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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 ?

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    $\begingroup$ Do you mean that $177\frac{1}{4}\pi$ is not the same thing as $\frac{1}{4}177\pi$ ? $\endgroup$ – Emperor_Udan Sep 29 at 13:05
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    $\begingroup$ No, the former is additive while the latter is multiplicative. $\endgroup$ – poetasis Sep 29 at 13:51
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    $\begingroup$ To me (schooled in 1960's UK), $177\frac14$ means unambiguously $177+\frac14$, just like $1\frac12$ means $1+\frac12$. $\endgroup$ – TonyK Sep 30 at 1:16
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    $\begingroup$ 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 ($\frac{709}{4}\pi$). $\endgroup$ – NotThatGuy Sep 30 at 8:46
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    $\begingroup$ @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. $\endgroup$ – Eric Duminil Sep 30 at 9:57
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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.

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    $\begingroup$ I did not know about mixed fractions, thank you for letting me know. $\endgroup$ – Emperor_Udan Sep 29 at 14:04
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    $\begingroup$ It is mostly taught in elementary school (at least in Canda). We do not use them in upper-years---for this very reason. $\endgroup$ – Shon Verch Sep 29 at 14:13
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    $\begingroup$ I would say that in the US, mixed fractions are still universally used. $\endgroup$ – Nick Matteo Sep 30 at 2:15
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    $\begingroup$ In fact, a fraction with a numerator greater than the denominator was historically called an improper fraction $\endgroup$ – DJohnM Sep 30 at 5:18
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    $\begingroup$ I edit high school mathematics textbooks for the Australian market, and I still see mixed fractions there quite often. $\endgroup$ – Geoffrey Brent Sep 30 at 11:33
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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.

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  • $\begingroup$ Thank you for your answer, I think you're right this must be a misprint of $(177 + \frac{1}{4})\pi$ $\endgroup$ – Emperor_Udan Sep 29 at 12:59
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    $\begingroup$ 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) $\endgroup$ – Henry Sep 29 at 23:38
  • $\begingroup$ @Henry In the original question it was written as $\frac{1}{4}177$ so it was a misprint, but maybe not in the texbook. $\endgroup$ – Elad Sep 30 at 8:21
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    $\begingroup$ @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. $\endgroup$ – Emperor_Udan Oct 1 at 16:05

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