# Order of precedence, multiplication vs. division

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?

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 $10 \cdot 5$.
• Surprisingly I just met this particular case where ÷ have precedence over ⋅ while calculating the taxes I have to pay in France as a freelancer. The formula is in a legal text (legifrance.gouv.fr/eli/decret/2017/3/8/ECFS1700138D/jo/texte/fr) and is Taux = T - 3,50 % × (1 - R/0,7 PSS). I could not make it work until I modified it to Taux = T - 3,50 % × (1 - R/ (0,7 PSS)). – Daishi Aug 22 '18 at 18:44
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.