# How do you subtract two rational numbers with different denominators?

Aug 3, 2018

Generic explanation given using algebra. Followed by a numeric example.

#### Explanation:

$\textcolor{b l u e}{\text{Initial concept}}$
A fractions construct is such that we have:

$\left(\text{numerator")/("denominator") -> ("count")/("size indicator of what is being counted}\right)$

This may be further considered as:

" count" color(white)("d")ubrace(" of ")color(white)("ddd")1/("size indicator of what is being counted")
$\textcolor{w h i t e}{\text{ddddddd.}} \downarrow$

"count "color(white)("d.")xxcolor(white)("ddd")ubrace(1/("size indicator of what is being counted"))
$\textcolor{w h i t e}{\text{dddddddddddddddddddddddddd.}} \downarrow$

$\text{count "color(white)("d.")xxcolor(white)("dddddddddd")" unit of measurment}$

$\textcolor{red}{\text{To directly add counts the units of measurement must be the same}}$
$\textcolor{g r e e n}{\text{To directly add numerators the denominators must be the same}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Answering the question}}$

Let the first rational number be $\frac{a}{b}$
Let the second rational number be $\frac{c}{g}$

Consider the context:
$\frac{a}{b} - \frac{c}{g} \leftarrow \text{ denominators are not the same}$

I opted for $g$ instead of $d$ as its difference in look to $b$ if very obvious. $b \mathmr{and} d$ are easily confused with each other.

Multiply by 1 and you do not change the actual value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\frac{a}{b} - \frac{c}{g} \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} \left[\frac{a}{b} \textcolor{red}{\times 1}\right] - \left[\frac{c}{g} \textcolor{red}{\times 1}\right]}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddd")->color(white)("dddd}} \left[\frac{a}{b} \textcolor{red}{\times \frac{g}{g}}\right] - \left[\frac{c}{g} \textcolor{red}{\times \frac{b}{b}}\right]}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddd")->color(white)("dddddd")[(ag)/(bg)]color(white)("d")-color(white)("d}} \left[\frac{b c}{b g}\right]}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddd")->color(white)("ddddddddd}} \frac{a g - b c}{b g}}$

So basically you change the denominators to the same value and then directly subtract the suitably adjusted numerators.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Example}}$

$\frac{3}{5} - \frac{7}{10}$

$\left[\frac{3}{5} \times \frac{2}{2}\right] - \frac{7}{10}$

$\left[\frac{3 \times 2}{5 \times 2}\right] - \frac{7}{10}$

$\textcolor{g r e e n}{\frac{6}{10} \textcolor{red}{- \frac{7}{10}}}$

But $\textcolor{red}{- \frac{7}{10}}$ may be split into $\textcolor{red}{- \frac{6}{10} - \frac{1}{10}}$ giving:

$\textcolor{g r e e n}{\underbrace{\frac{6}{10} \textcolor{red}{- \frac{6}{10}}} \textcolor{red}{- \frac{1}{10}}}$
$\textcolor{w h i t e}{\text{d.d}} \downarrow$
$\textcolor{w h i t e}{\text{ddd")0color(white)("dd}} - \frac{1}{10}$

So $\frac{3}{5} - \frac{7}{10} = - \frac{1}{10}$