Comparing improper fractions and mixed numbers

I've got pairs of mixed numbers and improper fractions and I want to think about which of the two is larger so one in seven eights 39 over 10 so you could do this in your head you could say 10 goes into 39 I'll even write out 10 goes into 39 3 times 3 times 10 and you want to find the largest number of times 10 goes into this without going over so if you couldn't write a 4 over here because then that would be 40 that would go over 39 3 times 10 is 30 and then you have a remainder of 9 so you could rewrite this expression right over here instead of 39 over 10 you could write it as 30 over 10 plus 9 over 10 and 30 over 10 is just 3 so this is equal to 3 and 9/10 and you could do that in your head you could say 10 goes into 39 3 times and the remainder is 9 you have your 9/10 and that's essentially just doing this in your head so now we can compare and we can literally just look at the whole number part this is 1 and something 1 in 7 eighths and we're comparing it to essentially 3 and nine-tenths 3 and 9/10 is clearly a larger number we have a 3 out here instead of a 1 so we will write less than and the way I always remember it is the opening always faces the larger number the opening is large it faces the larger number and the point is small it always points to the smaller number now let's do this next one four and 7/8 versus 49 over 9 so let's convert this to a mixed number nine goes into 49 five times and 5 times 9 is 45 so the remainder is going to be 4 the remainder is 4 so this is 5 and 4/9 once again we can literally just look at the whole number parts 5 is clearly larger than 4 so once again less than point facing the smaller number opening facing and the larger number now two-and-a-half versus 11 tense 10 goes into 11 only one time and if you care about the remainder it's 1 so it's 1 and 1/10 which is clearly smaller than 2 and 1/2 you just look at the whole number parts 2 is clearly larger than 1 so we want our opening the of our less-than or greater-than sign to face the the larger number so we would write it like this and this is greater than so two and a half is greater than eleven tenths the little point facing the smaller number 5 and 4/9 versus 47 47 7 goes into 40 so let me rewrite this 7 goes into 40 five times and then you're going to have a remainder of 5 because 7 times 5 is 35 you have a remainder of 5 to get to 40 so it's five and five sevens and if that looks like I'm doing some type of Voodoo just remember I'm really just breaking it up I'm just really saying that 40 over seven is the same thing as 35 plus five over seven the largest multiple of seven that is less than this number and this is the same thing as 35 over seven plus five sevens and then this 35 over seven is five and five sevenths is just five sevenths there this one is interesting because we have the same whole number out front on our mixed numbers five versus five so now we actually do have to pay attention to the fractional part of our mixed number we essentially have to compare 4/9 for nine to five sevens and there's a couple of ways to do this you could get them to have the same denominator that's probably the easiest way to do it so you could rewrite so what's the least common multiple of nine and seven they share no factors so really the least common multiple is going to be their product so if we want to rewrite 4/9 we would write 63 in the denominator that's nine times seven if we multiply the denominator by seven we also have to multiply the numerator by seven so that will be 28 now 5/7 we're going to make the denominator 63 we're multiplying the denominator times nine then we have to multiply the numerator times nine as well five times nine is forty-five so here it's easy to see 45 over 63 is clearly larger than 28 over 63 and so we could write this and because the whole number of parts are the same and 5/7 is the same thing as 45 45 60 thirds and 4/9 is the same thing as 2860 thirds we can write it that 5 and 4/9 is less than 40 over 7 another way that you could have thought about 4 ninths versus 5/7 you could have said well how does 4/9 how does 4/9 compared to 4/7 we have the same numerator the denominator here is large in the denominator here but when you have number in the denominator the larger it is a smaller the fraction the smaller the absolute value of the fraction so this right over here is a smaller quantity than four sevens and 4/7 is clearly a smaller quantity than 5/7 so 4/9 is clearly smaller than 5/7 so we would have gotten the same result