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let's now think about different ways to represent a mixed number and let's say that our mixed number is 2 and 2 and 1/8 2 and 1/8 actually let's make a little bit more interesting let's make it 2 and 1/4 2 and 1/4 so let's first think about the whole number part the two well the 2 is going to be the 2 is literally two holes you could literally view that if you want right here we've drawn each hole we've cut it up into sections of 8 so it literally is 8 8 so it's so let me just do it like this so the 2 is this whole region right over here that's 1 so this right over here is 1 and then this right over here is 2 2 wholes let me paint that in so that is 2 wholes and then I have 1/4 so this last piece this last hold is divided into 8 sections so let me divide it into fourths first so lets 1/4 2/4 and 3/4 so we want one of those four to be filled in one of those 4 in orange so one of those 4 to be filled in so just like that you might notice that I filled in two of the eight so that's because 1/4 and two eighths is the same thing so there I've represented this mixed number 2 and 1/4 let's see how we can decompose this so let's get our grid zip back so how else could we do it and I'm just going to throw a bunch of fractions up there and see what I get see what I get so what the first thing I'm going to throw out is 1/2 so let's say that I want to throw out 1/2 so how would I represent 1/2 here well if I take one of these wholes and I put it into two sections right over here 1/2 would be this section right over there so let me color that in so we have 1/2 so I'm first going to add 1/2 which is the same thing as 4 eighths and you see that I just filled in 4 out of the 8 sections which is exactly 1/2 this of this first hole so we're making some progress now let's throw in now let's throw in I don't know let's throw in 3/8 so what would 3/8 look like well 3/8 that's going to be these are literally each of these boxes are literally 1/8 and I could fill it in however I want but let me just put this is 1 2 & 3 and then let's fill in let's fill in plus I don't know plus another 8 8 8 8 now what's 8 8 well 8 8 is a hole and I'll do that over here I still haven't filled this one in yet but I'll fill in this one right over here so let's do that so 8/8 so that's one eight two eight three eight four eight five eight six eight seven eight and eight eighths and it's a hole so I have a hole hole here so that's eight eighths I want to make this one a hole because I want to get to two so let me put in a 1/8 there so plus 1/8 so plus 1/8 well that's going to be this one right over here so that's my 1/8 and then let's add another I don't know let's add another two eighths plus plus another two eighths plus two eighths well this is 1/8 right over here so two eighths is going to be two of these and notice you see that the two eighths is the same thing as 1/4 if you took each of these if you took this 1/4 and and split it into two so you have two two times as many pieces it becomes two ways and you see that if 1 times 2 is 2 4 times 2 is 8 so that 1/4 is the same thing as two eighths you see the eight eighths is the same thing as a hole and you can make another hole out of one half plus 3/8 plus 1/8 and they they add up to a hole and just to make sense of why that worked this is 1/2 could be written 1/2 is as for eights because you see that we filled in the for eight then you have three eights and then you have one eight one eight and if you add all of these together for eight plus three eighths plus one-eighth you will going to you are going to get in terms of eight four eighths plus 3/8 plus one-eighth is going to be eight eight four plus three plus 1 is 8 so you get 8 8 which is this entire whole so hopefully that helps give you a visual understanding of what we're doing when we're adding and decomposing these fractions a little bit more