If x is a positive number less than 10, is z greater than

[liwh_gd]

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

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 :-)

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

very nice explanation!

if they're huge (like x = 1000 and z = 5000)
Is X should be 0＜X＜10?

[tim]

for this problem, yes..

[ratheesh2306]

I tried it in a different way. I just want to know whether my method is correct.
Given: x<10 (so 1 to 9)
Is Z>(x+10)/2?
So Z can vary between 5.5 to 9.5 (because x can take value between 1 to 9)
So Is Z>5.5?

Statement 1: If X=1 then Z has to 6 or greater than 6. So z>5.5 is answerable. So sufficient.
Statement 2: If X=1 then Z=5 (so z is not greater than 5.5)
If X=2 then Z=10 (so z is greater than 5.5)
Ambiguity
So insufficient.

So answer is A.

[RonPurewal]

I tried it in a different way. I just want to know whether my method is correct.

No.
You're assuming that the numbers in the problem are integers. They don't have to be integers.

[rakshaki]

Hi Ron,

Reading the first statement, I treated this as a weighted average problem with x and 10 as the data points. Since z is closer to 10, it has to be more than the average. Hence Sufficient. Is this approach correct?

[RonPurewal]

Hi Ron,

Reading the first statement, I treated this as a weighted average problem with x and 10 as the data points. Since z is closer to 10, it has to be more than the average. Hence Sufficient. Is this approach correct?

yes, that's correct.

it's also closely related to what i posted earlier on this thread, here:
post18085.html#p18085