My way of doing long-hand division is a combination of the two ways I was taught, but it makes the most sense to me. For example, for 5/278, I would ask my self how many times 5 goes into 2 which is 0, so I ask myself instead how many times 5 goes into 27 using multiplication tables in my head as a guide if I need to. For this case, I'm already either counting in multiples of 5, or thinking 5 times 5 = 25 which is the closest under 27. I might be asking myself "5 times what, equals the closest number?" or going through all my times tables in my head that are relevant to the number 5 until I find the closest number that is less than or equal to 27. Then, I would write everything down the same way I was taught to write long division. My thinking is the only thing different here.
     Thus, I would write "5" above the line lining it up above the 7. I already figured out that 5 times 5 = 25 which is why I wrote the 5 down to begin with, so then I write "25" underneath 27 and subtract to get "2." I draw an arrow to show the 8 coming down next to the "2" that I just wrote. I write "8" next to the 2 so that I have "28." Then, I either go through the same process I did before but this time with "28" instead of "27," or I just know that the next multiple of 5 is 30, leading me to know that 25 is still the right product (under "28"). Since it is the exact same calculation, I write a "5" above the line with the 8 beneath. After the subtraction of "28-25," "3" is left over. You may stop here with "55 r3" or "55 remainder 3."
     If you continue, you put a decimal point and carry down a zero to make "30." I already figured out that 30 was a multiple of 5. Since 6 times 5 is 30, "6" is what I write above the zero. 30-30=0. "55.6" is the answer in decimal format.