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# Is Y < Z ? 1. Y + Z = 1 2. Y^2 < Z^2

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Director
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Is Y < Z ? 1. Y + Z = 1 2. Y^2 < Z^2 [#permalink]

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15 Jun 2005, 16:12
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Is Y < Z ?

1. Y + Z = 1
2. Y^2 < Z^2
Director
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15 Jun 2005, 16:55
I pick E

Nothing says X and Y are integers.
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15 Jun 2005, 17:08
WinWinMBA wrote:
Is Y < Z ?

1. Y + Z = 1
2. Y^2 < Z^2

this one is a bit inolved

C.

1) insuff, even if X,Z >0 it can be clearly seen
2) insuff, obviously

1)+2) Z^2 > (1-Z)^2 => Z must be greater than 0.5, graph it to see it clearly, either as a line or two intersecting parabolas facing up.

so Z>0.5 and Y<0.5 sufficient
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15 Jun 2005, 17:32
One more for C
Statement 1: y = 1-z
when z is negative y is not < z
when z is positive y may be < z
so not sufficient

Statement 2: Depends on whether y is negative or positive we will get 2 different answers, so not sufficient.

Combining these two: (1-z)^2 < y^2
when z is negative y is not < z
when z is between 0 and 1 y is not < z
when z is positive y is < z

sparky wrote:
1)+2) Z^2 > (1-Z)^2 => Z must be greater than 0.5, graph it to see it clearly, either as a line or two intersecting parabolas facing up.

so Z>0.5 and Y<0.5 sufficient

sparky, Just curious, how did you hone in on 0.5
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15 Jun 2005, 17:54
sparky wrote:
1) + 2) Z^2 > (1-Z)^2 => Z must be greater than 0.5, graph it to see it clearly, either as a line or two intersecting parabolas facing up.
so Z>0.5 and Y<0.5 sufficient
sparky, Just curious, how did you hone in on 0.5

i think, to hold the both equation, Z must be more than 0.5. i also agree with him/her.
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15 Jun 2005, 18:17
Z^2 > (1-Z)^2 = 1 + Z^2 - 2Z

Z>0.5

sparky is a he by the way
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15 Jun 2005, 18:18
HIMALAYA wrote:
i think, to hold the both equation, Z must be more than 0.5. i also agree with him/her.
Any calculations?

WinWinMBA wrote:
Is Y < Z ?
1. Y + Z = 1
2. Y^2 < Z^2

we know, from i, y and z could be anything, but their sum must be 1.
from ii, y and z could be both +ve and -ve.

from i and ii, z cannot be smaller than y because if z is smaller than y, the equation in i and ii do not hold.

so lets plug innnnnnnnnnnnnn......

if z = 1, y = 0=> possible
if z = 10, y = -9=>possible
if z = 0, y = 1 = > not possible because the eq z^2>y^2 doesnot satisy. so any value for z less than the y"s value doesnot satisfy the eqs.

so now lets plug in for some other values that are lowest for z:
if z = 0.9, y = .1, both equations are satisfied.
if z = 0.6, y = .4, both equations are satisfied.
if z = 0.51, y = .49, both equations are satisfied.
if z = 0.5, y = .5, both equations are not satisfied. so we cannot go for any value for z that is not more than 0.5 because sq of z must greater than the sq of y.

hope it is clear..............
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15 Jun 2005, 18:20
sparky wrote:
Z^2 > (1-Z)^2 = 1 + Z^2 - 2Z

Z>0.5. sparky is a he by the way

thanks man.............
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15 Jun 2005, 18:23
Clearly each (1) and (2) alone are not sufficient.

Together we have

Y+Z=1
Y^2<Z^2

or

Y^2<(1-Y)^2
Y^2<Y^2 - 2Y +1
1-2Y>0 => 1>2Y => Y < 1/2

=> Z > 1/2

=> Z>Y

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15 Jun 2005, 18:24
1) Y and Z can be any value and Y can be > or < Z and both still add up to 1

2) Y^2-Z^2 < 0
(Y+Z)(Y-Z)<0
-Z < Y < Z
- not sufficient since dependent on sign of Y

Using 1 and 2,
if -z = -1 and y = 0, thn Y+Z = 1 and y<z
if -z = 0 and y=1, then Y+Z = 1 y>Z

I take E
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15 Jun 2005, 18:30
HIMALAYA wrote:
so lets plug innnnnnnnnnnnnn......

hope it is clear..............

HIMALAYA, Thanks for your time. Your thought process is similar to mine. plugging innnnnnnnnnnn. But I was curious to know of a less time consuming approach.

Sparky's approach looks a lot quicker than our plugging innnnnnnn.
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15 Jun 2005, 18:38
HIMALAYA wrote:
so lets plug innnnnnnnnnnnnn......

hope it is clear..............

HIMALAYA, Thanks for your time. Your thought process is similar to mine. plugging innnnnnnnnnnn. But I was curious to know of a less time consuming approach.

Sparky's approach looks a lot quicker than our plugging innnnnnnn.

yes, sparky's approach is indeed a lot faster. Nice one
Director
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15 Jun 2005, 18:52
I just want to say something. When dealing with inequialities, try to solve them.

If you are plugging numbers on inequalities questions, you are pretty much doomed, unless you know what numbers to plug. But to know what numbers to plug, you need to have a general notion/sense how that particular inequality works which you might not have unless you have a lot of experince or are in a genious category.

Since it's all about signs try to get them in the following form

()*()/()*() >< 0

I bet you the things in brackets will correspond closely to what is given in 1) and 2)
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15 Jun 2005, 19:00
ywilfred wrote:
HIMALAYA wrote:
so lets plug innnnnnnnnnnnnn......

hope it is clear..............

HIMALAYA, Thanks for your time. Your thought process is similar to mine. plugging innnnnnnnnnnn. But I was curious to know of a less time consuming approach.

Sparky's approach looks a lot quicker than our plugging innnnnnnn.

yes, sparky's approach is indeed a lot faster. Nice one

ohhh yes, i agree with you guys. just trying to show how values of y and z are <0.5 and more than 0.5 respectively.........

we can do in many ways, but, of course, we need a shortest one. this could be one of them:

from i, z=1-y ..............................1
from ii, y^2<z^2 .........................2

lets put the value of z in 1 to eq 2: y^2 <(1-y)^2
solving the eq ....... y^2 < 1-2y+y^2
2y<1
so y<1/2. if y<1/2 or 0.5, then z >1/2 0r 0.5.
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15 Jun 2005, 20:12
WinWinMBA wrote:
Is Y < Z ?

1. Y + Z = 1
2. Y^2 < Z^2

No confusion about 1 or 2 being insufficient.

Combined:
From 2:
(Y+Z)(Y-Z)<0
From 1:
Y+Z=1
=>Y-Z<0
=>Y<Z

Should be finished in 30 seconds.
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16 Jun 2005, 04:57
The OA is C, HongHu's answer cannot be bettered !!!!
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16 Jun 2005, 06:03
HongHu, That's a kickass solution
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