Writing mixed numbers as improper fractions

right 5 and 1/4 as an improper fraction and an improper fraction is just a pure fraction where the numerator is greater than the denominator this right here it's not a pure fraction we have a whole number mixed with the fraction so we call this a mixed number so let's think about what 5 and 1/4 represents and let me rewrite it so if we're talking about 5 and 1/4 5 and 1/4 and you can literally think of this as 5 and 1/4 or 5 plus 1/4 that's what 5 and 1/4 represents so let's think about 5 5 is 5 holes or if you're thinking of pie we could draw literally 5 pies we could draw 5 pie so let me just cut up the pies from the get-go into 4 pieces since we're dealing with fourths so let me just cut up the pies right over here so that's 1 pie right over there let me copy and paste this copy and paste so I have two pies and then I have three pies and then I have four pies and then I have five pies so this is what the five represents five literally represents so let me circle all of this together that is the five part right there that is what five literally represents it represents five whole pies five whole pies now I have cut up the pies into four pieces so you can imagine each piece represents a fourth now how many pieces do I have in these five pies well I have four pieces per pie I have four pieces per pie four pieces per pie four pieces well let me just write it here four pieces per pie times times five pies is equal to 20 20 pieces or another way to think of it since each piece is 1/4 this is also equal to 20 times 1/4 or you could just write this as be equal to 20 fourths so we have five whole pies is equal to 20 24th let me write it like that twenty-fourths or we could write it as we can write it as 20 over 4 I've kind of done the same thing twice so that's what the five pies represent twenty-fourths or 20 pieces where each piece is 1/4 now the 1/4 right here represents literally one more fourth of a pie or one more piece of a pie so let me draw another pie here so let me draw another pie here so that is another pie cut it into 4 pieces but this 1/4 only represents one one of these pieces right this is one of the 4 pieces the 4 the denominator tells us how many pieces the one tells us how many of those pieces were dealing with so it's just this one piece over here that right there is the 1/4 now if we write 5 and 1/4 we just saw that the 5 the 5 right here is 20 fourths so we could write this so we could rewrite this so let me write it like this 5 and 1/4 5 and 1/4 can be re-written as this is the same thing as 5 plus 1/4 5 plus 1/4 which is the same thing as we just saw that 5 whole pie is the same thing as 20 fourths twenty-fourths and to see that these are the same thing you literally just divide 4 into 20 you get 5 and nothing is left over so 5 is the same thing is twenty-fourths and then this plus 1/4 the same thing as plus 1/4 so if I have twenty-fourths and I add one more fourth to it how many fourths do I have well I have 21 I have 21 fourths or another way of thinking about it this 5 is so this right here is that right there is 20 pieces 20 pieces of and you can even count it 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 but the quicker way is to say well we have 5 pies each of them have 4 pieces 5 times 4 is 20 this 1/4 right here represents one piece plus one piece so total we're going to have 21 21 pieces so we have 21 pieces where each piece is 1/4 so you could say we have 21 times 1/4 or 21 fourths pieces of pie however you want to think of it but we've we've solved the problem we're at an improper fraction we've written 5 and 1/4 is an improper fraction now I've gone through great pains to give you the intuition of what 5 and 1/4 means but there is a fairly straightforward process for getting straight to the improper fraction if you have if you have let me color code it so if you have 5 + 1 / 5 + 1 / 4 to convert it into an improper fraction you literally you're going to keep the same denominator so you're going to have the / 4 there but your numerator is going to be your your de not your your numerator of the fraction part before so it's going to be 1 plus your whole number times your denominator so 1 plus or actually let me do it the way I tend to think of it what I do is I take 4 times 5 so let me write that down 4 and I want to color code it 4 times 5 4 times 5 and then to that I add to that I add this numerator so I literally do 4 times 5 plus 1 and which is so this is equal to 4 times 5 is 20 plus 1 is 21 and that's over 4 so it's 21 forts and although this is kind of a fast way to do it we're literally doing the exact same thing that we did here in kind of a slower way we're saying ok 5 holes is the same thing as twenty-fourths so you take 5 and I figure that out times 5 times 4 and then I have one more fourth there so 4 times 5 plus 1 gets 20 one