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we're told to look at the rational numbers below order them from least to greatest they really didn't have to tell us this first sentence I would have known to look at the rational numbers to order them from least to greatest well anyway they tell us 1/2 negative 5 3 point 3 0 21 / 12 negative 5 point 5 & 2 & 1 eighths so the easiest way to visualize this might just be to make a number line that's long enough that it actually can contain all of these numbers and then we could think about how we can compare them so let me just draw a huge number line over here so take up almost the entire screen I'll stick let's see we have some negative numbers here we go as low as negative 5.5 and we have some positive numbers here looks like we go as high as three point three this thing is still a little less than two so we go about as high as three point three so I can put I could safely I think put zero right here in the middle I can go a little bit to the right since we have our negative numbers go more negative so zero and let's make this negative one negative two negative three negative four negative five and well that should be enough negative five and then in the positive direction we have one two three in the positive direction and let's see if we can plot these so to start off to start off let's look at one half where does one half sit so it sits let me actually let me make the scale a little bit a little bit better so this is one two and three and four all right so let's start with one half one half is directly in between zero and one it is half of a whole this right here would be one whole this would be one whole let me label that this over here is one so one half is directly between zero and one so one half is going to sit right over here so that is that is let me write that a little bit bigger you probably have trouble reading that all right one over two which is also 0.5 so this is also 0.5 anyway that's where it sits then we have negative five negative five well this is negative 1 negative 2 negative 3 negative 4 negative 5 negative 5 sits right over there and then we have 3 point 3 positive 3 point 2 3 I'll do that in blue positive 3 point 3 so this is 1 2 3 and then we want to do another point 3 so point 3 is about 1/3 of the way a little less than a third of the way it'd be 3 point 3 3 3 3 3 3 forever if it was a third of the way so 1/3 of the way that looks like about right over here this is 3 this right over here would be 3 point 3 let me label what I'm going to do is I'm going to label the numbers on the number line up here so it's 1 2 3 4 this is 0 negative 1 negative 2 negative 3 negative 4 negative 5 and so on and so forth and then we get to 0 which is one of the numbers that we've already written down 0 is obviously right over there on the number line so I'll just write this 0 and orange to make it clear to this 0 then we have 21 over 12 which is an improper fraction and to think about where we should place that on the number line to think about where to place it on the number line let me do this in this blue color to think about where to place it on the number line let's change it into a mixed number makes it a little bit easier to visualize at least for my brain so 12 goes into 21 well it goes into it one time 1 times 12 is 12 if you subtract you get a remainder of what we could actually simply could regroup here or borrow if you won't don't want to do this in your head you would get 9 but let's do it let's do this so if we borrow 1 from the two the 2 becomes a 1 this becomes 11 or we're really regrouping a 10 anyway 11 minus 2 is 9 1 minus 1 is 0 so we have a remainder of 9 so this thing written as a mixed number 21 over 12 written as a mixed number is 1 & 9 12 so you get 1 12 12 sinned there and then you get 9 12 left over so 1 in 9 12 so we can also write that actually we could have simplified this right from the get-go because both 21 and 12 are divisible by are divisible by 3 but now we could just we could just divide 9 we can simplify 9 12 divide both the numerator and the denominator by three we then get one and three over four one and three-fourths and just make a clear I could have simplified this right from the get-go 21 divided by 3 is equal to 7 and 12 divided by 3 is equal to 4 so this is the same thing as 7/4 and if you were to if you were to divide 4 into 7 4 goes into 7 one time subtract 1 times 4 is 4 subtract you get a remainder 3 1 and 3/4 so going back to where do we plot this well it is it's 1 and then we have 3/4 so we're going to go 3/4 of the way this is half way this is 1/4 2/4 3/4 would be right over there so this is our 21 over 12 which is the same thing as which is the same thing as 7/4 which is the same thing as 1 and 3/4 and then we have negative 5 point 5 negative 5 point 5 I'll do that in magenta again running out of colors negative 5.5 well this is negative 5 so negative five point five is going to be between negative 5 and negative 6 so let me add like negative 6 to our number line right here just to make it clear let me go a little bit further let's say this is negative 6 negative 6 and our number line we'll keep going to smaller values scroll to the left a little bit negative 6 so then if we go to negative five point five it's smack dab in between negative 5 and negative 6 so this is negative five point five all right over there and then finally we have two and one eighths I'll do that in I'll do that in orange again I'll do it in blue 2 and 1 eighths so it's 2 and then 1/8 and so if we want to find the exact place we could divide this into eighths this would be 4 eighths this would be two eighths and that would be 6 eighths and then one eighth would sit right over here so that right over there is two and 1/8 so we've actually plotted as best as we could the exact locations you didn't have to plot the exact locations if you're trying to just order them but it doesn't help it doesn't hurt to see exactly where they sit when we order them so now we've essentially ordered them because we stuck them all on this number line they the order is negative five point five is the smallest then negative 5 then 0 and then positive 1/2 then 21 over 12 then 2 and 1/8 and then 3.3 and we're done