Mistake 5 - Simplifying Complex Fractions Incorrectly

The common mistake number five is that students often flip the denominator or multiply by the reciprocal too early. Let me go ahead and show you the mistake that you look at a complex fraction and say, "Well I know that this is 1 over x and I know that I keep that. Then I change the sign to multiplication from a division and then I just flip this last fraction here. So that's going to be x over 2 plus 1. So now I’ve got, of course  - I can cancel out the x's. And then I can cancel out, let’s see, nothing else. So, it looks like 2 plus 1 is 3 so that equals one-third." Be careful of this. This is definitely the error. What happens is that students will often take the reciprocal of this without actually realizing that the time to take the reciprocal is when you have one fraction on top and one fraction in the denominator. Notice, there are two things here. You do have a one fraction here but then there’s also a whole number there, which if you wanted to, you can make into a fraction to say "2 over 1." And so really, there’s two fractions. So, you're going to have to put it as one whole fraction, meaning one denominator not two different denominators (where this would be an x and that would be a 1). So, this is what is the correct way to do. So the correct way is to go ahead and get this into one fraction. So let's simplify it to that. We've got 1 over x divided by (our least common denominator is going to be x), so we have 2x plus 1. And so now, since we have this top part here into one fraction and then this denominator into another fraction, we can go ahead and apply the reciprocal rule. We've got to keep the first one, change the sign to multiplication and then flip the last one (or multiply by the reciprocal). Notice I can cancel out these x's. And what I am left with is simply 1 over 2x plus 1.