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How do you know the order of operations when dividing by fractions? Recently I've been messing up problems involving dividing fractions. The division is written as fractions over fractions and not as some number divided by a fraction. I was just wondering what the order of operations is when dividing fractions by fractions as $$\frac{\left(\cfrac{3}{1}\right)}{3} = 1$$ isn't the same as $$\frac{3}{\left(\cfrac{1}{3}\right)} = 9$$ Obviously the parentheses control order of operations but what would it be if they weren't there. Do you need to know the context of the equation or is it division left to right. What if there are multiple fractions layered in a vertical format?

Do you base it on the biggest fractions and build off that or is it top to bottom? Will it always specify the width of the lines or could you have fractions with the same width of line for all of them?

I'm sorry if this is confusing but I don't know how to phrase this question without pictures.

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  • $\begingroup$ With no other context, it is left to right. If it's vertical, typically one bar will be larger than the other, with the larger bar being the "outer" fraction. Typically, when in doubt, adding additional parentheses is considered common practice. $\endgroup$
    – user261214
    Sep 21, 2015 at 1:49
  • $\begingroup$ Thanks. This was confusing me because I would have fractions such as (very simplified) 1/((1/x)+(x/1)) where you had to combine denominators on the bottom and I wouldn't know what the order of operations was after combining the fractions in the denominator. This was formatted in a vertical format and not inline like the example I've shown. $\endgroup$
    – Sam Sabin
    Sep 21, 2015 at 1:54
  • $\begingroup$ Good point. Just to clarify, the order of operations would be the dividing of the simplified fraction in the denominator followed by the outer fraction. $\endgroup$
    – user261214
    Sep 21, 2015 at 1:56
  • $\begingroup$ That's what I discovered after I got the answer wrong :/ $\endgroup$
    – Sam Sabin
    Sep 21, 2015 at 1:57

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Always use brackets if you can $$\frac{\cfrac{3}{1}}{3} = 1$$ otherwise like Alias mentioned the larger bar is considered to be the outer fraction $$\frac{3}{\cfrac{1}{3}} = 9$$ I just thought I would format it so you could see it more clearly.

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