Problem Solving

This is a WEIGHTED AVERAGE question.

With any mixture, the resulting solution will be closer in composition to whichever component we added more of. In other words, if we add more of a 50% solution, the resulting mixture will be closer to 50%, and if we add more of the 30% solution, the resulting mixture will be closer to 30%. If we add exactly half of each, the result will be 40%. (It could never be less than 30% or more than 50%, if those were the only two solutions added).

Since in this case the resulting mixture of 35% is closer to 30% than 50%, there must be more of solution Y in the overall mixture; Y must account for more than half of the 16 liters. Right away, we can eliminate anything 8 or less: A, B, and C.

In a weighted average, the ratio of PART : PART will be in the same proportion as the ratio of the differences from the original amounts (30 and 50) to the actual (35). 50 - 35 = 15 and 35 - 30 = 5. Thus, the ratio is 15 : 5, or 3 : 1.

If the ratio of Y to X is 3 to 1, then Y is 3/4 of the total mixture. 3/4 of 16 is 12.

The answer is E.

swerve wrote:Solution X, which is 50% alcohol, is combined with solution Y, which is 30% alcohol, to form 16 liters of a new solution that is 35% alcohol. How much of solution Y is used?

A. 4 liters
B. 6 liters
C. 8 liters
D. 10 liters
E. 12 liters

The OA is E.

Source: Veritas Prep

When solving mixture questions, I find it useful to sketch the solutions with the ingredients SEPARATED:


Since we want to determine the volume of solution Y needed, let's...
...let y = volume (in liters) of solution Y needed
This means 16 - y = volume (in liters) of solution X needed (since the combined volume of both amounts is 16 liters)

So, we get:


Now let's determine the volume of alcohol in each container.

Solution Y is 30% alcohol. We have y liters of solution Y.
So, the volume of alcohol = 0.3y

Solution X is 50% alcohol. We have 16 - y liters of solution X.
So, the volume of alcohol = 0.5(16 - y) = 8 - 0.5y

The combined solution is 35% alcohol. There are 16 liters of this solution.
So, the volume of alcohol = 0.35(16) = 5.6

So, our sketch looks like this:


At this point, we can focus on the volume of alcohol in each container.
We know that: (volume of alcohol in 1st container) + (volume of alcohol in 2nd container) = volume of alcohol in combined solution.
In other words: 0.3y + (8 - 0.5y) = 5.6
Simplify: 8 - 0.2y = 5.6
Subtract 8 from both sides to get: -0.2y = -2.4
Solve: y = 12

Answer: E