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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.

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  • $\begingroup$ Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, on this site we use MathJaX to format our maths. Here you can find a basic tutorial. $\endgroup$
    – gebruiker
    Commented Mar 22, 2016 at 10:34
  • $\begingroup$ Especially. What do you know about simplifying fractions? How would you apply it here? What part is it that makes you stumped? $\endgroup$
    – Ove Ahlman
    Commented Mar 22, 2016 at 10:35
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    $\begingroup$ I have made an edit to your question. Please check that I did not unintentionally changed what you were trying to asked. $\endgroup$
    – gebruiker
    Commented Mar 22, 2016 at 10:38
  • $\begingroup$ Note that for any integer $n$ we have $\frac{a}{b}=\frac{na}{nb}$. So a good start would be to multiply top and bottom by $760 = 8\times 5\times 19$ $\endgroup$
    – almagest
    Commented Mar 22, 2016 at 10:38
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    $\begingroup$ Although this question is fine at its core, I am voting to close, because we really cannot help you, until you indicate what you do and what you do not yet understand about (simplifying) fractions. $\endgroup$
    – gebruiker
    Commented Mar 22, 2016 at 11:02

2 Answers 2

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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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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.

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