What is the average (arithmetic mean) of x and y?

What is the average (arithmetic mean) of x and y?

The average (arithmetic mean) of x and y is \(\frac{x+y}{2}\). Thus we need to find the value of x+y.

(1) The average of x and 2y is 10 --> \(\frac{x+2y}{2}=10\). Not sufficient.

(2) The average of 2x and 7y is 32 --> \(\frac{2x+7y}{2}=32\). Not sufficient.

(1)+(2) We have two distinct linear equations with two unknowns, thus we can solve for both and get the value of x+y. Sufficient.

Answer: C.

Just a quick question somewhat related to this problem:

When you're asked for x+y, is it safe to assume that you need either x and y individually, or the value of x+y combined to achieve sufficiency? Will things like x+2y, 3x+4y, etc. always be insufficient? Whenever I'm faced with these problems, I find myself spending time trying to manipulate the x+2y into x+y, so it would be nice to know if I could just quickly say insufficient and move on.

Solution:

Question Stem Analysis:

We need to determine the average of x and y, i.e., the value of (x + y)/2. Notice that if we can determine the values of x and y, then we can determine the value of (x + y)/2.

Statement One Alone:

We are given that (x + 2y)/2 = 10. However, since we can’t determine either the value of x or of y, we can’t determine the value of (x + y)/2. Statement one alone is not sufficient.

Statement Two Alone:

We are given that (2x + 7y)/2 = 32. However, since we can’t determine either the value of x or of y, we can’t determine the value of (x + y)/2. Statement two alone is not sufficient.

Statements One and Two Together:

From the two statements, we have two linear equations in two variables. Note that neither equation is dependent on the other, which means that one equation is not a linear multiple of the other. Thus, we can solve for the values of x and y, and hence we can determine the value of (x + y)/2. Both statements are sufficient.

Answer: C­

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