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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.

jds
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    The first fails because you are changing what you divide by. Dividing by B is not the same as dividing by BxC. In the second you are dividing by B in both. The difference is that in one you are dividing by B before multiplying the result by C, and the second you are dividing by B after multiplying by C. It doesn't actually matter what order you multiply and divide by but it does matter what you multiply by. The parenthesis aren't really about order as grouping. – fleablood Nov 29 '16 at 04:09

1 Answers1

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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.

  • I see. So when you change the order of operations without moving the operands, you're potentially changing intermediate values. For example, $(80 \times \frac{1}{10}) \times 5 \neq 80 \times \frac{1}{10 \times 50}$. – jds Nov 29 '16 at 02:34
  • Yep.$\ \ \ \ \ $ –  Nov 29 '16 at 02:35
  • Can you do similar tricks with division? What would the rule be to cover all operands (+, -, , /)? As a guess, I'd say something like "+ and - can be reordered within their own group (parens). can be reordered within a group of * and /. / can be converted to 1/x and then reordered within a group of * and /. / cannot be reordered." I'm not sure that's right, but please correct where necessary. – Whothehellisthat Jun 23 '18 at 08:54