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

Lets approach the easier statement (2) first: -

z = 5x and x < 10 lets pick nos

x = 4; Arithematic Mean = (4+10)/2 = 7 and z = 5x4 = 20, So z>arithematic mean.

No lets assume x = 1/4, Arithematic mean = (1/2+10)2 = 5.25, and z = 5x1/2 = 2.5, so z< arithematic mean.

So (2) is insufficient, the answer choices boils down to A & D

Now look at (1)

On number line Z is closer to 10 than x.

Assume x =2, then A.M = 6. If Z has to be closer to 10 then the Z > 6 so Z > A.M
Assume x = 6 then A.M = 8, then Z >9 so Z > A.M

for all +ve values of x<10, Z > A.M Hence (1) is sufficient.

Answer is (A)

Great explanation!! Thanks a lot. (A) is the right answer.

Hello! thanks so much for going through the problem.

MGMAT staff! I have a question re: rephrasing the original statement. How would you rephrase this problem?
Thanks so much :-)

Rephrase for 1:

AM = Equidistant from both 10 and X
If Z is closer to 10 and all 3 quantities are positive then obviously z > AM


Rephrasing for 2:

z > (x+10)/2
z> x/2 + 5

By 2:

5x > x/2 +5

9/2 x >5
x could be anything. within 0 and 10.

there are a couple of ways.

(1) VISUAL
this is the best way. in general, if you have visual 'number line' rephrases, those are pretty much the best way to go, pretty much all the time.
the average of x and 10 is the midpoint on the number line between x and 10.
therefore, greater than that average means to the right of that midpoint.
if you're to the right of the midpoint, then you're closer to the greater number, which, in this case, is 10.
therefore, you can rephrase the question, using this visual approach, as:
is z closer to 10 than to x?

this is dynamite, because it makes statement #1 absolutely trivial.

(2) ALGEBRAIC
question:
is z > (x + 10)/2 ?
is 2z > x + 10 ?

statement 1:
either z > 10, or 10 - z < z - x (i.e., the distance between z and 10 is smaller than the distance between x and z -- notice that "closer" can be rephrased to "smaller distance")
if z > 10, then 2z is more than 20, and x + 10 is less than 20 (because x is less than 10). therefore, YES in that case.
if 10 - z > z - x, you can rearrange that to get x + 10 < 2z, which means YES to the prompt question (that's the exact form of the prompt question).
therefore, unilaterally YES, so, sufficient.

--

statement 2:
no indication of the size of x and z is given, so just consider the extreme possibilities.
if they're tiny (like x = 0.0001 and z = 0.0005), then the average of x and 10 is about 5, which is colossally huge compared to z. so, NO.
if they're huge (like x = 1000 and z = 5000), then the average of x and 10 is less than x (and therefore WAY less than z). so, YES.
insufficient.

answer = a