In all of these recurring decimals, not all of the decimals recur.

While there is a full method to determine the fractions, here is the short cut rule which can be applied.

From the example given, note the following:

#1color(blue)(.4)barcolor(red)(2) = 1color(blue)(.4)color(red)(2222222.... )larr 2 " recurs", color(blue)(4) " does not"#

(the whole number is #1#)

#= 1 (42-4)/90 = 38/90 = 19/45#

Form a fraction as:#" whole number" ("all decimals - non-recurring")/(color(red)(9)" for do and "color(blue)(0) " for don't")#

#0.2bar7 = (27-2)/90 = 25/90 = 5/18#

#4.6bar5 = 4 (65-6)/90 = 4 59/90#

#8.23bar7 = 8 (237-23)/900 = 214/900= 8 107/450#

#816.14bar(35) = 816 (1435-14)/9900 = 816 1421/9900#

#200 79bar(125) = 200 (79125-79)/99900 = 200 79046/99900 = 200 39523/49950#