x, y, and z are positive integers. The average (arithmetic mean) of x, y, and z is 11. If z is two greater than x, which of the following must be true?
I. x is even
II. y is odd
III. z is odd

A)I only
B)II only
c)III only
D)I and II only
E)I and III only

OA is B)
Explanation given is :
We know that the average of x, y, and z is 11. We can therefore set the up the following equation:

(x + y + z)/3 = 11

Cross-multiplying yields

x + y + z = 33

Since z is two more than x, we can replace z:

x + y + x + 2 = 33
2x + y + 2 = 33
2x + y = 31

Since 2x must be even and 31 is odd, y must also be odd (only odd + even = odd). x and z can be either odd or even. Therefore, only statement II (y is odd) must be true.

The correct answer is B.

I understand the above solution, but my logic was slightly different. I figured if 11 was the average, and the z = x+2, the only solution was

x y z
10 11 12

Using this logic, I thought the right answer was x= even and y = odd. Where am I making a mistake? i.e I and II Option D)

Hi,

You are right until the step where you are assuming that the only possible solution is 10,11,12.

Once you have deduced that y is definitely an odd number, we must try and check for x or z. From the question, we know that z = x + 2. This means that if x is odd, z is also odd and if x is even, z is also even. In short, we just need to check for either x or z. Lets try testing for x since we have the equation: 2x + y = 31

If you select x = 9, which is odd and put it in equation 2x + y = 31 you get y = 13.

If you select x = 8, which is even and put it in equation 2x + y = 31 you get y = 15.

Therefore, in each of the two cases, we are satisfying the criteria that y is odd and the mean of the three numbers is 11 and we can see that x and z can be both, odd and even.

So answer is B. The only small mistake that you made was assume that x, y, z are in ascending order or that y is between x and z. The question only mentions that they are positive and that they are integers.

Don't worry about such mistakes, practice will make you better. With GMAT quant, get into the habit of reading the question properly and working with exactly what is stated. Any assumptions, especially on Data Sufficiency will cost you dearly.

Regards

Sunil

So sorry. Should have realized earlier!

Nothing to be sorry about at all. This is what forums are for.

Regards

Sunil

We are so lucky to have Sunil here. Again, excellent explanation!

_________________
Jamie Nelson
ManhattanGMAT Instructor