How do you divide #\frac { 3} { 4} \div \frac { 5} { 4} #?

Dividing is the same as multiplying by the inverse.

So
#color(white)("XXX")3/4div5/4#
is the same as
#color(white)("XXX")3/4xx4/5#

#color(white)("XXX")=3/5#

The complex fraction method makes sense of the rule of invert and multiply.

#3/4-:5/4# can be written as

#(3/4) / ( 5/4) # This is a complex fraction

To simplify the fraction multiply both the numerator ( top fraction) and the denominator ( bottom fraction) by the inverse of the denominator .

The inverse of # 5/4 = 4/5# Now multiply both fractions by # 4/5#

#{ ( 3/4) xx (4/5)} / {(5/4) xx ( 4/5)} #

# ( 5/4) xx ( 4/5) = 1 # This causes the bottom fraction to disappear. This leaves only the top fraction which gives.

# ( 3/4) xx ( 4/5) = ( 12/20) # Now factor out the common factor of 4

#( 4 xx 3)/ (4 xx 5)# This leaves

# 3/5#

After the bottom fraction divides out it leaves the invert and multiply method. Using the complex fraction makes mathematical sense of the method and avoids reliance on memory.

A fraction consists of #("count")/("size indicators of what you are counting")#

Using the allocated names we have:

#("count")/("size indicator")->("numerator")/("denominator")#

Consider whole numbers. For example 6 and 3

These can, and may, be written as #6/1 and 3/1# They are rational numbers. It is not normally done but never the less it is correct.

Now consider # 6-:3->6/1-:3/1#

#color(blue)("You just divide the counts as the 'size indicators' are the same")#

#6-:3->6/1-:3/1->6/3=2#

#color(magenta)("You have been applying this principle for a long time without even realizing that you were.")#