When dividing you are seeing how many of one #ul("count")# you can fit into another #ul("count")#.
A fraction consist of #" "("count")/("size indicator")->("numerator")/("denominator")#
You can only #ul("directly")# divide counts in fractions if the 'size indicators' are the same. If they are not the same you have either make them the same or multiply the division of counts by a correction factor. This correction factor compensates for the difference in the size indicators.
#color(blue)("Demonstrated by examples with numbers")#
Suppose we had: #color(red)(2)/color(green)(4)-:color(red)(3)/
color(green)(8#
#=color(red)(2)/color(green)(4)xxcolor(green)(8)/color(red)(3)" " =" "(color(red)(2)xxcolor(green)(8))/(color(green)(4)xxcolor(red)(3))" "=" "(color(red)(2)xxcolor(green)(8))/(color(red)(3)xxcolor(green)(4))#
#" "=" "color(red)(2/3)xxcolor(green)(8/4)#
#" "=(color(red)(2)xxcolor(green)(8/4))xxcolor(red)(1/3)#
The #8/4# converts the 2 into the count number you would have if you hade changed #2/4# into #8^("ths")# so you have:
( Note that #-:3# is the same as #xx1/3#)
#" "(color(red)(2)xxcolor(green)(8/4))" "xx" "color(red)(1/3)#
#" "uarr" "uarr#
#" "color(white)(.)|" "color(white)(.)|#
#"The count of "2/4" converted divided by the count of"3/8#
