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?
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What exactly are my B (= 3) equal "partial groups" here?
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What exactly is my full group that's "a full group is actually A [= 2] of these partial groups"?
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How do I round "up A [= 2] partial groups to get a full group"?
1 answer
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
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
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 the boss give out in total?
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The "full group" are the items distributed to the team.
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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.
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"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).
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