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11 Jul 2025, 08:00
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D
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Difficulty:
95%
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Question Stats:
49% (02:17) correct
51%
(01:54)
wrong
based on 285
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For non-zero numbers a, b, c, and d, what is the range of values of the expression \(\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}\)?
A. 0
B. 13
C. 15
D. 28
E. 30
A. 0
B. 13
C. 15
D. 28
E. 30
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11 Jul 2025, 08:30
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Bunuel
GMAT Club Official Explanation:
Notice that each term in the expression can take only two possible values: ±1, ±2, ±3, ±4, and ±5, respectively. For example, if a > 0, then |a|/a = a/a = 1, and if a < 0, then |a|/a = -a/a = -1.
To find the range, we need to determine both the maximum and minimum possible values of the expression.
Maximum Value
We start by making the term that contributes the most to the positive sum, 5|abcd|/abcd, equal to 5. This requires abcd to be positive, which happens when an even number of the variables a, b, c, and d are negative.
Next, we consider the term with the next largest possible contribution, -4d/|d|. To make this equal to +4, d must be negative. We continue similarly with the other terms, choosing variable signs that make each expression contribute positively.
Let’s set:
- d < 0, so -4d/|d| = +4
- c > 0, so 3|c|/c = +3
- b < 0, so -2b/|b| = +2
- a > 0, so |a|/a = 1
That’s two negative values (b and d), so abcd > 0, and the last term becomes +5.
Expression becomes: 1 + 2 + 3 + 4 + 5 = 15
Minimum Value
Similarly here, we begin by making the term that contributes the most to the negative sum, 5|abcd|/abcd, equal to -5. This requires abcd to be negative, which occurs when an odd number of the variables a, b, c, and d are negative.
Next, we look at the term with the next largest potential contribution in magnitude, -4d/|d|. To make it equal to -4, d must be positive. We continue choosing signs to ensure that each term contributes negatively to the overall sum.
We can choose:
- d > 0, so -4d/|d| = -4
- c < 0, so 3|c|/c = -3
- b > 0, so -2b/|b| = -2
At this point we notice that for abcd to be negative, a must be positive, so:
- a > 0, so |a|/a = 1
Now abcd = (+)(+)(-)(+) = negative, so the last term is -5.
Expression becomes: 1 - 2 - 3 - 4 - 5 = -13
Range = 15 - (-13) = 28
Answer: D.
General Discussion
11 Jul 2025, 08:11
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The given expression involves sign functions that evaluate to either +1 or -1 because all the variables involved are non-zero. When simplified, the expression becomes a sum of these sign values, each multiplied by a constant. By testing different combinations of positive and negative values for the variables, we find that the minimum possible value of the expression is –15 and the maximum is 15. Therefore, the total range of values the expression can take is 30. The correct answer is option E, 30.
