Does $\frac{2.42+15.6}{4.8}$ mean the same as $\frac{(2.42+15.6)}{4.8}$, or is it shorthand for $\frac{2.42}{4.8}+\frac{15.6}{4.8}$?

Question

I have a question here that I feel like I'm overthinking. I understand PEMDAS and because of that I know that if there is something like $$ \frac{(2.42 + 15.6)}{4.8} $$ then I would add 2.42 and 15.6 before dividing by 4.8.

My confusion mostly comes in here. If I see the fraction written without the parentheses $$\frac{2.42 + 15.6}{4.8}$$ does this form of writing imply the parentheses because they are together on top. OR is this shorthand for writing the addition without splitting into two fractions like $$\frac{2.42}{4.8} + \frac{15.6}{4.8}$$

I am inclined to think the latter but I don't want to get this wrong.

Answer 1

As noticed in the comments, all the three ways to write the expression represent the same number, that is

$$\frac{(2.42 + 15.6)}{4.8}=\frac{2.42 + 15.6}{4.8}=\frac{2.42}{4.8} + \frac{15.6}{4.8}$$

In some sense parentheses could suggest to perform first the addition in the numerator and then the fraction but, as noticed, this step is not relevant for the final result.

To have a correct intuition about that let at first try also with simpler numerical expression, as for example

$$\frac{1+1}{2}=\frac 2 2=\frac 1 2 + \frac 12$$

$$\frac{1+2}{3}=\frac 3 3=\frac 1 3 + \frac 23$$

to check the rules you have learned and also at the time you need to recall them.

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Edit

Note also that the preference for one form respect to another might depends by the specific case, for example in this case that one

$$\frac{21+4}{5}=\frac{25}5=5$$

is more concise as notation than

$$\frac{21+4}{5}=\frac{21}{5}+\frac{4}{5}$$

but

$$\frac{49}{12}=\frac{48+1}{12}=\frac{48}{12}+\frac{1}{12}=4+\frac{1}{12}$$

is very nice in the latter form.

Answer 2

The correct answer is given by the comments of Arturo Magidin and Blue.

It was noted in comments that the two proposed interpretations give the same number. Then Arturo Magidin made the comment:

That said, generally if you have a vinculum (the technical name for the horizontal line between numerator and denominator), the expectation is that you will perform all operations in the numerator, and separately all operations in the denominator, and leave the division until the end. So $\dfrac{a+bc}{xy-z}$ would require you to calculate $a+bc$, separately calculate $xy-z$, and then divide the former calculation by the latter.

Blue's comment:

As an earlier comment suggests, it happens that $\dfrac{2.42 + 15.6}{4.8}$ is equivalent to $\dfrac{2.42}{4.8} + \dfrac{15.6}{4.8}$ (as an application of the Distributive Property), but it's important to note explicitly that this is not because the first is "shorthand" for the second.