# Problem of the Day

If 50 mL of an alcohol solution is mixed with 30 mL of a alcohol solution, what percentage of the final solution is alcohol?

(1)
(2)

A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

See the Solution

### Solution

[latexpage]

The simplest way to compute the strength of the final solution is by using the concept of a weighted average. The final solution has a total volume of 80 mL, of which 50 mL is $x\%$ alcohol and 30 mL is $y\%$ alcohol. Therefore, $\frac{5}{8}$ of the final solution is $x\%$ alcohol and $\frac{3}{8}$ of the final solution is $y\%$ alcohol. The overall percentage of the final solution, then, is $\frac{5}{8}x+\frac{3}{8}y$ or $\frac{5x+3y}{8}$. So, in order to answer the question, we need to know the value of $5x+3y$.

Statement (1): SUFFICIENT. This tells us the value of $5x+3y$ directly.

Statement (2): INSUFFICIENT. We cannot deduce what we need from the fact that $2x+7y=460$. It is clearly not enough to determine the values of $x$ or $y$, and since it is not a multiple of $5x+3y$, there is no way to determine the value of $5x+3y$.

This is a fairly straightforward problem if you take the time to work it out. However, there is a trap. The trap is to think, “I could definitely compute the overall percentage if I knew the values of $x$ and $y$. I need both statements to set up and solve a system of equations to find $x$ and $y$, so the answer must be ‘C’, both statements are needed.” It is certainly true that knowing the values of $x$ and $y$ is enough to solve the problem, but you definitely don’t need to know their exact values. All you need to know is the value of $5x+3y$. It could be that $x=20$ and $y=60$ or it could be that $x=50$ and $y=10$. There are infinitely many possibilities for the exact values of $x$ and $y$. However, as long as $5x+3y$ always equals $280$, the overall strength of the final solution will always be the same.