Simple fraction question $\frac{5}{x+5}$?

Question

I'm in the Geometry class and it's been a while since I took math, I realized I had tons of gaps in fractions. I was really confused by these two:

$$\frac{5}{x+5}$$ Why doesn't $\frac{5}{5}$ turn into 1, so that we have $\frac{5}{x}$ remaining? Also at $$\frac{x-1}{x^2-1}$$ What am I doing wrong here by canceling the 1's, having x to the neg 1 power, using the reciprocal to get $\frac{1}{x^1}$?

Answer 1

$$\frac{5}{x+5}$$ Why doesn't $\frac{5}{5}$ turn into 1, so that we have $\frac{5}{x}$ remaining?

Because fractions don't work like that. You can split the numerator: $$\color{green}{\frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}}$$ but not the denominator: $$\color{red}{\frac{a}{b+c} \ne \frac{a}{b}+\frac{a}{c}}$$ Try it with a few numerical examples to convince yourself of this.

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Also at $$\frac{x-1}{x^2-1}$$ What am I doing wrong here by canceling the 1's, having x to the neg 1 power, using the reciprocal to get $\frac{1}{x^1}$?

You cannot cancel the 1's because in a fraction, you can only cancel (common) factors, not (common) terms; so: $$\color{green}{\frac{ac}{bc} = \frac{a}{b}}$$ but you can't do: $$\color{red}{\frac{a+c}{b+c} \ne \frac{a}{b}}$$ In fact, 'cancelling' is what we say when we actually, more formally, mean that we are dividing (or multiplying) numerator and denominator by the same (non-zero) number.

Answer 2

I think, more fundamentally, you need to recognise the invisible brackets in fractions.

$$\frac{5}{x+5} = (5) \div (x+5)$$

As BIMDAS says (or whatever order of operations acronym you use), do your brackets before your division. And clearly, you can't divide $5$ by $x+5$. Hence, you can't simplify it like you did.

Answer 3

So your way would result in, say: $$\frac13=\frac{5}{10+5}=\frac5{10}=\frac12.$$