#color(blue)("Shortcut method")#
Turn what you are dividing by upside down (invert) and then multiply.
#3/5-:6/9" " ->" " 3/5xx9/6#
Cancel out what you can
#(cancel(3)^1)/5xx9/(cancel(6)^2)" "=" "9/10#
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#color(blue)("First principle method")#
When you divide you are dividing counts.
A fraction's structure is#" "("count")/("size indicator") " "->" "("numerator")/("denominator")#
You can not #color(brown)(ul("directly divide"))# the counts unless the size indicators are the same.
#color(green)("NOT NORMALLY WRITTEN THIS WAY BUT IT IS CORRECT.")#
#color(green)(" "darr)#
#color(green)("Example "4-:2" works because really they are "4/1-:2/1)##color(green)("Observe the size indicators are both the same value of 1.")#
Multiply by 1 and you do not change the value. However 1 comes in many forms.
#color(brown)([3/5color(blue)(xx1)] -:[6/9color(blue)(xx1)]" "->" "[3/5color(blue)(xx9/9)]-:[6/9color(blue)(xx5/5)]#
#color(brown)(" "[27/45]" "-:" "[30/45])#
This gives the same answer as:
#color(brown)(" "27-:30 " "->" "27/30#
#color(brown)(" "(27-:3)/(30-:3) " "=" "9/10)#
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#color(blue)("Why does the short cut turn upside down and multiply work?")#
#("count 1")/("size indicator 1")" " -: " "("count 2")/("size indicator 2") #
#("count 1")/("size indicator 1")" " xx" "("size indicator 2")/("count 2")#
#" "("count 1")/("count 2")" "xx" "("size indicator 2")/("size indicator 2")#
#color(red)(" "uarr" "uarr)#
#color(red)("Dividing counts compensates for difference in size indicator")#
