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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?

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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.

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  • $\begingroup$ The note about associativity helps, thank you. $\endgroup$ – PeteUK Oct 24 '12 at 9:51
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$$\frac{49}{\frac{4}{3}} = \frac{147}{4}$$

While,

$$\frac{\frac{49}{4}}{3} = \frac{49}{12}$$

I think you just misinterpreted the question...

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