here, it can help to write #0.396396# (or #0.dot3dot9dot6#) as #x#.
#x = 0.dot3dot9dot6#
if you move the decimal point #3# places forward, you'll go from #0.396396396...# to #396.396396...#, or #396.dot3dot9dot6# .
this means that #1000x = 396.dot3dot9dot6#
you can then subtract #x# from #1000x# to get #999x#,
and subtract # 0.dot3dot9dot6# from #396.dot3dot9dot6# to get #396#.
this means that #999x = 396#
dividing both sides by #999# gives you #x = 396/999#.
#396/999# can be simplified by dividing both numerator and denominator by #9#.
#396/9 = 44#
#999/9 = 111#
#396/999 = 44/111#
hence, #x = 44/111#.
(and if you type #44/111# into a calculator, you'll get #0.396396...#, your recurring decimal.)
