What’s ##42 2/3## divided by ##1/3##?

##42 2/3 -: 1/3=128##
##color(blue)(“For a moment consider just the “42 2/3)##
write this as ##42+2/3##
This is the same as:
##” “[42color(magenta)(xx1)]+2/3##
But we can write 1 as ##3/3## giving:
##” “[42color(magenta)(xx3/3)]+2/3##
##” “[126/3]” “+2/3 = 128/3##
‘~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
##color(blue)(“Putting it all together”)##
##42 2/3 -: 1/3″ “=” “128/3-:1/3##
‘~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
##color(blue)(“Shortcut method“)##
Turn the ##1/3## upside down and multiply
##128/3xx3/1 ” “=” “3/3xx128/1 ” “=” “1xx128” “=” “128##
Or by cancelation (which is the same thing really):
##128/(cancel(3))xx(cancel(3))/1 = 128/1=128##