Simplifying Fractions involving negative numbers

Question

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.

Answer 1

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.

Answer 2

Just simplify the numerator first and then simplify the denominator.

$$\frac{7}{-10} \times \frac{-15}{6} = \frac{-105}{-60} = \frac{-7}{-4} = \frac{7}{4}$$

leads us to

$$\frac{\frac{7}{4}}{\frac{7}{-19} + \frac{-17}{-8}}.$$

Very similar principle with the denominator.

$$\frac{7}{-19} + \frac{-17}{-8} = \frac{267}{152}$$

gives us

$$\frac{\frac{7}{4}}{\frac{267}{152}}.$$

To bring this home, rewrite this division of two fractions as a multiplication:

$$\frac{\frac{7}{4}}{\frac{267}{152}} = \frac{7}{4} \times \frac{152}{267} = \frac{1064}{1068} = \frac{532}{534} = \ldots$$

You get the idea.