Sarai Davenport

2022-06-22

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.

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.

Turynka2f

Beginner2022-06-23Added 17 answers

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:

$\begin{array}{rl}\frac{\frac{7}{-10}\times \frac{-15}{6}}{\frac{7}{-19}+\frac{-17}{-8}}& =\frac{\frac{7}{10}\times \frac{15}{6}}{\frac{-7}{19}+\frac{17}{8}}\\ & =\frac{\frac{7}{4}}{\frac{-7\times 8+17\times 19}{19\times 8}}\\ & =\frac{\frac{7}{4}}{\frac{267}{152}}\\ & =\frac{7}{4}\times \frac{152}{267}\\ & =\frac{1064}{1068}\end{array}$

You can simplify this last expression by yourself if you wish.

$\frac{-a}{b}=\frac{a}{-b}=-\left(\frac{a}{b}\right)$

$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$

$\frac{n\times a}{n\times b}=\frac{a}{b}$

$\frac{a}{b}=a\times \frac{1}{b}$

In particular, by the last identity above:

$\frac{a}{\frac{b}{c}}=a\times \frac{c}{b}$

Now it follows that:

$\begin{array}{rl}\frac{\frac{7}{-10}\times \frac{-15}{6}}{\frac{7}{-19}+\frac{-17}{-8}}& =\frac{\frac{7}{10}\times \frac{15}{6}}{\frac{-7}{19}+\frac{17}{8}}\\ & =\frac{\frac{7}{4}}{\frac{-7\times 8+17\times 19}{19\times 8}}\\ & =\frac{\frac{7}{4}}{\frac{267}{152}}\\ & =\frac{7}{4}\times \frac{152}{267}\\ & =\frac{1064}{1068}\end{array}$

You can simplify this last expression by yourself if you wish.

Reginald Delacruz

Beginner2022-06-24Added 7 answers

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}=\dots $

You get the idea.

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

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}=\dots $

You get the idea.

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