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we have three expressions here this is 2/3 times 7/8 the second expression is 8 7 times 2/3 this third expression is 5 times 2 over 3 times 5 and what I want you to do is pause this video right now and think about which of these expressions is the largest which is 1 which one is in the middle in terms of in terms of value and which one is this smallest and I want you to think about it without actually doing the calculation if you could just look at them and figure out which of these is the largest which of these is the smallest and which of these is in the middle so pause the video now now you might have taken a shot at it and I'll give you a little bit of a hint in case you might in case you had trouble with it all of these involve multiplying something by two-thirds and you see a 2/3 here you see a 2/3 here and it might not be as obvious but you also see a 2/3 here let me rewrite that to make a little bit clearer so this first expression could be re-written as 7 over 8 times 2/3 this second expression here could be written as what's already written as 8 over 7 times 2/3 times 2/3 and then this is the last expression we could write it as in the numerator 5 times 2 5 times 2 and then in the denominator it's over 5 times 3 5 times 3 which is of course the same thing as 5 over 5 times 2/3 times 2/3 so you see all three of these expressions involved something times 2/3 now I'm looking at it this way does it become easier to pick out which of these are the largest which of these are the smallest and which are these are someplace in between I encourage you to pause it again if you haven't thought about it yet so let's visualize each of these expressions by first trying to visualize 2/3 so let's say the height of what I'm drawing right now let's say the height of this bar right over here is 2/3 so this right over here represents 2/3 the height the height here is 2/3 is two thirds so first let's think about what this one on the right here represents this is five over five times two thirds well what's five over five five over five is the same thing as one this is literally just one times two thirds this whole expression is the same thing as one times two thirds or really just two thirds so this the height here two thirds this is the same thing as this thing over here this is going to be equal to this could also be viewed as five times two over three times five which was this first expression right over here now let's think about what these would look like so this is 7/8 times two-thirds so it's less than eight eighths times two-thirds it's less than one times two thirds so we're going to scale two thirds down this is going to be less than two thirds it's going to be seven eighths of two thirds so this one right over here this one right over here would look something like this would look something let me see if I could draw it yeah it would look something like something like this if the yellow height is two thirds then this right over here then this height this height right over here we draw make it clear this height right over here would be seven eighths times two thirds likewise let's look at this one right over here let's look at this one in the middle eight seven times two thirds well eight seven is bigger than seven sevens it's more than one this is more than two thirds this is one and 1/7 times two thirds so it's going to be the same height as two-thirds plus another seventh so it's going to look something it's going to look something like this it's going to look something like this so it's height now we scaled the two thirds up because eight seven is greater than one so this right over here this height is going to be eight over seven times two-thirds so the way that you could have spotted which of these is the largest in which of these is the smallest is to say well what how are they scaling two thirds this one right over here you're essentially multiplying two thirds by one so you're you're or you're just going to get 2/3 you're not scaling it up or you aren't scaling it down this one right over here you're scaling 2/3 down you are multiplying it by something less than one if you multiply it by something less than one then you were going to be scaling it down I should say a positive number or a number between 0 & 1 less than 1 then you're going to be scaling it down so this thing is scaled down it's going to be the smallest and here you're multiplying the 2/3 times the number bigger than 1 by 1 and 1/7 so you're going to scale it up so this expression is the largest 8 7 times 2/3 the smallest is 2/3 times 7/8 and this one right over here is in between