repeating decimal

Question

How do you write 0.142857 in standard form? 

Answer 1

x=0.142857142857142857...

the repetend is 6 digits long so multiply both sides of the equation by 10⁶ or 1,000,000

1,000,000x=142857.142857142857142857...

               x=         0.142857142857142857...

subtract the equations

  999,999x=142857 (the decimal parts cancel)

              x=142857/999,999

simplify by dividing both terms by 9

              x=15,873/111,111

now divide both terms by 3

              x=5291/37,037

now divide both terms by 37

              x=143/1001

now divide both terms by 11

              x=13/91

now divide both terms by 13

              x=1/7 is the answer

whenever there is no lag (the lag is what precedes the repetend) you can simply put the repetend over 9, 99, 999, 9999, 99,999, 999,999, etc depending on how many digits are in the repetend

in 0.142857142857142857... there are 6 digits that repeat so you use 6 nines

142857/999,999 and then simplify from here...

if the decimal were 0.727272... the answer would be 72/99=...

if the decimal were 0.459459459... the answer would be 459/999=...

but remember that there cannot be a lag, otherwise you have to solve using the top method or another method that involves separating the lag and the repetend but I won't illustrate that now unless you need more help