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If x<y<z and u<v, is x<u<y<v<z?

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Source: — Data Sufficiency |
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by GMATGuruNY » Thu Nov 06, 2014 6:10 am
RJCN wrote:Hi guys, can you help me with this one?

If x<y<z and u<v, is x<u<y<v<z?

(1) y-u = u-x
(2) z-v = v-y
Statement 1: y-u = u-x
Combining like terms, we get:
x+y = 2u
u = (x+y)/2.
Since u is the AVERAGE of x and y, u is HALFWAY BETWEEN x and y.
Case 1: x=1, u=2, y=3, v=4, z=5.
In this case, x<u<y<v<z.
Case 2: x=1, u=2, y=3, z=4, v=5.
In this case, it is not true that x<u<y<v<z.
INSUFFICIENT.

Statement 2: z-v = v-y
Combining like terms we get:
y+z = 2v
v = (y+z)/2.
Since v is the AVERAGE of y and z, v is HALFWAY BETWEEN y and z.
Case 1: x=1, u=2, y=3, v=4, z=5.
In this case, x<u<y<v<z.
Case 3: u=1, x=2, y=3, v=4, z=5.
In this case, it is not true that x<u<y<v<z.
INSUFFICIENT.

Statements combined:
Since u is hallway between x and y, and v is halfway between y and z -- and x<y<z -- it must be true that x<u<y<v<z.
SUFFICIENT.

The correct answer is C.
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by [email protected] » Thu Nov 06, 2014 11:11 pm
Hi RJCN,

Mitch's approach (TESTing potential VALUES) is a great way to gather the necessary evidence to answer this DS question. With a bit of note-taking and some logic, you can answer this question in another way:

We're told that X < Y < Z and U < V. Looking at these two sets of inequalities, there's no way to know how the two sets interact yet. Maybe U and V are relatively small or relatively huge (compared to the other variables) or maybe U and V are equal to (or interspersed) among X, Y and Z.

We're asked if X < U < Y < V < Z? This is a YES/NO question.

Fact 1: Y - U = U - X

This means....

X + Y = 2U

Since X < Y, then Y must be GREATER than U and X must be LESS than U.

So, X < U < Y

Since we know that Y < Z, we now have....

X < U < Y < Z

BUT we don't know anything about the V. It might be....
between U and Y (the answer is NO)
equal to Y (the answer is NO)
between Y and Z (the answer is YES)
equal to Z (the answer is NO)
or greater than Z (the answer is NO)
Fact 1 is INSUFFICIENT

Fact 2: Z - V = V - Y

This means....

Y + Z = 2V

Since we know that Y < Z, then Z must be GREATER than V and Y must be LESS than V.

So Y < V < Z

Since we know that X < Y, we now have...

X < Y < V < Z

BUT we don't know anything about the U. It might be....
less than X (the answer is NO)
equal to X (the answer is NO)
between X and Y (the answer is YES)
equal to Y (the answer is NO)
between Y and V (the answer is NO)
Fact 2 is INSUFFICIENT

Combined, we know....
X < U < Y
and
Y < V < Z
So we can combined (based on the common "Y") and get...
X < U < Y < V < Z
The answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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