Let's say I have $3 \div 2 \times 5$. Why can't $2 \times 5$ be the first step in solving it?
Because the result would end up being $3 \div 10 = 0.3$ When the proper result would obviously be $1.5 \times 5 = 7.5$.
I know, for a fact, that the ans is obviously $7.5$. Just want to know why $2 \times 5$ can't be the first step
$3 \div 2 \times 5 = x$
$3 \times 5 = x \times 2$
$x = 15 \div 2$
$x = 7.5$
Note that $$ (3 \div 2) \times 5 \neq 3 \div (2\times 5) $$ and so we need some convention to distinguish between these two expressions. The convention that is commonly adopted is to evaluate expressions left-to-right, meaning that $$ 3\div 2\times 5 $$ is generally interpreted as $$ (3\div2)\times 5 \, . $$ However, it is better to write $$ \frac{3}{2}\times 5 \, , $$ which is completely unambiguous.
Order of operations matters because division and subtraction are not associative. So for example $5-(3-2)=4 \neq (5-3)-2=0$. Division has a similar problem and it is for this reason in higher math we get rid of subtraction and division all together and instead add negatives and multiply by reciprocals. This is because for addition and multiplication associativity does hold, meaning $(a+b)+c=a+(b+c)$ and $(ab)c=a(bc)$.
This problem occurs because the notation we use (infix) cannot distinguish what is the proper order of operations to do, so we had to assign one by fiat, otherwise statements would be ambiguous! There's nothing fancy about the one we chose, any other would be valid.
There's another notation, postfix or reverse polish notation that doesn't need parenthesis or order of operation rules as every expression has a unique unambiguous way of doing it. The way it works is you put the operator first (like +, -, etc.), and then the inputs afterwards. You read from left to right until you get the right number of inputs for whatever the operator currently most recently read requires.
So for example, in RPN:
$$+,* 3,2,4$$ you would read the +, then the *, then the 3 and the 2. Oh look, * has two inputs, so you do $3*2$ and replace it getting $$+,6,4$$ Now you have + followed by two inputs, so you get $$6+4=10$$ If I wanted a different order, I could just change the sequence. For example,
$$+,3,*,2,4$$ the first operator that reaches two inputs is *, with inputs of 2 and 4, so you multiply and get 8. THen the + has two inputs of 3 and 8, giving 11
The downside of RPN is...we didn't grow up learning it
Sometimes you can clear things up by replacing $``\div x"$ with $``\times \frac 1x"$. Or, more generally, by replacing $``\div \frac ab"$ with $``\times \frac ba"$.
If you rewrite $3 \div 2 \times 5$ as $3 \times \frac 12 \times 5$, it should be pretty clear that $2 \times 5$ is a not really there, and at the same time, the order things are done in makes no difference now.
