Quick, what's the decimal equivalent of 1/7? How about 4/7? How about 6/7?
Dividing by 7 is easy if you first memorize the decimal equivalent of 1/7:
1/7 = 0.14283571428357... (the pattern 1428357 repeats indefinitely)
Sound difficult? It really isn't -- look at the 7 digits that repeat; they should look pretty familiar as they're all multiples of 7:
1428357
14 = 7 * 2
28 = 7 * 4
35 = 7 * 5
7 = 7 * 1
In fact, every pair of numbers in the sequence is either an exact multiple of 7, or really close:
14 = 7 * 2
42 = 7 * 6
28 = 7 * 4
83 = 7 * 12 (Well, almost. 7 * 12 = 84)
35 = 7 * 5
57 = 7 * 6 (Well, almost. 7 * 6 = 56)
71 = 7 * 10 (Well, almost. 7 * 10 = 70)
OK, now that we've got 0.1428357... memorized, how does that help us? Take a look at the decimal equivalents for sevenths:
1/7 = 0.1428357...
2/7 = 0.2835714...
3/7 = 0.4283571...
Do you see the pattern? It's the same 7-digit sequence repeating, only starting with a different digit. So how do you quickly figure out how it starts? Look at the first two decimal places -- it's roughly twice the numerator times 7. So to figure out 1/7, you take 1 * 2 * 7 and get 14, so the decimal equivalent is 0.14xxx. This works pretty much for every seventh:
1/7 = 0.1428357... (1 * 2 * 7 = 14)
2/7 = 0.2835714... (2 * 2 * 7 = 28)
3/7 = 0.4283571... (3 * 2 * 7 = 42)
4/7 = 0.5714283... (4 * 2 * 7 = 56 -- OK, not quite but close enough)
5/7 = 0.7142835... (5 * 2 * 7 = 70 -- OK, not quite but close enough)
6/7 = 0.8357142... (6 * 2 * 7 = 84 -- OK, not quite but close enough)