How to visualize division as splitting Dividend into B equal “partial groups”, then rounding up A partial groups to get a full group?
Because $10 \div \dfrac{4}{3}$ isn't an integer, I changed the numbers in this Reddit post. My $X = 6, A = 2, B = 3$. Undoubtedly, I know $6 \div \dfrac{2}{3} = 6 \times \dfrac{3}{2} = 9$, but don't use $\dfrac{a}{b} \div \dfrac{c}{d} \equiv \dfrac{a}{b} \times \dfrac{d}{c}$ here to explain.
But you could replace 6, 3, and 4 with any numbers X, A, and B. If you do that, the same sort of logic holds. $X \times \dfrac{A}{B}$ means taking a full group of X items and splitting it apart into B equal parts but only taking A of these parts as your final amount that you have. Meanwhile, $X \div \dfrac{A}{B}$ means starting out with a total of X items, splitting it up into B equal "partial groups" (where a full group is actually A of these partial groups), and then rounding up A partial groups to get a full group. Again, the actions you do are the same in both cases  you start with X, split it into B equal groups, and take A of these groups as your final answer.
I don't understand the bolded sentence. Can you please picture all this?

What exactly are my B (= 3) equal "partial groups" here?

What exactly is my full group that's "a full group is actually A [= 2] of these partial groups"?

How do I round "up A [= 2] partial groups to get a full group"?
1 comment
You've typoed the bolded sentence. It is $X \times (B/A)$, not $X \times (A/B)$. If you misread it this way, that might explain your confusion. Also, as with your other question, here "rounding up" seems to be being used in the sense of "rounding up cattle" as opposed to rounding up a number to an integer. Its a pretty poor choice of wording in this context. — Derek Elkins 18 days ago