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GMAT Algebra
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Author:  questioner [ Fri Jul 15, 2011 6:34 pm ]
Post subject:  GMAT Algebra

T is the set of all numbers that can be written as the following sum involving distinct non-zero integers a, b, c and d: |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd). What is the range of T?
A. 15
B. 20
C. 28
D. 29
E. 30

(C) If a > 0, |a| / a = 1, but if 0 > a, |a| / a = –1 (because you have a positive number being divided by a negative number).
This is also true for |b|/b, |c|/c and |d|/d. What about abcd|abcd? It all depends on the number of negative variables among {a, b, c, d}: if one or three of these variables are negative, |abcd|abcd = –1. Otherwise, |abcd|/abcd = 1.

To find the range we are going find the largest possible value, and then find the smallest possible value.
If each of these four variables is positive, we get 1 + 2 + 3 + 4 + 5 = 15, the greatest possible value of an element in T.
To find the smallest possible value of T, we ideally need to get all the fractions negative. Regarding 5(|abcd|/abcd), it needs to be made negative, because of the five terms in the sum; it’s the one that has the greatest impact on the sum. This term will be –5 only if one or three variables are negative. If all four variables are negative, 5(|abcd|/abcd) will be 5. So, to find the smallest element in T, pick a positive value of a and let the other three variables be negative. The values for the fractions are then (+1), (-2), (-3), (-4), (-5). The corresponding value of the element in T is 1 – 2 – 3 – 4 – 5 = –13, and thus the range of T is 15 – (–13) = 28 (the difference between the highest value and the lowest one). The correct answer is C.
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Shouldn't "0" be counted in the range, therefore making the answer 29?

Author:  Gennadiy [ Fri Jul 15, 2011 6:46 pm ]
Post subject:  Re: GMAT Algebra

No, the range of a set is the difference between its largest element and the smallest one.

EXAMPLES:
The range of {1, 2, 3, 4, 5} is 5 – 1 = 4
The range of {1, 4, 5} is 5 – 1 = 4
The range of {-2.5, 0, 1} is 1 – (-2.5) = 3.5

In our case the set is {-13, ... some elements ... , 15}, so its range is 15 – (-13) = 15 + 13 = 28. The main idea is that we do NOT have to find all the elements, but just the smallest and the largest ones.

Note:
- when a question deals with integers in its statement, it might NOT be all about integers.
- the "range of a function" is a completely different term, which means the set of all the values a function possesses.

Author:  ccheng [ Mon Oct 17, 2011 2:25 pm ]
Post subject:  Re: GMAT Algebra


Author:  Gennadiy [ Mon Oct 17, 2011 3:16 pm ]
Post subject:  Re: GMAT Algebra


Author:  fedana [ Fri Mar 23, 2012 4:44 pm ]
Post subject:  Re: math (t.4, qt.22): sets, algebra

This is a very good question.

Author:  Robert.Delane [ Wed Aug 22, 2012 4:35 pm ]
Post subject:  Re: GMAT Algebra

Why are we assuming the real numbers can be 1, 2, 3, 4, 5 only? Why not 6, 7, 8, 9, 10 or any other number combination?

Author:  Gennadiy [ Wed Aug 22, 2012 4:41 pm ]
Post subject:  Re: GMAT Algebra


Author:  dave [ Fri Nov 02, 2012 5:53 pm ]
Post subject:  Re: GMAT Algebra

If we take a positive value for a, |abc|/abc is going to become positive and 3(|abc|/abc) = 3 not -3.

Author:  Gennadiy [ Fri Nov 02, 2012 6:02 pm ]
Post subject:  Re: GMAT Algebra


Author:  questioner [ Sun Aug 18, 2013 6:52 am ]
Post subject:  Re: GMAT Algebra

The correct answer should be 29 instead of 28: 15 - (-13) + 1, because 15 and -13 are both in the range inclusive.

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