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805+ Level|   Absolute Values|   Min-Max Problems|         
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 01:10
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|x|/x = 1 if x > 0
|x|/x = -1 if x < 0

So, every expression will be +-1 only,
in order to maximize we want all positives; so putting values
max = 1+2+3+4+5 = 15

For minimum, ideally it should have been -15 but if we check carefully, there are no such values as if we make a and c negative abcd will become positive by default.
So we need to keep one expression of either a or c positive.
Min. = 1 -2 -3 -4 -5 = -13

Range = 15 - (-13) = 28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 03:15
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Bunuel
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


 


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IMO D is correct

See, let the equation contain 5 deciding terms which has strong impact on value.

A = |a|/a
B= -2b/|b|
C=3|c|/c
D=-4d/|d|
E=5|abcd|/abcd


Now, to create a maximum valuue we need a situion so all portion generate a positive number right. So, when we take condition when a>0, b<0, c>0,d<0

then since it has even number of negative signs then abcd also is a positive number. so |abcd| = abcd. and ALL the terms will become like, A= 1, B= 2, C= 3, D= 4 and E=5

so total is 15.

Now, for minimum it can give us when all terms are negative, (B,D will give negative term only when mod value is same as number due to having a negative sign from before itself) so this can happen only when a is also < 0 and c is also <0 but then product (E will become positive) will become positive due to even negaative terms so to create negative value what we can do is we can sacrifise sign of one term lets say A, since it has just 1 as multiple and no other higher number so even if it has +1 it is better than having +2 which will not give us lowest value.

So now when we consider, a>0, b>0, c<0 and d>0 then product is also having 1 negative sign so E= -5, D=-4, C=-3, B=-2 and A=+1. So final total is -13

so range will be, maximum - minimum = 15-(-13) = 28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 03:16
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The highest value from the expression is

1-(-2)+3-(-4)+5= 15

The lowest value from the expression is
1-2-3-4-5=-13


so the range lies from -13 to 15

15-(-13)
=28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 03:43
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Bunuel
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
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when a>0 ---> |a|/a = 1
a<0 ---> |a|/a = -1

when b>0 ---> −2b/|b| = -2
b<0 ---> −2b/|b| = 2

when c>0 ---> +3|c|/c= 3
c<0 ---> +3|c|/c= - 3

when d>0 ---> −4d/|d|= -4
d<0 ---> −4d/|d|= 4

when, either 2 or 4 of a,b,c,d are positive ---a*b*c*d >0
+5|abcd|/abcd = 5
when either 1 or 3 of a,b,c,d are negative ---a*b*c*d <0
 +5|abcd|/abcd = - 5


to get maximum value of the given expression, all terms should be positive.... hence, let's a,c>0 , b, d<0
|a/|a−2b/|b|+3|c|/c−4d/|d|+5|abcd|/abcd = 1+2+3+4+5 = 15


to get minimum value of the given expression, all terms should be negative.... hence, let's a,c<0 , b, d>0 , however in this case abcd>0 since exponent of abcd is greater than the exponent of a, we will consider a>0 to get the minimum value ---> a,b,d>0 c<0
|a/|a−2b/|b|+3|c|/c−4d/|d|+5|abcd|/abcd = 1-2-3-4-5 = -13

Range = 15-(-13)=28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 04:11
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Option D is the correct answer.

Lets understand the information mentioned in the question before trying to solve for it.

So the question starts by telling us that "a, b, c and d are not zero numbers" then asks us to find the range of value of expression: |a|/a - 2b/|b| + 3|c|/c - 4d/|d| + 5|abcd|/abcd.

As mentioned in the question a, b, c and d are non-zero numbers which would mean that either they all can be negative, positive or some number negative or some positive. Lets understand it arithmetically: So a, b, c and d could be, a,b,c&d>0, a,b,c&d<0, a&b<0<c&d and so on.

And we all know that mod function never be <0, which would mean that |a|, |b|, |c| & |d| are all greater than 0 i.e. positive numbers.

Now to find the range we need to find the minimum and maximum possible value of the expression: Range = Maximum value - Minimum value

So in order to find the minimum possible value we need to assume c<0 and a,b,&d>0.
As we have assume c<0 which would mean that (-c)>0.
=>|a|/a - 2b/|b| + 3|c|/c - 4d/|d| + 5|abcd|/abcd
=>a/a - 2b/b + (3*(-c)/c) - 4d/d + (5*(-abcd)/abcd)
=>1 - 2 - 3 - 4 - 5
=> -13

In order to find the maximum value of the expression lets assume b&d<0 and a&c>0, which would mean that (-b)&(-d)>0.
=>|a|/a - 2b/|b| + 3|c|/c - 4d/|d| + 5|abcd|/abcd
=>a/a - (2*(-b)/b) + 3c/c - (4*(-d)/d) + 5*(abcd)/abcd
=>1 + 2 + 3 + 4 + 5
=>15

So the range of the expression = 15 - (-13)
Range of the expression = 15 + 13
Range of the expression = 28 (Option D)

Bunuel
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


 


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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 04:31
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To find min, we need fourth term <0, fifth term <0 at the expense of first term
min = 1 -2 -3 -4 -5 = -13 (b>0, c<0, d>0, acbd<0 ; so a>0)

To find max, we need fourth term >0, fifth term >0 at the expense of first term
max = 1 + 2 + 3 + 4 + 5 = 15 (b<0, c>0, d<0, abcd>0 ; so a>0)

Range = 28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 04:45
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After cancelling out the variables, we get ->

\((+-1_a)-(+-2_b)+(+-3_c)-(+-4_d)+(+-5_{abcd})\)

\(+-\) means the coefficient can either be \(+ve\) or \(-ve\) depending on the sign of the variable it was attached to. The subscripts denote the variable it was attached to.

Restructure to have \(+ve\) and \(-ve\) terms grouped together ->

\((+-1_a)+(+-3_c)+(+-5_{abcd})-(+-2_b)-(+-4_d)\)

To get the maximum value from the expression, let's make it such that all terms add to each other and none subtract ->

When \(a\), \(c\) are positive and \(b\), \(d\) are negative, all terms add up (\(abcd\) is also positive)

\(Max=(+1_a)+(+3_c)+(+5_{abcd})-(-2_b)-(-4_d)=1+3+5+2+4=15\)

To get the minimum, make it such that majority terms subtract ->

We can make \(a\), \(c\) negative and \(b\), \(d\) positive, but that will make \((+-5_{abcd})\) positive. We need to make one of \(a\) or \(c\) negative to get \((+-5_{abcd})\) to be negative. The term with the smallest magnitude in the expression is \((+-1_a)\). We can make \(a\) positive and \(c\) negative

\(Min=(+1_a)+(-3_c)+(-5_{abcd})-(+2_b)-(+4_d)=1-3-5-2-4=-13\)

Range\(=Max-Min=15-(-13)=28\)
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 04:55
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To maximize (or minimize), we need to ensure the max/min value of the highest magnitude item first.
Start from -5, then -4, then, -3, then -2.

The highest in magnitude is 5|abcd|/abcd - To maximize this, we want abcd to be + and to minimize this abcd has to be -ve

For max:
|a| = a
|b| = -b
|c| = +
|d| = -d

Max value of the expression = 1+2+3+4+5 = 15

For min:
|d| = d
|c| = -c
|b| = b

Since, we want |abcd| to be -ve, |a| should stay a not -a.

Min. value of the expression = 1-2-3-4-5 = -13

Range = Max - Min = 15-(-13) = 28

Option D
Bunuel
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


 


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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 05:09
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The expression we have is,
|a|/a - 2b/|b| + 3|c|/c -4d/|d| + 5|abcd|/abcd

|a|/a = +- 1 = s1
b/|b| = +- 1 = s2
|c|/c = +- 1 = s3
d/|d| = +- 1 = s4
|abcd|/abcd = +- 1 = x5 = s1s2s3s4

|a|/a - 2b/|b| + 3|c|/c -4d/|d| + 5|abcd|/abcd = s1 -2s2 + 3s3 - 4s4 +5s5
We need a range of t he possible values and to do this lets find the max and min possible values

Maximum value of s1 -2s2 + 3s3 - 4s4 +5s5 is possible when all of them are adding up with each other
=>s2 and s4 should be negative,s1 and s3 positive, which will make s5 positive too
So, s1 = 1 , s2 = -1 ,s3 = 1, s4 = -1 => s5 = 1*-1*1*-1 = 1
Max = s1 + 2s2 + 3s3 + 4s4 + 5s5 = 1 + 2 + 3 + 4 + 5
=> Maximum = 15

Minimum value of s1 -2s2 + 3s3 - 4s4 +5s5 is possible when all of them are subtracting with each other up with each other, but this is not possible since that would make s5 positive, so we make the smallest number s1 positive to make s5 negative
=>s3 should be negative,s1, s2 and s4 positive,which will make s5 negative
So, s1 = 1 , s2 = 1 ,s3 = -1, s4 = 1 => s5 = 1*1*-1*1 = -1
Minimum = s1 - 2s2 - 3s3 - 4s4 - 5s5 = 1 - 2 - 3 - 4 - 5
=> Minimum = -13

Range = |Max - Min| = |15 -(-13)| = 28

D. 28
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 05:16
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This equation simply means the functions of the non-zero numbers can either take -1 or 1 since the numerator and denominators are equal in terms of absolute values
Min value is when a=1, b=1, c=-1 and d=1. This gives us -13
Max value is when a=1, b=-, c=1 and d=-1. This gives us 15
Range is therefore 15-(-13)= 28
Bunuel
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


 


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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 05:17
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Let's look at each term as its coefficient:
|a|/a = 1 or -1
2b/|b| = 2 or -2
3|c|/c = 3 or -3
4d/|d| = 4 or -4
5|abcd|/(abcd) = 5 or -5

To maximize the expression, we can use a= 1 , b= -1, c= 1, d= -1
-> 1 + 2 + 3 + 4 + 5 = 15
To minimize the expression, we can use a= 1 , b= 1, c= -1, d= 1 (if we would take a = -1 then abcd will be positive , expression won't be minimum then)
-> 1 - 2 - 3 - 4 - 5 = -13

Range = Max - Min
-> 15 - (-13)
= 28
Ans. D
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 05:50
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Bunuel
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


 


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Option D is my answer


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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 05:55
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The components of the expression are:
|a|/a = -1 or 1
-2|b|/b = -2 or 2
3|c|/c = -3 or 3
-4|d|/d = -4 or 4
5|abcd|/abcd = -5 or 5

If all the components are positive, the highest value: 15 = 1+2+3+4+5
It's impossible that all the components are negative because a and c should negative, b and d positive and then 5|abcd|/abcd would be positive. The first component must be positive (a positive) and the other four negative (b positive, c negative, d positive).
Lowest value: -13 = 1-2-3-4-5

Range=15-(-13)=28

Answer D
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 07:05
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to find the range, we need to find the maximum and minimum value of the expression. For the maximum value, all values that are added should be maximum and all values that are subtracted, should be minimum.

For maximum:
a>0, c>0 and b<0, d<0 , abcd>0
\(\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}\)
= 1 - 2(-1) + 3 - 4(-1) + 5(1)
= 1 + 2+ 3 + 4 + 5 = 15

For minimum:
c<0 and a>0, b>0, d>0, abcd <0 (we take a>0 as we want the number with the biggest coefficient to be negative)
= 1 - 2 - 3 - 4 - 5 = -13

Range = 15 -(-13) = 28

Option D

Bunuel
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


 


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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 07:10
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All the addends can be positive or negative:
- addend with a can be -1 or +1
- addend with b can be -2 or +2
- addend with c can be -3 or +3
- addend with d can be -4 or +4
- addend with abcd can be -5 or +5

The greatest value is obtained when all the addens are positive: 1+2+3+4+5=15
The smallest value would be obtained when all the values are negative. But the fifth addend is based on the other four and it is not possible, one of them must be positive. Choose the one with the smallest absolute value, a.
smallest value: 1-2-3-4-5=-13

The range between 15 and -13 is 28.

Correct answer is D
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 07:25
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|a|/a is 1 if a is positive and -1 if a is negative.

So the sign of the monomials can be:
a: -1 or +1
b: -1 or +1
c: -1 or +1
d: -1 or +1
abcd: -1 or +1

To find the minimum expression the ideal would be that all of them were negative. But this is not possible because the last one depends on the other four. The most we can do is to do the b, c, d and abcd monomials negative and the a monomial positive:

minimum value = 1-2-3-4-5=-13

The maximum expression occurs when all the monomials are positive:

maximum value = 1+2+3+4+5=15

The range is maximum value minus minimmum value = 15-(-13) = 28

The right answer is D
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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d 12 Jul 2025, 07:31
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abs(x)/x or x/abs(x) can be 1 or -1.

The expression will have the greatest positive value if all the terms are positive. It's possible if b and d are negative and a and c are positive.
Maximum value: 1+2+3+4+5=15

The expression will have the least negative value if the terms with b, c, d and abcd are negative. It's possible if c is negative and a, b and d are positive. It's not possible to do the term with a also negative because in this case abcd would be positive and the fifth term would be positive.
Minimum value: 1-2-3-4-5=-13

Range = 15-(-13) = 28

IMO D
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