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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
31 Jul 2025, 12:10
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Official Solution:
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
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 \(\frac{|a|}{a} = \frac{a}{a} = 1\), and if \(a < 0\), then \(\frac{|a|}{a} = \frac{-a}{a} = -1\).
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, \(\frac{5|abcd|}{abcd}\), 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, \(\frac{-4d}{|d|}\). 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 \(\frac{-4d}{|d|} = +4\)
• \(c > 0\), so \(\frac{3|c|}{c} = +3\)
• \(b < 0\), so \(\frac{-2b}{|b|} = +2\)
• \(a > 0\), so \(\frac{|a|}{a} = +1\)
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, \(\frac{5|abcd|}{abcd}\), 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, \(\frac{-4d}{|d|}\). 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 \(\frac{-4d}{|d|} = -4\)
• \(c < 0\), so \(\frac{3|c|}{c} = -3\)
• \(b > 0\), so \(\frac{-2b}{|b|} = -2\)
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
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
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 \(\frac{|a|}{a} = \frac{a}{a} = 1\), and if \(a < 0\), then \(\frac{|a|}{a} = \frac{-a}{a} = -1\).
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, \(\frac{5|abcd|}{abcd}\), 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, \(\frac{-4d}{|d|}\). 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 \(\frac{-4d}{|d|} = +4\)
• \(c > 0\), so \(\frac{3|c|}{c} = +3\)
• \(b < 0\), so \(\frac{-2b}{|b|} = +2\)
• \(a > 0\), so \(\frac{|a|}{a} = +1\)
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, \(\frac{5|abcd|}{abcd}\), 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, \(\frac{-4d}{|d|}\). 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 \(\frac{-4d}{|d|} = -4\)
• \(c < 0\), so \(\frac{3|c|}{c} = -3\)
• \(b > 0\), so \(\frac{-2b}{|b|} = -2\)
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
