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December 8th, 2014, 08:02 AM  #1 
Newbie Joined: Mar 2013 Posts: 9 Thanks: 0  algebra to invert fractional equation?
so.. if a/b=c/d then b/a=d/c because you can use each side of the equation do divide one by to invert. how do i represent this as an explicit step in place of just flipping the fractions? would a/b=c/d 1/(a/b)=1/(c/d) b/a=d/c be legit algebra? this seems wrong somehow. 
December 8th, 2014, 08:18 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,142 Thanks: 726 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Yep. Doing "1 over" is a valid mathematics operation on both sides of an equation provided you don't have any zeroes. It's okay to do $\displaystyle x = y$ $\displaystyle \frac{1}{x} = \frac{1}{y}$ (if x and y are nonzero), but not $\displaystyle x+y = 0$ $\displaystyle \frac{1}{x+y} = \frac{1}{0}$ because 1 divided by 0 is undefined. Also, you need to pay attention to the fact that in the first case presented above, x and y must be nonzero. Otherwise you're doing the same thing as the second case. As for your specific example, another way of showing that they are the same is to multiply both sides by $\displaystyle \frac{bd}{ac}$ $\displaystyle \frac{a}{b} = \frac{c}{d}$ $\displaystyle \frac{a}{b} \times \frac{bd}{ac}= \frac{c}{d} \times \frac{bd}{ac}$ $\displaystyle \frac{\cancel{a}}{\cancel{b}} \times \frac{\cancel{b}d}{\cancel{a}c}= \frac{\cancel{c}}{\cancel{d}} \times \frac{b\cancel{d}}{a\cancel{c}}$ $\displaystyle \frac{d}{c} = \frac{b}{a}$ which is okay if a, b, c and d are nonzero. 
December 8th, 2014, 10:32 AM  #3 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361 
Another way of simplifying 1/(a/b) with a*b is nonzero, is as follows: $\displaystyle \frac{1}{\frac{a}{b}} = \frac{1}{\frac{a}{b}} \cdot 1 = \frac{1}{\frac{a}{b}} \cdot \frac{b}{b} = \frac{b}{\frac{a}{b} \cdot b} = \frac{b}{\frac{ab}{b}} = \frac{b}{ \frac{b}{b} \cdot a} = \frac{b}{1 \cdot a} = \frac{b}{a}$ where you might like to leave out some steps. 

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algebra, equation, fractional, invert 
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