I asked this question on another page but haven’t received a response as it’s not as active.
Last night I was fiddling with some equations and admittedly, I made a careless mistake because I was exhausted. However, in doing so, I began to question the process of what I used to understand as "cancelling" in fractions. The equation was:
(6 ÷ 2)/2
I cancelled the divison by 2 with the 2 in the denominator.
After some research, I came to realise that a better term is reduction to 1 as it is a better indication of what we are actually doing when simplifying fractions.
However, I would like to know the reason why we can't do what I did because if a student asks, I would like to see if there is an explanation beyond BODMAS.
Also, I do believe that we can only cancel factors but if a student were to say ((6 ÷ (2)(1))/2, we now have factors that we seem to be able to cancel, but because of the division in the numerator, we cannot. Is there a more articulate and mathematically correct way of putting this?
Looking forward to this discussion!
EDIT: Please explain why this has been downvoted. If there is an issue with the question, please provide some constructive criticism.
An expression of the form $\Large{\frac{a\cdot x}{b\cdot x}}$ can be simplified to $\Large{\frac{a}{b}}$ assuming $x\ne 0$.
@AndreasLenz’s suggestion to simplify nested divisions is a good one.
Often I found it useful to rewrite $\LARGE{\frac{\frac{a}{b}}{\frac{c}{d}}}$ as $\Large{\frac{a\cdot d}{b\cdot c}}$.
I remember this rule as: Dividing by a fraction is the same as multiplying by its reciprocal.
I hope this helps.