What are the rules for when I can assume implicit multiplication?

Question

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?

Answer 1

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 $\frac{27}4$.

Answer 2

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.