Who Are We to Question Why, Just Invert and Multiply!

At the end of last week's class, our instructor posed an interesting question...  She was talking about dividing fractions and, of course, we all know the rule to divide fractions is "invert and multiply".  That is, you keep the first fraction the same, invert (or flip) the second one, and change it from division to multiplication.  Like this:

(https://www.coolmath4kids.com/math-help/fractions/dividing-fractions)

But then she posed the question..."WHY?"  Why do we invert and multiply?  And none of us knew the answer right off the bat!  We were likely all thinking, "Because that's how it's done."  I mean, as the title of the blog says, "Who are we to question why, just invert and multiply!" Seriously though,    speaking for myself, I have never even really thought about why we invert and multiply to divide fractions.  So I decided to do some research and find out.  There is a wonderful page I found at Mike's Math Club (http://www.mikesmathclub.org/div_fractions.pdf) which explains it perfectly.  The images below are from that PDF.

Firstly, it's worth the question... If we can multiply fractions by going straight across, why can't we just divide fractions using the same method.  And the truth is...you can!  In fact, if the fractions have a common denominator, then it's a very efficient method!  For example:

And even if there isn't a common denominator, if the two numerators and two denominators divide nicely, then it's still faster to just divide straight across:

So, now then, what happens if the two numerators and two denominators don't divide nicely?  Well, you have two options:

1.  You could manipulate them to have common denominators, then use the method above.

2.  You could still divide them straight across using the following method.  Oh, and by the way, an "identity element" is something that, for a particular operation (like multiplication, division, etc), returns exactly the same answer that was there to begin with.  So the identity element for addition is 0; for multiplication, it's 1.

So there you have it! 

(http://www.funnyism.com/i/funnypics/when-you-finally-understand-a-mathematical-concept)

I truly love finding out the "why" of the simple math tricks we were taught in high school.  I actually felt quite excited when I realized that you totally can divide fractions straight across just like with multiplication!  Honestly, I'd just never really tried - when you're taught to "invert and multiply", or, as my math teacher said, "Keep Times Flip", it's just what you do.  But now I know why it works...and so do you!