdot \pi$. Just stop defending an ambiguous notation when the context is mathematics. – ilkkachu Oct 01 '19 at 08:20

2 Answers2

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In business and the trades, at least before everything went to decimal notation for fractions, you would almost never see someone write a number as (for example) $\frac 52.$ Instead they would write $2\frac12,$ which by convention was read as a single number equal to $2+\frac12.$ This notation is called a mixed fraction. It is highly discouraged in most mathematical settings, but you can still see it used sometimes, especially in old puzzle books.

While I was trying not to be U.S.-centric in this answer, I should acknowledge that mixed fractions are still extremely common in the U.S. for many kinds of measurements, and as noted in the comments are seen in some contexts in at least a few other countries.

David K
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    I did not know about mixed fractions, thank you for letting me know. – Emperor_Udan Sep 29 '19 at 14:04
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    It is mostly taught in elementary school (at least in Canda). We do not use them in upper-years---for this very reason. – Shon Verch Sep 29 '19 at 14:13
  • In the US I think it’s still common to see mixed fractions in some non-metric measurements. – David K Sep 29 '19 at 14:25
  • Even more confusingly, you used to sometimes see mixed fractions written with a hyphen, so that $3.5$ would be $3$-$\frac12$. Of course, it was easy to mistake the hyphen for a minus sign... – saulspatz Sep 29 '19 at 14:25
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    I would say that in the US, mixed fractions are still universally used. – Nick Matteo Sep 30 '19 at 02:15
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    In fact, a fraction with a numerator greater than the denominator was historically called an improper fraction – DJohnM Sep 30 '19 at 05:18
  • Also taught in UK schools up until 10/12 yo, discouraged later down the line, at least in formal mathematical settings – Gamora Sep 30 '19 at 11:06
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    I edit high school mathematics textbooks for the Australian market, and I still see mixed fractions there quite often. – G_B Sep 30 '19 at 11:33
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    -1 Because the assertion that mixed fractions are some archaic notation is false and baffling. E.g.: I just searched for a recipe site and clicked on the first recipe I could find, and that alone has two mixed-fraction items in the ingredients list: https://www.allrecipes.com/recipe/163229/scrumptious-frosted-fudgy-brownies/ – Daniel R. Collins Sep 30 '19 at 16:18
  • @DanielR.Collins I did note (although initially, only in a comment) that mixed fractions are still common in the U.S. Regarding recipes, I happen to have on my bookshelf two editions of a dessert cookbook, a French edition and its translation for sale in the U.S. The U.S. edition naturally has mixed fractions throughout; the French edition, not one mixed fraction in 100 recipes. Yet the recipes in each book are for the same desserts. So if I erred, perhaps it was in trying too hard not to be U.S.-centric. I apologize. – David K Sep 30 '19 at 18:52
  • Mixed fraction notation is common in some contexts, but I've never seen it used before juxtaposed with another number without expressly specifying multiplication (e.g., with $117 \frac{1}{4} \times \pi$---which still looks bad but at least is less ambiguous than $117 \frac{1}{4} \pi$) probably in part because of the ambiguity and confusion evident here. – Travis Willse Sep 30 '19 at 21:30
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As far as I can tell you’re answer is fine and the textbook is wrong. Maybe the misprint was $709/4=177 +1/4$. So the answers are typed almost the same.

Elad
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  • Thank you for your answer, I think you're right this must be a misprint of $(177 + \frac{1}{4})\pi$ – Emperor_Udan Sep 29 '19 at 12:59
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    This is clearly the intended meaning.

    Whether it is a misprint rather than ambiguous may be more subjective: it is not uncommon to write $2 +\frac12$ as $2½$ so the writer or publisher may have thought it a small step to writing $\frac{709}{4}$ as $177¼$ and thus $\frac{709}{4}\pi$ as $177¼,\pi$ (which could be even more ambiguous as it is not totally clear whether the $\pi$ is in the numerator or denominator)

     – Henry Sep 29 '19 at 23:38
  • @Henry In the original question it was written as $\frac{1}{4}177$ so it was a misprint, but maybe not in the texbook. – Elad Sep 30 '19 at 08:21
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    @Elad The textbook has $177\frac{1}{4}$, I made a typo when I wrote the OP because I was completely oblivious to the very existence of mixed fractions. – Emperor_Udan Oct 01 '19 at 16:05