 This video will show how to simplify fractions. So fractions can be written A over B, where B cannot be equal to 0. If we divide by 0, it's undefined, so B can't be 0. For a proper fraction, then, we have the numerator that is going to be smaller than the denominator. So that would be like 1 half or 3 fourths, the top number smaller than the bottom number. An improper fraction is just the opposite, where the numerator is going to be greater than the denominator. So that would be something like 5 thirds or 7 fifths. The top number is bigger than the bottom number. And then a mixed fraction is where you have a whole number and a fraction, usually a proper fraction. So it would be something like 1 and 1 half or 5 and 7 eighths, something like that. So we can go from improper to mixed or mixed to improper. If we start with improper to mixed, what you want to do is really just divide. 26, if we did it by long hand, divided by 3, that would go in there 8 times, leaving with 24 down here, and then 24 from 26 would be 2, so this would be 8 and 2, and then you put it over the same denominator because we want to know how many thirds we have. We have 8 whole parts plus 2 thirds left over. So if I take that 8 and 2 thirds, I should be able to get back to 26 over 3 if I go from mixed to improper. So to do that, we're going to take the big number, and we're going to multiply it by the bottom number, and then we are going to add the top number, and then we're really going to put that all over the bottom number because we want to know how many thirds do we have. Just like over here, we had thirds to end up with, 2 thirds. We still are talking about thirds in this problem. So if I do that here, I'm going to take 8, and then times my 3, which gives me 24, and then I'm going to add my 2, which gives me 26, and then we divide it by that bottom number of 3. And there we are at 26 over 3, just like we had when we went from improper to mixed, going from mixed back to improper, we got the same thing. So when we have simplifying fractions, one way to solve them is to prime factor it and then find common factors. So let's take 144 and let's prime factor that. So it goes in 2, and 2 into 14 would be 7, and then that would be 2 into 4 goes twice, so 2 times 72 is 144, and that goes in, that's an even number, so 2 goes into that, and that would be 36. So if we divide that by 2, that will give us 18, and 18 would be 2 times 9, and 9 would be 3 times 3. Here we have 2 times 2 times 2 times 2 times 3 times 3, all on the top. And then if we do 54, and prime factor that one, and that would be 2 times 27, and 27 would be 3 times 9, and then 9 would be 3 times 3. So the prime factors would be 2 times 3 times 3 times 3. What we're looking for is common factors between the top and the bottom. Well here's a common factor, that becomes 1, and then here's a common factor, 3 and 3. One on the top and one on the bottom becomes a factor of 1, and there's one more 3 that they have in common. So that leaves us with 2 times 2 times 2 on the top, and 3 on the bottom, or 8 over 3. Another way we can simplify fractions is just to think about common factors. We don't necessarily have to find the prime factors, but what about the common factor? So I'm looking at 12 and 15, and I'm saying I know they both have 3 as a factor. 12 is 3 times 4, and 15 is 3 times 5. So here I see my factor of 1, which leaves me with 4 over 5 as my reduced or simplified fraction. Well what happens when we have something really large like this? Well when you end up with a bunch of zeros in your numbers that are all lined up, then the ones are going to cancel, the 10s are going to cancel, the 100s are going to cancel, and it's really like if we had divided each of these by 1000, that would give me 110, 110,000, if you want to see it, 110,000. Divide it by 1000, give me 110, and same thing here, if I divide off the 1000, those three zeros are going to drop off, and I'm going to have 286. So now I have a simplified number, but it's still not completely simplified, so I have to go a little bit further, and I have to find out what they have in common. I could prime factor, or I could think about some common factors, and another way that you actually could do it is to come over into your calculator and take the number that you want to find factors for, so the bottom number is 286, and I can say 286 divided by X. That means that when I look at my table, it's going to, the X and Y columns are going to be my factors, and in 110 divided by X would be the top. So I go look at my table, second graph, and these are the one factor, and here's another factor, all those decimals don't work for me, but I see I have 11 and 26, and 13 and 22, up here I have 10 and 11, let's go down and see if we have anything else in common, and I know that I have, where did it go? 22 is also common, I knew that to be a fact. So for 286, that's my bottom number, I know I did this backward, but 286 is going to be 22 times 13, and when we did the other one, it was 22 times 5 is 110, and again looking for a common factor of 22 in this case, that leaves us with 5 over 13.