sevenths

The SCSon came down tonight asking how does he convert his fractional answer to a decimal he can input into the homework scoring app...

His answer was 6 and 3/7ths.

I quickly answered "well that's easy... Try 6.428571."

He looked at me as though I'd grown a horn.  (How can you be so certain, with such precision, without a calculator?)

"The SCDad is really smart" and left it at that.  But in reality, it's a simple mathematical trick that his problem played into.

Fractions of seven are a mathematical oddity.  If you can remember the sequence Fourteen, (double it) twenty eight, fifty-seven.  one-four-two-eight-five-seven (142857), you have all the whole fractions of seven at your finger tips.  You just have to know where to start the sequence. 

Here is how:
one seventh (1/7th) calculates out to a repeating decimal 0.142857 (decimal repeating to infinity)  but it turns out, every whole fraction of seven uses the same repeating 6 digits, save starting at a different digit in the sequence:

So 1/7th is 0.142857 (repeating)
2/7ths is (start with the two in the sequence) 0.285714(repeating)
3/7ths is (start w/the next highest digit in the sequence) 0.428571 (repeating)
4/7ths is (start w/the next highest digit in the sequence)  0.571428 (repeat)
5/7ths is (start w/the next hightest digit in the sequence) 0.714184 (repeat)
6/7ths is (you get the idea) 0.857142.
Six fractions, six repeating digits.  Easy Peasy.

Woh.... 
This is the kind of mathematical co-inky-dink that makes math interesting!

The SCSon tried it and it still came up wrong... turns out he'd dropped a sign.  Not my fault.