Precedence with division and multiplication matters. For example:
$$ (80 \div 10) \times 5 = 40\\ 80 \div (10 \times 5) = \frac{8}{5} $$
More generally, I might write:
$$ (A \div B) \times C \neq A \div (B \times C) $$
(Although sometimes that equation holds.) But it does seem like I can move around operands and their operations, for example:
$$ 5600 \div 8 = (56 \times 100) \div 8 = (56 \div 8) \times 100 = 7 \times 100 = 700 $$
Which makes me wonder:
$$ (A \div B) \times C = (A \times C) \div B $$
Why does one seem to work and the other doesn't? I'd really appreciate a small visual demonstration if possible.
Fact 1: Division by $b$ is the same as multiplication by $\frac 1b$: $a\div b = a\times \frac 1b$.
Fact 2: You can change the order of multiplication: $a\times b = b\times a$.
Fact 3: You can move the parentheses around however you like if all of your operations are multiplication: $((a\times b)\times c)\times d = a\times ((b\times c)\times d) = \cdots$.
So: $$\begin{align}(A \div B) \times C &= \left(A\times \frac 1B\right) \times C \\ &= A\times \left(\frac 1B \times C\right) \\ &= A\times \left(C\times \frac 1B\right) \\ &= (A\times C) \times \frac 1B \\ &= (A \times C) \div B\end{align}$$
To see why $(A \div B) \times C \neq A \div (B \times C)$ we use the same properties as above and compare the left (LHS) and right hand sides (RHS).
LHS: $$(A \div B) \times C = \left(A\times \frac 1B\right)\times C = A\times \left(\frac 1B\times C\right)$$
RHS: $$A \div (B \times C) = A \times \left(\frac{1}{B\times C}\right)$$
These are clearly different.