The first thing you need to keep in mind for fractions is that fractions are division rewritten.
(Sidenote: you scrubs in lower division math got it rough - nobody in higher division even cares about mixed fractions. I'm going to take a moment to point and laugh at you.)
What is a mixed fraction? That's something like
These words are in quotes because the idea is ridiculous and improper fractions are way easier to work with than proper fractions and as a math major I disagree with the classification NOBODY CARES MOVING ON
Improper fractions are fractions where the top (the numerator) is a bigger number than the bottom (the denominator). WHOA WHAT THE FUCK HOLD UP. Denominator? Numerator?
Look guys you gotta know these things, we can't do a vocab lesson every time. (Sigh, but we're going to.) The problem with these names is that they're really fucking easy to confuse with each other, so it helps to have a way to remember what the hell these words actually mean.
If the fraction bar is the surface of the ocean, the numerator is the number of dudes in a boat who are about to get dominated by the greatest great white that ever lived, the Denominator. FRACTIONS ARE JAWS.
Okay that was stupid let's move on. Improper fractions: top bigger than bottom. Proper fractions are the opposite: the denominator is bigger than the numerator.
Proper no longer means anything as a word. I hate this post.
Here is a thing which is true:
Why is this true? It is true because fractions are division, and to explain it I am going to have to bring up the good old standby of long division. DO YOU KNOW HOW TO LONG DIVIDE? BECAUSE SO HELP ME IF I HAVE TO EXPLAIN THAT TOO...
Fine. I can't even find a way to include a long division symbol so I'm just going to write it as a close-parenthesis I HOPE YOU'RE HAPPY.
You're dividing by the number on bottom, so
2 goes into 7 three whole times - that means you can subtract 2 from the original number three times before the number being subtracted from is bigger than 2.
Okay, so what the hell do we do with that 1 that's left over? That's smaller than 2, so you can't just divide into it, but remember what division really is? ~*Fractions*~! You can write the remainder as
The thing about mixed fractions, though, is that they are a pain in the ass to work with inside a problem. This is why people don't use them past Algebra II (you scrubs). So how do you turn a mixed fraction back into an improper fraction so you can actually work with it? You pretty just do what we did backwards. It's not that hard.
Hey guess what's really cool about the number 1? You can multiply it in anywhere and not change a damn thing about the problem's answer! Know what's even cooler? You can write 1 as anything over itself. And remember last post where I said that you can rewrite 3 as
(multiply by 1 in the form of
to get the same denominator)
(now you can add the numerators because the denominators are the same)
OH MAN WASN'T THAT AWESOME? We just worked backwards to get our original improper fraction!
Yeah we did a whole bunch of work to get back to exactly where we started. That's not important. What's important is that this applies to any mixed fraction you will ever meet, and also any case where you need to add a whole number to a fraction. This is how you write whole numbers into fractions with a specific denominator.
But I'm fucking sick of fractions, so we'll talk about lowest common denominators and shit next time.
* Again, codecogs.com is great for providing the equations EXCEPT THE LONG DIVISION ONE THAT I HAD TO MAKE MYSELF I HATE EVERYTHING
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