Alcohol mixture problem

Question

You need 390 mL of a 40% alcohol solution. On hand, you have a 10% alcohol mixture and a 75% alcohol mixture. How much of each mixture will you need to obtain the desired solution?

Answer 1

Let

r = 40% = 0.4 be the density of the desired mixture,

r1 = 10% = 0.1 be the density of one source,

r2 = 75% = 0.75 be the density of the other source,

V = 390 mL be the desired volume,

V1 (to be calculated) be the volume to use of the first source,

V2 (to be calculated) be the volume to use of the second source.

We have two constraints:

- constraint on the desired volume:

V1 + V2 = V

- constraint on the desired density

r1V1 + r2V2 = rV

I do not know whether you also need explanation of how to solve these two

equations, but let me show you one way.

Multiply the first equation by r2 and subtract the second equation.

r2V1 - r1V1 = r2V - rV

Divide both sides by (r2 - r1)

V1 = V(r2 - r)/(r2 - r1) = 390×(0.75 - 0.4)/(0.75 - 0.1) = 210 mL

Then use the first equation to calculate V2.

V2 = V - V1 = 390 - 210 = 180 mL