So in tutoring classes such as pre calc -calculus 1,2 I get a lot of confusion on fractions (obviously) however there is one mistake that I see a lot that I just say DON'T do that without a very good explanation.
Here is what I believe to be the root of the problem:
$$ \frac{a}{\frac{b}{c}} $$ we do the following operation $$ \frac{a \cdot\frac{c}{b}}{\frac{b}{c} \cdot \frac{c}{b}} $$ leading to $$ \frac{a \cdot c}{b} $$ all well and good and they get this however then comes the issue for example in using the definition of the derivative to calculate the derivative of $f(x)=\frac{1}{x}$ I see the following procedure all the time:
$$ \lim_{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h} $$ $$ \lim_{h \rightarrow 0} \frac{\frac{h-(x+h)}{(x+h)(x)}}{h} $$ $$ \lim_{h \rightarrow 0} \frac{\frac{-x}{(x+h)(x)}}{h} $$ $$ \lim_{h \rightarrow 0} \frac{\frac{-x}{x^2+hx}}{h} $$ then using the $\frac{a}{\frac{b}{c}}$ idea the flip the fraction $$
$$ \lim_{h \rightarrow 0} \frac{-xh}{x^2+hx} $$ which is of course wrong my question is how to explain this. Is our defintion of fractions a bit vague?
So we start here: $$ \lim_{h \to 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}. $$
The next step is $$ \lim_{h \to 0} \frac{\left(\frac{x-(x+h)}{(x+h)x}\right)}{h}. $$
Already there is an error in the derivation in the question, where $h-(x+h)$ is written instead of $x-(x+h).$ From the corrected expression we proceed to $$ \lim_{h \to 0} \frac{\left(\frac{-h}{(x+h)x}\right)}{h}, $$ which is the same as $$ \lim_{h \to 0} \frac{\left(\frac{-h}{x^2+hx}\right)}{h}. $$
And now we see there is no way to apply the $\frac{a}{\left(\frac{b}{c}\right)} = \frac{ac}{b}$ idea because it is the numerator of the larger fraction, not the denominator, that is also a fraction. What we have instead is $$ \frac{\left(\frac ab\right)}{c} = \frac{a}{bc}, $$ which we apply as follows: $$ \lim_{h \to 0} \frac{\left(\frac{-h}{x^2+hx}\right)}{h} = \lim_{h \to 0} \frac{-h}{(x^2+hx)h} = \lim_{h \to 0} \frac{-1}{x^2+hx}. $$
I would say that to make this not "vague," your students should make it a regular practice to never write a formula like either $\frac{a}{\frac{b}{c}}$ or $\frac{\frac ab}{c},$ where the formula does not explicitly show the parentheses around the fraction that is to be evaluated "inside" the other. No matter what format is used in the book or is shown in lecture, their study notes for homework and exams should always show the parentheses.