# Dividing by a fraction - why did I make this error?

Earlier I encountered $$\frac{\frac{49}{4}}{3}$$ and parsed it as $49$ divided by $\frac{4}{3}$. Using the knowledge that dividing by fraction is the same as multiplying by its reciprocal, I changed it to $\frac{147}{4}$.

It turns out this was wrong and I should've considered it as $\frac{49}{12}$. I understand the $3$ and the $4$ get multiplied together to get the $12$.

I'm almost embarrassed to ask this question as it seems too basic. Can someone explain why my original thinking didn't work. When you have two division bars, does it become ambiguous?

You have to distinguish $$\frac{\frac{49}{4}}{3}$$ from $$\frac{49}{\frac{4}{3}}$$ Unfortunately, the typeset difference my be subtle, so better think of it as $(49/4)/3$ vs. $49/(4/3)$. Division (like subtraction: $(49-4)-3=42\ne 48=49-(4-3)$, but unlike addition or multiplication) is not associative.
$$\frac{49}{\frac{4}{3}} = \frac{147}{4}$$
$$\frac{\frac{49}{4}}{3} = \frac{49}{12}$$