<s>[INST] Simplifying Fractions involving negative numbers I want to simplify $$\frac{\frac{7}{-10} \times \frac{-15}{6}}{\frac{7}{-19} + \frac{-17}{-8}}$$
I really don't understand how to do this, or even how to start? 
Negative numbers make it even harder for me to try and simplify it. 
 [/INST] A few important facts to know are the following. For all complex numbers $a,b, n$:
$$ \frac{-a}{b} = \frac{a}{-b} = -\left(\frac{a}{b}\right) $$
$$ \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} $$
$$ \frac{n\times a}{n \times b} = \frac{a}{b} $$
$$ \frac{a}{b} = a \times \frac{1}{b} $$
In particular, by the last identity above:
$$ \frac{a}{\frac{b}{c}} = a \times \frac{c}{b} $$
Now it follows that:
\begin{align}
\frac{\frac{7}{-10} \times \frac{-15}{6}}{\frac{7}{-19} + \frac{-17}{-8}} & = \frac{\frac{7}{10} \times \frac{15}{6}}{\frac{-7}{19} + \frac{17}{8}} \\
& = \frac{\frac{7}{4}}{\frac{-7\times8 + 17\times19}{19\times8}} \\
& = \frac{\frac{7}{4}}{\frac{267}{152}} \\
& = \frac{7}{4} \times \frac{152}{267} \\
& = \frac{1064}{1068}
\end{align}
You can simplify this last expression by yourself if you wish.
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