Sarai Davenport

2022-06-22

Simplifying Fractions involving negative numbers
I want to simplify
$\frac{\frac{7}{-10}×\frac{-15}{6}}{\frac{7}{-19}+\frac{-17}{-8}}$
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.

Turynka2f

A few important facts to know are the following. For all complex numbers $a,b,n$
$\frac{-a}{b}=\frac{a}{-b}=-\left(\frac{a}{b}\right)$
$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$
$\frac{n×a}{n×b}=\frac{a}{b}$
$\frac{a}{b}=a×\frac{1}{b}$
In particular, by the last identity above:
$\frac{a}{\frac{b}{c}}=a×\frac{c}{b}$
Now it follows that:
$\begin{array}{rl}\frac{\frac{7}{-10}×\frac{-15}{6}}{\frac{7}{-19}+\frac{-17}{-8}}& =\frac{\frac{7}{10}×\frac{15}{6}}{\frac{-7}{19}+\frac{17}{8}}\\ & =\frac{\frac{7}{4}}{\frac{-7×8+17×19}{19×8}}\\ & =\frac{\frac{7}{4}}{\frac{267}{152}}\\ & =\frac{7}{4}×\frac{152}{267}\\ & =\frac{1064}{1068}\end{array}$
You can simplify this last expression by yourself if you wish.

Reginald Delacruz

Just simplify the numerator first and then simplify the denominator.
$\frac{7}{-10}×\frac{-15}{6}=\frac{-105}{-60}=\frac{-7}{-4}=\frac{7}{4}$
$\frac{\frac{7}{4}}{\frac{7}{-19}+\frac{-17}{-8}}.$
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:
$\frac{\frac{7}{4}}{\frac{267}{152}}=\frac{7}{4}×\frac{152}{267}=\frac{1064}{1068}=\frac{532}{534}=\dots$
You get the idea.

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