Order of precedence, multiplication vs. division

Question

Recently I had this doubt about the order of precedence of mathematical operations multiplication and division. Given that we have a simple question like this

80 / 10 * 5

without parenthesis, what should be the answer?

Should it be 40 considering both multiplication and division has the same precedence and they should be operated in a left-to-right manner in this situation?

Or

Should it be 8/5 given that multiplication has precedence over division?

Answer 1

Think of it as $$ 80 \div 10 \cdot 5 $$ Since $\div$ and $\cdot$ have the same precedence. However, depending on the context is might also mean $\frac{80}{10 \cdot 5}$., but that is almost never the case unless you have brackets around $10 \cdot 5$.

Answer 2

Unfortunately, the acronym PEMDAS does not accurately reflect the order of operations as it suggests to those who are unfamiliar with how math actually works that - Multiplication has precedence over Division and that Addition has precedence over Subtraction; neither of which is true. There are really only 4 types of operations, and they are:

1 - Parentheses or brackets - P - used to override the order of precedence of operators 2 - 4. Beginning with the most deeply nested grouping and proceeding left to right inside each nested level. ( 2nd ( 1st ) 3rd )

2 - Exponentiation - E - operations from left to right take precedence over

3 - Multiplication (and Division) - M - operations from left to right take precedence over

4 - Addition (and Subtraction) - A - operations from left to right.

Acronyms are only there to remind of you of something you really should thoroughly understand already. That said, a better acronym to remember would be PEMA. I highly recommend that anyone who doesn't understand why M and D, for example, have the same precedence; to review the reciprocal nature of these operations. Once you thoroughly understand M/D then move on to exponents, negative exponents, and fractional exponents. (And btw, $x - y = x + (-y)$. (See, S is really just A with negative numbers.)
As always, if I made a mistake, please let me know. That's how peer review works.

Answer 3

When you have multiple binary operators written in 'horizontal form' such as in $80\div 10\times 5$, that are of the same precedence, you perform them in left-to-right order. So in this case, $80\div 10\times 5=8\times 5=40$ (and yes, the order does matter in this case and others like it).

An alternative way of thinking about this, is to "treat the division sign like you do a negative sign (it comes with the number to it's right)" So we can think of $80\div 10\times 5$ as $80\times \frac{1}{10}\times 5$, and the we have associativity and commutativity since it's all multiplication.

Answer 4

It's true that multiplication has the same priority as division; as does addition and substraction however in my case in high school ("preparatoria" in Mexico) they told us that multiplication has priority over divition, but i guess it is because its easier to do the multiplication first and then the divisions as long as we are careful, for example in:

80/10*5= ?

giving priority to the multiplication can be misunderstood as : 80/50 = 8/5 as it says above. However how i was thought and got used to, and works is multipling 80*5=400, giving priority to the multiplication, and then divide by 10 = 40. So, we give priority to the multiplication(s) and do the division(s) next. But it's just a convention that works, and trust me, it really helps making it easier in programming computers, where you can find things such as:

x=a/bc/d/ef can be confusing and lead to mistakes if we give the same priority to the division as to the multiplication; which equals: x=acf/b/d/e giving priority to the multiplication is easier to read and also equals acf/(bde)

I once programmed a cientific calculator and it also helps a lot.