… the book is solving it by division without [explaining] why …
?\(\;\) If the book does not explain division or how we use it, then maybe your book was written for students who have already learned the meaning of division. In your case, you probably need an introductory book (written for people learning arithmetic).
… what's the concept of division in mathematics? …
At your level, division is about separating a quantity into
equal-sized pieces.
In word problems, there are two ways 500÷2=250 is used.
(1) We separate the quantity 500 into two equal groups. The answer 250 is the
size of each group (i.e., two groups, with 250 in each group).
(2) We separate 500 into groups of size 2. The answer is the
number of groups (i.e., 250 groups, with 2 in each group).
For the question "How many 2 dollars in 500 dollars?", we use interpretation (2) above. That is, we want to know
how many groups will we have, if we separate $500 into groups of size $2. (500÷2=250 means the number of $2 in $500 is 250.)
Here is a
different question, where we would use interpretation (1) above: "How many dollars is one half of $500?" That is, we want to know
how big is each group, if we separate $500 into two groups. (500÷2=250 means half of $500 is $250.)
As a child, did you ever share your candy with friends? I had 21 chocolates in a package, and I wanted to share them equally with Matt, Sean and myself. As I took candies out of the package, I put them in three separate groups (one at a time) and said, "One for Matt, one for Sean and one for me … one for Matt, one for Sean and one for me…" After I had separated all the candies, there were three groups. Each group was the same size (seven chocolates). That's division.
21/3 = 7
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