we can do this in two ways, we can do it with a diagram, showing the different cups, or we can use simple division.

#color(white)(c)#

#ul color(black)("Method 1, diagramming:")#

original amount of trail mix: #5 1/2# cups

#color(red)("cup " 1: {3/4 " cup"} #

Amount of trail mix left: #5 1/2 - 3/4 = color(blue)(4 3/4 " cups remaining"#

#color(red)("cup " 2: {3/4 " cup"} #

Amount of trail mix left: #4 3/4 - 3/4 = color(blue)(4 " cups remaining"#

#color(red)("cup " 3: {3/4 " cup"} #

Amount of trail mix left: #4 - 3/4 = color(blue)(3 1/4 " cups remaining"#

#color(red)("cup " 4: {3/4 " cup"} #

Amount of trail mix left: #3 1/4 - 3/4 = color(blue)(2 1/2 " cups remaining"#

#color(red)("cup " 5: {3/4 " cup"} #

Amount of trail mix left: #2 1/2 - 3/4 = color(blue)(1 3/4 " cups remaining"#

#color(red)("cup " 6: {3/4 " cup"} #

Amount of trail mix left: #1 3/4 - 3/4 = color(blue)(1 " cups remaining"#

#color(red)("cup " 7: {3/4 " cup"} #

Amount of trail mix left: #1 - 3/4 = color(blue)(1/4 " cups remaining"#

From this, we can see that after #7# cups, there is only #1/4# of a cup left, not enough to fill another #3/4# cup. So Zoey can divide the #5 1/2# cups of trail mix into #7# sets of #3/4# full cups with #1/4# of a cup remaining.

#color(white)(c)#

#color(white)(c)#

#ul color(black)("Method 2, simple division:")#

splitting #5 1/2# cups of trail mix into #x# sets of #3/4# cups can be written algebraically as #x xx 3/4 = 5 1/2#

#x xx 3/4 = 5 1/2#

In this, we need to isolate #x#, to find it's value.

#(x xx color(red)(cancel(color(black)(3/4)))) / (color(red)(cancel(3/4)))= (5 1/2)/ (color(red)(3/4))#

#x = 5 1/2 -: 3/4#

#x = 11/2 -: 3/4#

Finding the reciprocal of the second fraction and replacing the #-:# with #xx#

#x = 11/color(red)(cancel(color(black)(2))1) xx color(red)(cancel(color(black)(4))2)/3#

#x = 11/1 xx 2/3#

#x = 22/3#

#x = 7 1/3#

This is represented as #7 1/3# sets of #3/4# cups, #1/3color(blue)("(remaining amount of " 3/4 " cup)" )# of #3/4 color(green)("(Serving size of cup)"# is #1/4#, so there is #1/4# of a full cup remaining and #1/3# of a #3/4# cup remaining.

#color(white)(c)#

Zoey can divide the #5 1/2# cups of trail mix into #7# sets of cups that are#3/4# full, with #1/4# of a #100%# full cup remaining.