You need a 40% alcohol solution. On hand, you have a 540 mL of a 10% alcohol mixture. How much pure alcohol will you need to add to obtain the desired solution?

270 mL of pure alcohol is needed.

Step-by-step explanation:

In 540 mL of the mixture, you have 540*10% = 54 mL of pure alcohol.

If you add x mL pure alcohol, then you have (540 + x) mL solution, in it there is (54 + x) mL of pure alcohol

so (54 + x)/(540 + x) = 40%

Then

54 + x = 0.4(540 + x)

54 + x = 216 + 0.4 x

0.6x = 216 - 54 = 162

x = 162/0.6 = 270 mL

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!