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Integers

by usfall13ivy » Mon May 28, 2012 5:46 am
If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
A-2
B-3
C-4
D-5
E-6

Source: Magoosh GMAT

OA is given a option D-5. The explanation assumes highest integer to be one of ab and every other integer to be negative. My doubt is, how can we know for certain that a or b is the highest integer, what is one of c,d,e,f is the highest integer, the entire premise of the Given solution is wrong. The only safe way to account for all the possibilities and yet satisfy the equation would be to assume the highest number of integers to be 4 Option C split as one of ab and three of c,d,e,f since in GMAT we have to find an answer that would account for all the possibilities. Please explain if you think my chain of thought with respect to this answer is wrong.[/spoiler]
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by Anurag@Gurome » Mon May 28, 2012 7:01 am
usfall13ivy wrote:If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
Let's first see in how many ways we can make (ab + cdef) negative and analyze each situation.

1. ab > 0 and cdef < 0 and |ab| < |cdef|
  • None or both of a or b are negative.
    Either one or three of c, d, e, and f are negative.
    Hence, maximum 5 negative inetgers
2. ab < 0 and cdef > 0 and |cdef| < |ab|
  • Eithera or b is negative.
    Either none or two or all of c, d, e, and f are negative.
    Hence, maximum 5 negative inetgers
3. ab < 0 and cdef < 0
  • Either a or b is negative.
    Either one or three of c, d, e, and f are negative.
    Hence, maximum 4 negative inetgers
Hence, maximum number of negative integers that can be negative is 5.

The correct aswer is D.
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by Anurag@Gurome » Mon May 28, 2012 7:06 am
usfall13ivy wrote:If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
A more easier and less time consuming approach...

All of the six integers cannot be negative as that would make the whole expression positive. Now can five of the six be negative?

Yes.
If all of c, d, e, and f are negative, then cdef > 0 and either of a or b but not both are negative, then ab < 0. Now if we choose |ab| > |cdef|, then (ab + cdef) < 0.

For example, take a = -2, b = 1, c = d = e = f = -1
Then, ab = -2 and cdef = 1
Hence, (ab + cdef) = (-2 + 1) = -1 < 0

Hence, maximum number of negative integers that can be negative is 5.

The correct aswer is D.
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by usfall13ivy » Mon May 28, 2012 7:29 am
Anurag@Gurome wrote:
usfall13ivy wrote:If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
A more easier and less time consuming approach...

All of the six integers cannot be negative as that would make the whole expression positive. Now can five of the six be negative?

Yes.
If all of c, d, e, and f are negative, then cdef > 0 and either of a or b but not both are negative, then ab < 0. Now if we choose |ab| > |cdef|, then (ab + cdef) < 0.

For example, take a = -2, b = 1, c = d = e = f = -1
Then, ab = -2 and cdef = 1
Hence, (ab + cdef) = (-2 + 1) = -1 < 0

Hence, maximum number of negative integers that can be negative is 5.

The correct answer is D.
But what if, a=-1, b =1, c,d,e=-1 and f = -2? The answer would then be 1, which is positive. That was my original doubt with the original explanations, we cant just assume values that would give us an answer we need, need to account for all factors right? or is it that, since the questions asks for Maximum possible values, we take 5 and if there is a scenario in which it fits, we use 5?
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by Anurag@Gurome » Mon May 28, 2012 7:42 am
usfall13ivy wrote:...or is it that, since the questions asks for Maximum possible values, we take 5 and if there is a scenario in which it fits, we use 5?
Exactly what I'd have typed.

The question simply asks for the maximum number of integers that can be negative. We don't need to satisfy any unnecessary conditions like what if a was greater than c etc.
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by slay3r » Mon May 28, 2012 8:04 am
Anurag@Gurome wrote:
usfall13ivy wrote:If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
A more easier and less time consuming approach...

All of the six integers cannot be negative as that would make the whole expression positive. Now can five of the six be negative?

Yes.
If all of c, d, e, and f are negative, then cdef > 0 and either of a or b but not both are negative, then ab < 0. Now if we choose |ab| > |cdef|, then (ab + cdef) < 0.

For example, take a = -2, b = 1, c = d = e = f = -1
Then, ab = -2 and cdef = 1
Hence, (ab + cdef) = (-2 + 1) = -1 < 0

Hence, maximum number of negative integers that can be negative is 5.

The correct aswer is D.
This could be arrived at two different ways


ab + cdef < 0


1) |ab| > |cdef| and ab being negative
or
2 ) ab<0 and cdef <0

in the case of (1) cdef should be positive that means that all c,d,e & f are negative with a or b being negative that makes the number as 5

or in the case of (2) ab <0 in which case either a or b can be negative
and cdef<0 which makes any of the three numbers be negative in which case the max number is only (3+1) 4

and hence the max number of negative numbers happen in case (1).
I hope this line of thought helps in reaching the solution a lot faster.
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