Understanding division of fractions

let's think about what it means to take 8/3 and divide it by one-third so let me draw a number line here so there is my number line this is 0 this is 1 and this is 2 maybe this is 3 right over here and let me plot 8/3 so to do that I just need to break up my each each hole into thirds so let's see it's one-third two-thirds three-thirds 4/3 5/3 6 thirds 7/3 8/3 so right over here and then of course 9/3 would get us to 3 so this right over here is 8/3 8/3 now one way to think about 8/3 divided by 3 is what if we take this length and we say how many jumps would it take to get there if we're doing it in jumps of 1/3 or essentially we're breaking this up if we were to break up 8/3 into sections of 1/3 how many sections would I have or how many jumps what I have well let's think about that if we're trying to take jumps of 1/3 we're gonna have to go 1 2 3 4 5 6 7 8 jumps so we could view this as we could you let me do this in a different color in this orange so we took these 8 jumps right over here so we could view 8/3 divided by 1/3 as being equal to 8 now why is this actually make sense well when you're dividing things in 2/3 for every hole you're now going to have 3 jumps so whatever value you're trying to get to you're going to have that number times 3 jumps so another way of thinking about it is that 8/3 8/3 divided by divided by 1/3 is the same thing as 8/3 8/3 times 3 now we could either write it like this we could write x 3 like that or if we want to write 3 as a fraction we know that 3 is the same thing three over one and when we already know how to multiply fractions a multiply the numerators eight times three so you have eight let me do that that same color you have eight times three in the numerator now eight times three and then you have three times 1 in the denominator three times three times one in the denominator which would give you 24 thirds which is the same thing as 24 divided by three which once again is equal is equal to is equal to is equal to eight now let's see if this still makes sense instead of dividing by one-third if we were divided by two-thirds so let's think about what 8/3 divided by two-thirds is divided by two-thirds well once again this is like asking the question if we wanted to break up this section from zero to 8/3 into sections of 2/3 or jumps of two-thirds how many sections or how many jumps would I have to make well let's think about it one jump just in a different color we could make one jump that's the same color as my 8/3 we could do one jump my computer is doing something strange we could do one jump two jumps three jumps and four jumps so we see 8/3 divided by 2/3 is equal is equal to four now does this make sense in this world right over here well if we take 8/3 and we do the same thing saying hey look dividing by a fraction is the same thing as multiplying by a reciprocal well let's multiply by three halves let's multiply by the reciprocal of 2/3 so we swap the numerator and the denominator so we multiply it times three halves and then what do we get in the numerator once again we get eight times three which is 24 and in the denominator we get 3 times 2 which is 6 so now we get 24 divided by 6 is equal to four now does it make sense that we got half the answer if you think about the difference between what we did here and what we did here these are almost the same except here we really just didn't divide or you could say you divided by one while here you / - well does that make sense well sure because here you jumped twice as far you jumped twice as far so you had to take half the number of steps and so in the first example you saw why it makes sense to multiply by 3 when you divide by a fraction for every hole you're making 3 jumps so that's why when you divide by this fraction you multiply or when you divide whatever's in the denominator you multiply by it and now when the numerator is greater than 1 every jump you're going twice as far as you did in this first one right over here and so you would have to do half as many jumps hopefully that makes sense it's easy to think about just mechanically how to divide fractions dividing 8/3 divided by 1/3 is the same thing as 8/3 times 3 over 1 or 8/3 divided by 2/3 is the same thing as 8/3 times 3 over 2 but hopefully this this video gives you a little bit more of an intuition of why this is the case