Question 846331
Joe will get {{{3/4}}} of {{{2/3}}} of the cake.
When you want to find {{{3/4}}} of something (like {{{3/4}}} of 48 apples),
you multiply times {{{3/4}}} ,
and the same goes if you are trying to find {{{3/4}}}  of {{{2/3}}} of a cake.
{{{(3/4)*(2/3)=3*2/(4*3)=3*2/(3*4)=(3/3)*(2/4)=1*(1/2)=highlight(1/2)}}}
Joe will get 1/2 of the cake.
 
ANOTHER WAY:
Since we have thirds and fourths,
{{{12}}} could be a common denominator,
so let's think of twelfths.
Supposed that the cake came conveniently cut into {{{12}}} equal slices.
Each slice is {{{1/12}}} of the cake.
If you have {{{2/3}}} of the cake left, you have {{{8}}} slices left,
because {{{4}}} slices would be {{{1/3}}} of the cake,
and {{{8}}} slices is twice {{{4}}} slices.
To figure those {{{8}}} slices, out of the original {{{12}}} ,
you divided by {{{3}}} to get {{{1/3}}} ,
and then multiplied times {{{2}}} to get {{{2/3}}} of the original {{{12}}} slices.
That is {{{12*(2/3)=12*2/3=(12/3)*2=8}}} .
You are multiplying times {{{2/3}}} when you 
first divide by {{{3}}} and then multiply times {{{2}}}, or
when you multiply times {{{2}}} and then divide by {{{3}}} .
It's all the same.
Now, {{{1/4}}} of those {{{8}}} slices left would be {{{2}}} slices,
and {{{3/4}}} of those {{{8}}} slices left would be {{{6}}} slices that you give to Joe.
Since the whole cake was {{{12}}} slices, those {{{6}}} slices are half of the cake.