When to use these fractions properly?

Question

I'm self-taught in mathematics, however, I have missed many basic ideas of arithmetic, where I'll usually use symbolab to help me get through some simplification of equations. although, I'm starting to get annoyed by this, as its impeding my learning. I would really like your support in helping me understand the distinction in fractions likeso:

$$[a] \space \cfrac{x}{1-x\over x}$$ when this could be simplified as either:

$$[1] \space\frac{x}{(1-x)x} $$ or $$[2] \space \frac{x^2}{1-x}$$

Though, I've also used symbolab and it's given me fractions likeso:

$$[b] \space \cfrac{x \over1+x}{1-x\over x}$$

Where it will be split like the first fraction into either (1) or (2). Are there rules that help me decide if its either (1) or (2) given equation (a), and when is it acceptable to use these for equation (b).

Or does it not matter for (b), and that the result should ensue likeso:

$$[b] \space \cfrac{x \over1+x}{1-x\over x} = \frac{x}{1+x}\frac{1-x}{x}$$

Thank you for the support! this should help me from relying on symbolab all the time for this.

Answer 1

$$\frac{x}{\frac{y}{z}}=\frac{(x)}{\left(\frac{y}{z}\right)}=x\left/\frac{y}{z}\right.=\frac{xz}{y}$$

$$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a}{b}\left/\frac{c}{d}\right.=\frac{ad}{bc}$$

In both cases, the middle fraction line is the "longest", so there are implicit brackets around the numerators and the denominators.

Answer 2

A good rule of thumb: to divide by a fraction $\frac{a}{b}$ is the same as to multiply by the reciprocal fraction $\frac{b}{a}$.

Thus:

$$\frac{x}{\frac{1-x}{x}}\left(\text{ which is }x\div\frac{1-x}{x}\right)=x\cdot \frac{x}{1-x}=\frac{x^2}{1-x}$$

as the reciprocal of $\frac{1-x}{x}$ is $\frac{x}{1-x}$.

Make sure, however, that you always write clearly which division is done first, and which division is done next! A very similarly looking expression:

$$\frac{\frac{x}{1-x}}{x}=\frac{x}{1-x}\div x$$

is actually completely different, the order of division is swapped, and the result is different: $\frac{x}{(1-x)x}$.

Hope this helps.

Answer 3

In general, $$\frac{a}{\bigg(\frac bc\bigg)}=\frac{ac}{b}$$ $$\frac{\bigg(\frac ab\bigg)}{c} =\frac{a}{bc}$$ $$\frac{\bigg(\frac ab\bigg)}{\bigg(\frac cd\bigg)}=\frac{ad}{bc}$$

I recommend using simple examples to show you, and recall that dividing by a fraction is the same as multiplying by its reciprocal. I don't know how many times I chucked $\frac{\frac 12}{\frac 35}$ into a calculator to make sure I was right about how to evaluate those fractions

Answer 4

In my opinion, making the reader figure out how to parse an expression is, at best, an aggravating speed bump. Order of operations is for idiotic questions on facebook and computer programmers. The expression $\frac{\frac{x}{1-x}}{x}$ could mean

$$\text{either $\frac{\left( \frac{x}{1-x} \right)}{x}$ or $\frac{x}{\left(\frac{1-x}{x}\right)}$}.$$

If you have to check yourself what an expression means, then some readers are going to have to do the same thing. So try not to do that. I can't say don't do that because there are always exceptions. But you should always strive for clarity and ease of reading.