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Algebra Operations

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Algebra Operations

by Rush_1982 » Tue Feb 02, 2010 1:21 am
I found these 2 questions in the Quant Review 2nd Edition. I am not able to quite follow the solution. Anyone interested in taking a stab at these? Please explain in detail.

A) For all z, [z] denotes the least integer greater than or equal to z. Is [x] = 0?
(1) -1<x<-0.1
(2) | x + .05 | = 1

B) For all z, [y] denotes the least integer less than or equal to y. Is d < 1?
(1) d = y- [y]
(2) d = 0
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by ajith » Tue Feb 02, 2010 1:35 am
Rush_1982 wrote:I found these 2 questions in the Quant Review 2nd Edition. I am not able to quite follow the solution. Anyone interested in taking a stab at these? Please explain in detail.

A) For all z, [z] denotes the least integer greater than or equal to z. Is [x] = 0?
(1) -1<x<-0.1
(2) | x + .05 | = 1

B) For all z, [y] denotes the least integer less than or equal to y. Is d < 1?
(1) d = y- [y]
(2) d = 0
A)

1. Sufficient [x] will always be 0 if (1) is true
2. insufficient since x+0.5 =1 or x+0.5 =-1 x is either 0.5 or -1.5 we cannot be sure about [x]

A is the answer

B) 1. Sufficient d will always be less than 1
2. sufficient d =0 and hence less than 1

D is the answer
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by Rush_1982 » Tue Feb 02, 2010 11:40 am
Hi Ajith,
Can you elaborate more? I kind of understand what's happening but don't fully grasp the concept. Can you pick some numbers to explain?


Thanks!
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by ajith » Tue Feb 02, 2010 11:54 am
Rush_1982 wrote:Hi Ajith,
Can you elaborate more? I kind of understand what's happening but don't fully grasp the concept. Can you pick some numbers to explain?
A) For all z, [z] denotes the least integer greater than or equal to z. Is [x] = 0?
(1) -1<x<-0.1
(2) | x + .05 | = 1

1. Sufficient [x] will always be 0 if (1) is true


say x = -0.5 [x] = 0 (least integer greater than or equal to -0.5)
say x = -0.11 [x] =0 (least integer greater than or equal to -0.11)

So for any number you choose between -1 and -.1 as x, [x] =0 and hence can answer the question


2. | x + .05 | = 1 can have two solutions

one corresponding to x + .05 =1 and other to x + .05 = -1

x+0.5 =1 gives x =0.5; [x] =1 (least integer greater than or equal to 0.5)
x+0.5 = -1 gives x = -1.5 [x] = -1 (least integer greater than or equal to -1.5)

Sufficient to answer the question whether [x] =0 (it is not; [x] is either 1 or -1)

D is the answer

B) For all z, [y] denotes the least integer less than or equal to y. Is d < 1?
(1) d = y- [y]
(2) d = 0

1) say y =1.9; [y] =1 (the least integer less than or equal to y)
y-[y] = 1.9-1 =0.9
say y = -1.1; [y] = -2 (the least integer less than or equal to y)
y-[y] = -1.1 -(-2) = 0.9

In any case d<1 sufficient

2) d=0 sufficient to answer whether d<1

Hence D

Sorry for the mistake in the earlier solution
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