Question

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?

Answer 1

Remaining 8 spot count is 225

Explanation:

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")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`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`