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.
If x is a positive number less than 10 . . .
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- Jay@ManhattanReview
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Hi Vincen,Vincen wrote: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.
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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Target question: Is z greater than the mean of x and 10?Vincen wrote: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
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