How do you evaluate #5\frac { 3} { 4} + 4\frac { 1} { 2} #?

Once well practised you will be able to solve this problem type in just a few lines.

Consider this as #5+3/4+4+1/2#

#color(brown)("The whole number part is "5+4=9)#

Consider the fraction part.

We have #3/4+1/2#
................................................................................................
A fractions structure is such that you have:

#("count")/("indicator the size of what is being counted")->("numerator")/("denominator")#

You can not #ul("directly add")# the counts unless the 'size indicators' are the same.

Multiply by 1 and you do not change the value. However, 1 comes in many forms so you can change the way something looks without changing its actual value.
..............................................................................................................

#color(green)(3/4" "+" "[1/2color(red)(xx1)])#

#color(green)(3/4" "+" "[1/2color(red)(xx2/2)])#

#3/4" "+" "2/4" "=" "(3+2)/4=5/4#

but #5/4# is the same as #4/4+1/4=1 +1/4#

#color(brown)("So the fraction part is "3/4+1/2=1 1/4#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all together")#

#color(blue)(9+1 1/4=10 1/4)#