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I found a post that mentioned this problem where students get confused when interpreting mixed numbers: Avoiding confusion between mixed fractions and multiplication

If a student ask "then when can I assume there is a multiplication sign between two numbers", what is the correct answer?

Cecilie
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2 Answers2

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I would say "Always except when the first number is an integer and the second number is a fraction":

  • $\frac 34 6 = \frac 34\cdot 6 = \frac 92$ just multiplication.
  • $5(7-3) = 5\cdot(7-3)=20$ just multiplication.
  • $6\frac 34$ could either be $6+\frac34$ or $6\cdot\frac34$ depending on context.

The "mixed fraction" notation ($6\frac34=6+\frac34$) is more common in "everyday life" (e.g. when buying pie, a sign may say "$2\frac12$ pie for only $10$ dollars") whereas "implicit multiplication" notation ($6\frac34=6\cdot\frac34$) is more common in mathematical contexts.

In my opinion, when you write $6\frac34$ and you want it to mean $6\cdot\frac34$, one should just write the dot there to avoid confusion. If you want it to mean $6+\frac34$, make sure you are not in a mathematical context. If you are, write a $+$ there, or write it as $\frac{27}4$.

student91
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The only way I know of to express the mixed number $6 \frac 34$ is $6 \frac 34$, whereas I was never taught to express the product of 6 and $\frac 34$ as such, but rather as either $6\cdot\frac34$ or, earlier in my education, as $6\times\frac34$. Either of these 2 ways of expressing products in this context seems good to me now and since I'm not aware of any other way to express mixed numbers, it seems best to me to stick to these rules I was taught. I realise that in the context of algebraic equations, no sign is used for multiplication, so ab means $a \times b$, but not in the context of mixed (non-algebraic) numbers (at least this is how I was taught) and this makes sense to me for the reason I have already mentioned.

George Tomlinson
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  • They say mixed numbers are never used in Continental European countries, so $6 \frac 34$ would not be understood as a mixed number. – GEdgar Oct 09 '23 at 13:49
  • @GEdgar: If you look at the comments to this answer, you will see that what they say is quite wrong! – TonyK Oct 09 '23 at 13:56
  • Wait what? Mixed numbers are never used in Europe? Are they living in the Dark Ages? Do you mean rather that this particular notation isn't used, but some alternative instead? – George Tomlinson Oct 09 '23 at 13:57
  • Yes, they use the alternative of decimals. – GEdgar Oct 09 '23 at 14:00
  • Oh this is interesting. I don't like the sound of it personally, but "whatever floats your boat" I guess. – George Tomlinson Oct 09 '23 at 14:04
  • It is OK to write "$2.7$ liters of water" in a recipe instead of $2\frac23$. It is OK to write "$2.7$ km" on a road sign instead of $2\frac23$. I am American; My information about lack of mixed numbers (which I found surprising) is from comments here: https://matheducators.stackexchange.com/q/24420/127 – GEdgar Oct 09 '23 at 14:04
  • If one were to use 2.7 instead of $2\frac23$ in Aeronautical Engineering, I imagine one might see more planes dropping out of the sky than usual, because the 2 expressions are not equal. I wouldn't recommend it. One might get away with 2.67, but I still wouldn't recommend it, when it's dead easy to use $2\frac23$ instead. – George Tomlinson Oct 09 '23 at 14:13
  • If you're using a dot or parenthesis to indicate multiplication, why resort to improper fractions instead of mixed? To avoid confusion created by those who don't follow this maxim I suppose. I see where you're coming from, but honestly, in general, if not always, it's going to be clear what is meant from the context, I would think. – George Tomlinson Oct 09 '23 at 14:17
  • @GeorgeTomlinson a bunch of Americans and Brits arguing about notations used in Continental Europe? In the US, isn't there is some principle for road signs like: intelligible to someone with a 5th grade education. So that allows mixed numbers, but not decimals. – GEdgar Oct 09 '23 at 14:25
  • I don't know about the US bro; you tell me. Also I just received the dreaded 'avoid extended chat conversations' warning – George Tomlinson Oct 09 '23 at 14:47