Confusion about order of operations

Question

The question is simple, why the following is wrong!?

24 $\div$ $\frac{9}{-3}$ = 24 $\div$ 9 $\div$ -3

Answer 1

$24\div\frac{9}{-3}=\frac{24}{\frac{9}{-3}}=24\times\frac{-3}{9}=24\times(-3)\div9$

Answer 2

$a \div \frac{b}{c} = a * \frac{c}{b}$

So $24 \div \frac{9}{-3} = 24 * \frac{-3}{9}$

Answer 3

Division, unlike addition or multiplication, is not associative.

So the result of: $$ a \div b \div c $$

will depend on the order you perform the operations in. Thus in most cases

$$ a \div (b \div c) \ne (a \div b) \div c $$ In your case, I think you have implied:

$$ 24 \div (9 \div -3) = (24 \div 9) \div -3 $$

which is not the case.

Answer 4

Fractions are thought of as being a single object. Here is something a little bit more complicated

$\frac {x^2 - 3x + 2}{x - 1}$

We evaluate everything above the line together and everything below the line together. No parenthesis are written but they are implied.

So,we see the $\frac 9{-3}$ and reduce the fraction before moving along.

$\div$ happens left to right.

Most math beyond middle school gets rid of the $\div$ notation precisely because it is frequently ambiguous.