Problem of the Day

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

(1) 5x+3y=280
(2) 2x+7y=460

Reveal Answer


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

See the Solution



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.

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