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I'm an instructor of a College Algebra course. The computer gave the following question, which I saw as ambiguous:

Computer question: Write the corresponding algebraic expression or equation for the verbal statement. Let x represent the unknown number. The quotient of one and five times a number.

The problem is where does one put in the pause in the English language. You could parse this as either "The quotient of one and five" "times a number" which gives you the answer of 1/5 * x, or the way the computer wants, "The quotient of one and" "five times a number", which gives 1/(5x). The computer only accepts the second interpretation, but according to my reading of the English language, either reading should be acceptable.

Now, someone else in a comment section noted an even third possible reading, that the times could modify both the 1 and the 5, giving (1x)/(5x)=1/5 as a third possible reading.

Does anyone have any reasoning/sources on if these are all valid interpretations, and if not, why one is more valid? Otherwise I'm going to contact our vendor and try to get this question removed/changed.

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    It's gibberish to me. – Hot Licks Apr 25 '19 at 22:56
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    It's complete nonsense and completely ambiguous! "The quotient of one and five times a number." is not even a proper sentence (there is no verb) and it's not clear what it's referring to. – TrevorD Apr 25 '19 at 23:08
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    But a "statement" still needs to be in the form of a sentence! If it had a verb in it, it might make more sense. – TrevorD Apr 25 '19 at 23:18
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    It's terrible. The 'question' defines the variable x as the unknown number and then doesn't use the defined variable. It would be marginally better if it said "Let x represent the unknown number then express the quotient of one and five times that number." However the best way to express what the computer accepts would be "...then express the reciprocal of five times that number". That would avoid the jump cut at the end of the definition and remove the cause of the ambiguity. Best of luck when arguing with the software vendor! – BoldBen Apr 26 '19 at 0:40
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    @AndyT, "one and five" could also be 17 pence in the right context (pre-decimal British currency). – Peter Taylor Apr 27 '19 at 6:43
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It's definitely ambiguous, and all three readings you mentioned are possible. I would contact the vendor. But as I'm a computer scientist and native English speaker, below I've given my opinion on what the problem should have said for each equation you listed.

  • "(1/5) * x", I would expect to read "The product of one-fifth and a number."
  • "1/(5x)", I would expect to read exactly what the problem states, "The quotient of one and five times a number."
  • "(1x)/(5x)", I would expect an even more ambiguous sentence: "The quotient of one times a number and five times a number."
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  • (1/5)x is not the quotient of one-fifth and a number, but the product of one-fifth and a number. – Rosie F Apr 26 '19 at 6:34
  • Whoops. You're right, that's what I meant to say. I'll update my answer. – alasher Apr 26 '19 at 13:07
  • Is (1x)/(5x) not merely 1/5 or 0.2? – Mick Apr 29 '19 at 10:05
  • I disagree with the second bullet: the computer question should read, "The quotient of one and the product of five and a number." This makes it clear the quotient is between one and some product, where the product is given by five multiplied by some number. – Clayton Apr 29 '19 at 17:03
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As alasher says, the statement as it stands is ambiguous and there are all those possible readings. Since you mentioned pauses, curiously the ambiguities can all be resolved by indicating the pause(s) with one or more commas:

  1. The quotient of one, and five times a number (must mean 1/(5x))
  2. The quotient of one and five, times a number (must mean 1/5 * x, or x/5)
  3. The quotient of one, and five, times a number (must mean x/5x)

That said, I dislike 3 and feel it would be argued by anyone who didn't write down the right answer.

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I just saw this in Mathematics StackExchange meta (here) and joined this group to point out a simple fix for the ambiguity in "The quotient of one and five times a number". First, I see two types of structural ambiguity -- the scope of the second conjunct of "and" (or equivalently in this situation, the scope of the first factor associated with "times"), and the use of "and" for a non-commutative operation (namely, division). The least invasive fix I can think of is the following:

the quotient of one by five times a number

Of course this doesn't answer the question asked, and as a nod towards that I recommend contacting the vendor with your concerns, which you said you're considering.

Second, the issue with this not being a complete sentence seems to be a non-issue to me, at least if the phrase to be mathematically translated is not presented as such, because the mathematical equivalent is an expression and not a mathematical statement.

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In a mathematical sense there is only one reasonable way to interpret this, as 1/5x. If you want to translate it directly, it would be 1/5*x, and multiplication supersedes division (though in most programming languages it would not).

That said, this is really only the case in writing. By intonation in speech you could easily make either meaning the clearly intended one.

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  • "multiplication supersedes division " False. And "1/5x" is ambiguous as to whether it means "(1/5)x" or "1/(5x)", unless you're taking a particular order of operations as given, which is the very thing being discussed. – Acccumulation May 6 '19 at 21:27
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The mathematical answer is that according to the order of operations, division and multiplication have the same precedence, and when you have operations with the same precedence, they are taken left to right. So it would be "divide one by five, and then multiply the result by x", or (1/5)x.

The common language answer is that if someone wanted to say "(1/5)x", the natural way of doing so would be to say "one fifth of a number", so the fact that they are not doing so implies that they mean 1/(5x).

BTW, it could be made unambiguous by writing it as "The quotient of one and the product of five times a number" or "The quotient of one and the quantity five times a number" for 1/(5x) or "The product of the quotient of one and five times a number" for (1/5)x

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