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M28-56

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If \([x]\) denotes the greatest integer less than or equal to \(x\) for any number \(x\), is \([a] + [b] = 1\)?


(1) \(ab = 2\).

(2) \(0 \lt a \lt b \lt 2\).
[Reveal] Spoiler: OA

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Given that some function [] rounds DOWN a number to the nearest integer. For example \([1.5]=1\), \([2]=2\), \([-1.5]=-2\), ...

(1) \(ab = 2\). First of all this means that \(a\) and \(b\) are of the same sign.

If both are negative, then the maximum value of \([a] + [b]\) is -2, for any negative \(a\) and \(b\). So, this case is out.

If both are positive, then in order \([a] + [b] = 1\) to hold true, must be true that \([a]=0\) and \([b]=1\) (or vise-versa). Which means that \(0 \leq a \lt 1\) and \(1 \leq b \lt 2\) (or vise-versa). But in this case ab cannot be equal to 2. So, this case is also out.

We have that the answer to the question is NO. Sufficient.

(2) \(0 \lt a \lt b \lt 2\). If \(a=\frac{1}{2}\) and \(b=1\), then \([a] + [b] = 0 + 1 = 1\) but if \(a=\frac{1}{4}\) and \(b=\frac{1}{2}\), then \([a] + [b] = 0 + 0 = 0\). Not sufficient.


Answer: A
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New post 26 Nov 2014, 08:04
For stmt 1, wouldn't the max value of [a]+[b] be -3? because (-1)(-2)=2, and (-1)+(-2)=-3?

Or am I missing some factors?

Thanks

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New post 26 Nov 2014, 08:39
makes sense. Thanks

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New post 24 Aug 2016, 03:51
I think this is a high-quality question and I agree with explanation.

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New post 29 Aug 2016, 07:27
I think this the explanation isn't clear enough, please elaborate.

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New post 05 Sep 2017, 14:15
Bunuel wrote:
JackSparr0w wrote:
For stmt 1, wouldn't the max value of [a]+[b] be -3? because (-1)(-2)=2, and (-1)+(-2)=-3?

Or am I missing some factors?

Thanks



It says that the maximum possible value of [a] + [b] is -2 (without the restriction ab=2).


Hello Bunuel
What combination would give us value of (-2)?
the max i could think of is [-1.4] + [-1.4] = (-2) + (-2) = (-4)
[?] + [?] = (-1) + (-1) = (-2) ?
Can you please let me know.
Thanks

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New post 05 Sep 2017, 20:07
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manishtank1988 wrote:
Bunuel wrote:
JackSparr0w wrote:
For stmt 1, wouldn't the max value of [a]+[b] be -3? because (-1)(-2)=2, and (-1)+(-2)=-3?

Or am I missing some factors?

Thanks



It says that the maximum possible value of [a] + [b] is -2 (without the restriction ab=2).


Hello Bunuel
What combination would give us value of (-2)?
the max i could think of is [-1.4] + [-1.4] = (-2) + (-2) = (-4)
[?] + [?] = (-1) + (-1) = (-2) ?
Can you please let me know.
Thanks


What I meant was that generally if a and b are negative, then the maximum value of [a] + [b] is -1+(-1)=-2. But you cannot get -2, if ab=2.

For example, [-0.5] + [-0.5] = -1 + (-1) = -2.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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New post 06 Sep 2017, 16:45
It says that the maximum possible value of [a] + is -2 (without the restriction ab=2).[/quote]

Hello Bunuel
What combination would give us value of (-2)?
the max i could think of is [-1.4] + [-1.4] = (-2) + (-2) = (-4)
[?] + [?] = (-1) + (-1) = (-2) ?
Can you please let me know.
Thanks[/quote]

What I meant was that generally if a and b are negative, then the maximum value of [a] + [b] is -1+(-1)=-2. But you cannot get -2, if ab=2.

For example, [-0.5] + [-0.5] = -1 + (-1) = -2.[/quote]


Got it thanks [b]Bunuel

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M28-56   [#permalink] 06 Sep 2017, 16:45
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