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The bolded sentence confuses $A$ and $B$, it should be:

**Meanwhile, $X  \div \frac{A}{B}$ means starting out with a total of X items, splitting it up into A equal "partial groups" (where a full group is actually B of these partial groups), and then rounding up B partial groups to get a full group.**

Visualization 1
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First a completely integer example: There are two teams who receive 6 items, how many items does each team get? Clearly one team gets $3 = 6 \div 2$ items.

Visualization 2
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Now to your numbers: Two thirds of a team receive 6 items, how many items does the whole team get? The answer is $9 = 6 \div \frac23$.

More wordy:
A team has $B=3$ members, $A=2$ are in one room and the third is in another room. The boss leaves $X=6$ items in the first room and tells them that every one gets the same number of items. How many items did he give out in total?

1. The "full group" are the items distributed to the team.

2. The "partial groups" are the items each team member recieves (imagine the 6 items in 2 bags of three items each, one bag is a partial group). Since there are three members in the team, it takes $B=3$ partial groups to make the full group.

3.  "Round up" should be read as "gather" rather than in the mathematical rounding sense.


**Meanwhile, $6  \div \frac23$ means starting out with a total** *(for the first room)* **of 6 items, splitting it up into 2 equal "partial groups"** *(splitting it among the two members of the first room)* **(where a full group is actually 3 of these partial groups)** *(where the whole team has actually 3 members)* **, and then rounding up 3 partial groups to get a full group.** *(and then gathering the items for 3 members to get the total number of items distributed to the full team)*.