Martha is playing with Lego. She has 300 of each type - 2spot, 4spot, 8spot. Some bricks used to make zombie. Uses 2spots, 4spots, 8spots in ratio 3:1:2 when finished has twice as many 4spots left as 2spot. How many 8spots are left?

Let the identifier for type 2 spot be #S_2 larr# 300 at start
Let the identifier for type 4 spot be #S_4 larr#300 at start
Let the identifier for type 8 spot be #S_8larr # 300 at start

Zombie #-> S_2:S_4:S_8 -> 3:2:1 #

Left over: #S_2:S_4:S_8 -> 1:2:? #
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Notice that we have:
#color(Brown)("As a guess")#

zombie#color(white)("dd")->3:2:1#
remaining#ul(->1:2:?)#
#color(white)("ddddddd")->4:4:?#

As the vertical sum of all the different type ratios had the same value I suspect the last ratio value for the remaining will have to be 3. Giving a remaining of #1:2:3#. As it turns out that is correct.

Building the system in Excel has shown that there is 75 zombies to be made to give a remaining 1:2 ratio of the 2 spot to 4 spot. How this works will be demonstrated later. The remaining 8 spot is 226.

#color(white)("d.")1:2#
#obrace(75: 150): 225# remaining. Note that #3xx75=225#
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#color(blue)("The mathematical solution")#

#color(brown)("Method")#

Accumulate the 'zombie' ratio count and subtract it from the total.

The remaining has to have the ratio #("2 spot")/("4 spot")-> 1/2#

#color(brown)("Building the relationship")#

#color(brown)("Let the count of 'zombies' be "x)#

The accumulated zombie #(S_2)/(S_4) -> 3/2xx1=3/2xx x/x =(3x)/(2x)#

The remaining is # (300-3x)/(300-2x) #

As a ratio this has to match: # (300-3x)/(300-2x) =1/2#
Cross multiply

#(2(300-3x)=1(300-2x)#

#600-6x=300-2x#

#4x=300#

#x=300/4=75 color(red)(larr "The total zombie count")#
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#color(blue)("Remaining count of 8 spot")#

75 zombies at: #S_2:S_4:S_8 -> 3:2:1#

Have used a count of 75 as 8 spot bricks for the zombies

Remaining 8 spot #-> 300-75 = 225#