Question 715070
(Note, the graph below shows lines but no inequalities shading)

Letting x be how many fruit cakes and y be how many sponge cakes, and all measures as kilograms, here's an accounting for flour and sugar quantities:


Which cake_____________flour_________________sugar
Fruit Cake___________0.5x____________________0.1x
Sponge Cake__________0.2y____________________0.2y
Total Materials____0.5x+0.2y______________0.1x+0.2y


The Required Cakes Inequality:
x+y>=5, gives us {{{y>=-x+5}}}


Flour Sum Inequality:
{{{0.2y<=2-0.5x}}}
{{{y<=(2/0.2)-(.5/.2)x}}}
{{{y<=-(5/2)x+10}}}


The Sugar Sum Inequality:
{{{0.2y<=-0.1x+1.2}}}
{{{y<=-(1/2)x+1.2/0.2}}}
{{{y<=-(1/2)x+6}}}


Summary of the Inequalities:
{{{highlight(y>=-x+5)}}}
{{{highlight(y<=-(5/2)x+10)}}}
{{{highlight(y<=-(1/2)x+6)}}}

Graphing those and shading the proper regions seem to point out x=1 fruit cake and y=4 sponge cakes.  Another possiblility is 3 fruit cake and 3 sponge cake.  Or another way to use the flour and sugar is 5 or 6 sponge cakes but none of any fruit cakes.  A few other answers are found in the graph.

I present here the graph alone without any shading for inequalities:
{{{graph(400,400,-1,11,-1,11,y=-x+5,y=-(5/2)x+10,y=-(1/2)x+6)}}}