chemical mixing 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

Hi Devon,

This is a fairly tricky question that requires a bit of algebra. Here is how I would tackle this:

Set up two variables for the volume of 10% and 75% mixtures

A = ml of 10% mixture

B = ml of 75% mixture

We know that at the end of the mixing, the total volume must equal 390 ml:

Eqn 1: A + B = 390 ml

We also know that at end of the mixing, the concentration must equal 40%. So we need to calculate the amount of "pure" alcohol contained in the mixtures. Here is how that looks in algebra:

Eqn 2: 40% x 390 ml = 10% x A + 75% x B

Multiplying 40% x 390 on the left side gives us:

Eqn 2A: 156 = 0.1 x A + 0.75 x B (converted percents to decimals and dropped the "ml")

We now have two equations with two unknowns (a "system of equations")

Eqn 1: A + B = 390 ml

Eqn 2A: 156 = 0.1 x A + 0.75 x B

We can subtract B from both sides of Eqn 1 to isolate A on the left side:

A + B - B = 390 - B

A = 390 - B

We can then substitute "390-B" for A in the second equation:

156 = 0.1 x (390-B) + 0.75 x B

Multiplying out:

156 = 39 - 0.1 x B + 0.75 x B

Combining the B terms (0.1xB and 0.75xB):

156 = 39 + .65 x B

Subtracting 39 from both sides:

156 - 39 = 39 - 39 + 0.65 x B

117 = 0.65 x B

And dividing both sides by 0.65 provides a value for B.

180 = B

Now that we know how much 75% mixture to use (180 ml), we can use the original Eqn 1 to determine how much 10% mixture to use:

390 ml = A + 180 ml

Subtracting 180 ml from both sides:

390 ml - 180 ml = A + 180 ml - 180 ml

210 ml = A

That's the preliminary solution. You should always check your answer by substituting the values you found for A and B into the second equation:

40 % of 390 ml = 10% of 210 ml + 75% of 180 ml

0.4 x 390 = 0.10 x 210 + 0.75 x 180

156 = 21 + 135

156 = 156 SOLUTION CHECKED!

Final answer:

210 ml of a 10% mixture plus 180 ml of a 75% mixture will produce 390 ml of a 40% mixture

Hope this helps!

Bob