If x is a positive number less than 10 . . .

[Vincen]

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

The OA is A.

I thought the right option should be E. I need a clarification here. Thanks.

[Jay@ManhattanReview]

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

The OA is A.

I thought the right option should be E. I need a clarification here. Thanks.

Hi Vincen,

Let's list down the given information.

1. 0 < x < 10
2. Average of x and 10 = (x + 10)/2
3. z is any number, we do not have any information.

You must know that average of two numbers lies in the mid of the two numbers.

Thus, x ≤ (x + 10)/2 ≤ 10

This implies that the average cannot be less than the smaller number and greater than the larger number.

We have to determine whether z is greater than (x + 10)/2.

Statement 1: On the number line, z is closer to 10 than it is to x.

Case 1: Say z > 10.

Refer to the following depiction in the number line.

0----x-----Av.----10----z

Since Av. would lie between x and 10, and z > 10, we see that z > Av. The answer is Yes.

Case 2: Say z ≤ 10.

Refer to the following depiction in the number line.

0----x-----Av.------z--10

Since Av. would lie exactly in mid of x and 10, it (Av.) would be equidistant from x and 10. Again, since it is given that z is closer to 10 than it is to x, we can ascertain that z > Av. The answer is Yes. Sufficient.

Hope this helps!

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-Jay
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[Brent@GMATPrepNow]

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

Target question: Is z greater than the mean of x and 10?

Statement 1: On the number line, z is closer to 10 than it is to x.
IMPORTANT: On the number line, the mean of two numbers will lie at the midpoint between those two numbers.
So, the mean of x and 10 will lie halfway between x and 10.
So, if z is closer to 10 than it is to x, then z must lie to the right of the midpoint between x and 10.
This means that z must be greater than the mean of x and 10
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z = 5x
There are several pairs of numbers that meet this condition. Here are two:
Case a: x=1, z=5, in which case z is less than the mean of x and 10 (mean = 5.5)
Case b: x=4, z=20, in which case z is greater than the mean of x and 10 (mean = 7)
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent