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GMAT Club Olympics 2025 (Day 10): For non-zero numbers a, b, c, and d

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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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That is a cool question!

So, what we can do here is the following: We can ignore the variables given completely and just assume, that these allow us to vary the signs in front of the numbers.
That said, we have as a max:
1+2+3+4+5 = 15
and as a min:
-1-2-3-4-5 = -15

As a result, we have a range of 30.

The result is E)

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Bunuel

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The equation consists of 5 terms.

Term 1 outcome is positive if a>0 , and -ve if a <0

Term 2 = -2*b / (mod b) = -2 , when b>0 and 2 , when b<0 .

Term 3 = 3*(mod c)/ c = 3, when c >0 and -c when c<0 .

Term 4 = -4*d / (mod d) = -4, when d> 0 and 4, when d <0.

Term 5 = 5*(mod abcd) / (abcd) , = 5 when all +ve, or -5, when either odd terms are negative.

Case 1:

When a is positive , b is positive, c is negative and d is positive, hence abcd is negative .

Then the equation becomes 1-2-3-4-5 = -13

This is the maximum negative value that can be generated.

Case 2:

When a is positive, b is negative, c is positive and d is negative , and abcd is positive .

Then the equation becomes 1+2+3+4+5 = 15

This is the maximum value.

Range = 15-(-13) = 15+13 = 28.

Option D

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To find the answer we need to first find highest and lowest value.

To make it lowest we need to keep max negative values.

Fraction will give one so assuming numbers wont affect we just have to concentrate on the negative and positive sign.

A for now lets assume it to be negative
B since a negative sign is present before it if b is -ve then the whole value will turn positive so B will be positive only.
C has a positive sign before so this will be negative.
D has a negative sign so this will be positive

Now we have
-a,+b,-c,+d when multiplied, this will give out +ve making 5 +ve. But we need it to be negative for lowest value.

and a is smallest so, we will make it positive.

Number will be:
+1-2-3-4-5=-13

To make it highest we will have to make each integer positive.

So, A=+ve, B=-ve, C=+ve, D=-ve (Product will also give positive)

Number will be:
+1+2+3+4+5=15

Range=highest-lowest
15-(-13)=15+13=28

Answer D

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	mod(a)/a	-2b/mod(b)	3*mod(c)/c	-4mod(d)/d	total
greater than 0	+1	-2	+3	-4	-2
less than zero	-1	+2	-3	+4	+2

according to the value of variable greater than or less than 0 above are the values of the first four terms which total up to the values
mentioned in the fifth column(total)
now we know that {5*mod(abcd)/abcd} can have only two values that are +5 and -5
to find out the range we must add up the extreme values
i.e. +5+2=7 and -5-2=-7
hence the range should be -7 to +7 a total of 15 values

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Topic(s)- Absolute Value, Max/Min, Range
Strategy- Max/Min
Variable(s)- Max Expression Value = "y"; Min Expression Value = "x"
Notation- absolute value = "abs"
Rephrase the Question: What is y - x?

1. Determine what each term could be
i. Note that the variables in each fraction will reduce to (+/- 1)/1

abs(a)/a	-2*b/abs(b)	3*abs(c)/c	-4*d/abs(d)	5abs(abcd)/(abc*d)
1/(1)	-2*(1)/1	3*1/(1)	-4*(1)/1	5*1/(1)
1/(-1)	-2*(-1)/1	3*1/(-1)	-4*(-1)/1	5*1/(-1)

2. Find the max and min value of the expression
i. Select the maximum value of each term
y = (1) + (2) + (3) + (4) + (5) = 15

ii. Select the minimum value of each term
x = (-1) + (-2) + (-3) + (-4) + (-5) = -15

3. Find the range of the expression
y - x = (15) - (-15) = 30

Answer: E

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We know that the fraction $\frac{|x|}{x} $ equals 1 if x is positive and equals -1 if x is negative.
So each expression equals the constant multiplied by either 1 or -1.

To find the highest limit of the total expression, we should see if it's possible for a,c, and abcd to be positive and b and d to be negative.
If a,c are positive and b,d are negative, then abcd would be positive, and the result is: 1+2+3+4+5=15

To find the lowest limit of the total expression, we should see if it's possible for a,c, and abcd to be negative and b and d to be Positive.
If a, c are negative and b,d are positive, then abcd will be positive. So, to still minimize the whole expression, we can assume that a is also positive, since it has the lowest constant multiplied by it. So, a,b,d are positive, and c,abcd are negative. And the result of the expression is: 1-2-3-4-5=-13

The question asks for the range of the expression, which is: 15-(-13)=28

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For non-zero numbers a, b, c, and d, what is the range of values of the expression
|a| / a − 2b / |b| + 3|c| / c − 4d/|d| + 5|abcd| / abcd

First aknowledge that every fraction can be simplified

for every pos integer => |a| / a = 1
for every negative integer => |a| / a = -1

for every pos integer => 2b / |b| = 2
for every negative integer => 2b / |b| = -2

for every pos integer => 3|c| / c = 3
for every negative integer => 3|c| / c = -3

for every pos integer => 4d/|d| = 4
for every negative integer => 4d/|d| = -4

for every pos integer => 5|abcd| / abcd = 5
for every negative integer => 5|abcd| / abcd = -5

if we combine al the lowest possible values, lowest value would be = -1 + -2 + -3 + -4 + -5 = -15, but following the expression this would entail that we need to have an uneven amount of negative values so that the last term would be negative (-5) => therefore c would need to be negative and a,b and d would be positive.

=> resulting in 1 - 2 + (- 3) - 4 +(- 5) = -13 = lowest value for the expression

=> if b and d would be negative and a, c positive than the highest value for the expression will result in = 1 - (-2) + 3 - (-4) + 5 = 15

range = highest value - lowest value = 15 - (-13) = 28

answer D

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For non-zero numbers a, b, c, and d, what is the range (Max - Min) of values of the

expression: $\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$

(a = +ve, b = -ve, c = +ve, d = -ve, then abcd = +ve)
Max: 1 + 2 + 3 + 4 + 5 = 15

(a = +ve, b = +ve, c = -ve, d = +ve, then abcd = -ve)
Min: 1 - 2 - 3 - 4 - 5 = -13;

Range: 15 - (-13) = 28;

A. 0
B. 13
C. 15
D. 28
E. 30

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In the expression abcd has been used
Now a can have 2 values +ve and -ve
Similarly b can have 2 values
c can have 2 values
d can have 2 values
Total combinations that can be made =2*2*2*2= 16
But options are 13 and 15 which are less tahn 16, options 0,28,30 are not possible
1st possible values
So some values will be repeating we have to check for all 16 values
a,b,c,d all are positive then abcd is also positive
Sum,S0=3

2nd possible values
one is negative three are positive
abcd becomes negative
Possible cases are 4C1
-abcd, a-bcd,ab-cd,abc-d
S1= -1-2+3-4-5= -9
S2=1+2+3-4-5=-3
S3=1-2-3-4-5=-13
S4=1-2+3+4-5=1
S1= -9, S2= -3, S3=-13,S4=1

3rd possible cases
2 are negative 2are positive
4C2= 6 possible cases and abcd is positive, substituting +ve and -ve
We get
S5=5, S6=-5,S7=9,S8=1, S9=15, S10=5

4th possible cases
Three are negative and one is positive
4C1 possible cases abcd is negative
S11=-11 ,S12=3 ,S13=-1 ,S15= -7

5th possible case
All are negative abcd is positive
S16= -1+2-3+4+5= 7
Total 16 values are there but S8,S4=1 have same value, S12, S0=3 has same values and S5, S10=5 has same values

So correct answer is 13 option B

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Bunuel

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the expression $ \frac{|a|}{a} = -1 $if $a < 0$, and $+1$, if $a>0 $
similarly, $\frac{b}{|b|} = -1$ if $b<0$, and $+1 $if $b>0 $
We can see that each of the fractions in the expressions depend on the sign of the number involved, if number is negative, then that fraction will be -1, if number is positive, then that fraction will be +1

To find maximum number
We need to get as many fractions to be positive as possible
$\\
a = +1, \frac{|a|}{a} = +1\\
b = -1, \frac{2b}{|b|} = -2\\
c = +1, \frac{3|c|}{c} = +3 \\
d = -1, \frac{4d}{|d|} = -4 \\
abcd = +1, \frac{5|abcd|}{abcd} = +5 \\
$

Sum = +1 -(-2) + 3 -(-4) +5 = 1+2+3+4+5 = 15, this is the max we can get because all components are positive here
Max = 15

To find minimum number, we need to get as many components to be negative as possible,
trying to get all components negative not possible because>>
$a = -1, \frac{|a|}{a} = -1\\
b = +1, \frac{2b}{|b|} = +2\\
c = -1, \frac{3|c|}{c} = -3 \\
d = +1, \frac{4d}{|d|} = +4 \\
abcd = +1, \frac{5|abcd|}{abcd} = +5 \\
\\
$

then sum= -1-2-3-4+5 = 5-10 = -5

To get minimum, we can afford having a to be positive, so that we can get abcd = -1,
$a = +1, \frac{|a|}{a} = +1\\
b = +1, \frac{2b}{|b|} = +2\\
c = -1, \frac{3|c|}{c} = -3 \\
d = +1, \frac{4d}{|d|} = +4 \\
abcd = -1, \frac{5|abcd|}{abcd} = -5$

then sum = +1-2-3-4-5 = 1-14= -13

Range = 15-(-13) = 15+13 = 28
Answer is D

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Correct Answer: Option E 30

Here we will find out minimum and maximum values to find out range:

1) Let's find out maximum values first,
Here we keep,
a= 1
b= 2
c= 3
d= 4

|a|/a - 2b/|b| + 3|c|/c - 4d/|d| + 5|abcd|/abcd
|1|/1 - 2*2/|2| + 3*|3|/3 - 4*4/|4| + 5*|1*2*3*4|/1*2*3*4
1+2+3+4+5= 15

2) Now find out minimum values,
Here we keep,
a= -1
b= -2
c= -3
d= -4

|a|/a - 2b/|b| + 3|c|/c - 4d/|d| + 5|abcd|/abcd
|-1|/-1 - 2*-2/|-2| + 3*|-3|/-3 - 4*-4/|-4| + 5*|-1*-2*-3*-4|/-1*-2*-3*-4
-1-2-3-4-5= -15
Now if we need to find range= Maximum-Minimum
= 15- (-15)
= 30

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To maximise we need to ensure that all values are of same sign otherwise they will decrease the final value.
Now the absolute value of coefficients is 1+2+3+4+5=15
Let us try to make minium value as zero because if we make any value negative it will have an effect on the 15 we calculated earlier.
If signs are a,b,c,d=pos,neg,pos,neg it will eventually come as zero.
So range of values is 15.

Bunuel

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For any non-zero number x, the term $\frac{|x|}{x}$ has only two possible values:

If x > 0, then $\frac{|x|}{x}$ = $\frac{x}{x}$ = 1.
If x < 0, then $\frac{|x|}{x}$ = $\frac{-x}{x}$ = -1

From given expressions:

==>$\frac{|a|}{a}$: The value is either 1 (if a>0) or -1 (if a<0).
==> $-\frac{2b}{|b|}$: The value is either -2(1) = -2(b>0) or -2(-1)=2(b<0).
==>$\frac{3|c|}{c}$: The value is either 3(1) = 3(if c>0) or 3(-1)=-3(if c<0)
==>$-\frac{4d}{|d|}$:The value is either -4(1) = -4(d>0) or -4(-1)=4(d<0)
==>$\frac{5|abcd|}{abcd}$: This term depends on the sign of the product abcd. If abcd > 0, the value is 5(1) = 5. If abcd < 0, the value is 5(-1) = -5

Now finding the maximum value:
To find the max values we need each term to be positive:
From the above expressions the max possible value is = 1(a>0)+2(b<0)+3(c>0)+4(d<0)+5(abcd>0)= 15
and product of the signs is also positive which means abcd>0

Now finding the minimum value:
To minimize the expression, assign signs to make each term as small as possible, ensuring consistency with abcd(abcd<0):
From the previous analysis, the minimum occurs when(a>0),(b>0),(c<0),(d>0):
Now from above expressions:
$\frac{|a|}{a}$= 1 ,$-\frac{2b}{|b|}$= -2,$\frac{3|c|}{c}$= -3, $-\frac{4d}{|d|}$= -4
Sum of first 4 terms = 1-2-3-4= -8
Product abcd < 0 (c<0), so
$\frac{5|abcd|}{abcd}$ = -5
Total = -8-5 = -13

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

Final answer Option D

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When all of $a,b,c,d$ are positive the expression will result in $1-2+3-4+5=3$

When all of $a,b,c,d$ are negative the expression will result in $-1+2-3+4+5=7$

The maximum value of expression will occur when all the terms in expression is positive. When $b & d$ are negative, the value is $1+2+3+4+5=15$

To minimize the value, all the terms of expression must be negative. For that $\text{a & c}$ must be negative. If that's the case the 5th term of expression will be positive.

When only $c$ is negative, the value of expression is $1-2-3-4-5=-13$

Range of values $=15-(-13)=28$

Answer: D

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Bunuel

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Range = Maximum - Minimum

For Maximum value each term should be positive. For that to happen,

a should be positive
b should be negative
c should be positive
d should be negative

abcd should be positive

Max Value = 1 + 2 + 3 + 4 + 5 = 15

For minimum value we should try to have maximum negative value as abcd should be negative.

In that case, we can have

d positive
c negative
b positive
a positive

Minimum value = -5 - 4 - 3 -2 +1 = -13

Range = 15 + 13 = 28

Option D

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Bunuel

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We know that |a| = -a if a<0,
= a if a>=0.

Using this principle, let's find the max and min values of this equation so that we can find the range.

Max values:
Let a>0, b<0, c>0, d<0 , In this case abcd>0. Now let's simplify the equation using above above-stated principle.

$\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$ = 1+2+3+4+5 = 15.

The maximum value of this expression is 15.

Min values:
Let a>0, b>0,c<0,d>0, So abcd<0. Now let's simplify the equation using above above-stated principle.
$\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$ = 1-2-3-4-5 = -13.

The minimum value of the expression is -13.

Hence range is 28. IMO D

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Range is max. value - min. value
Finding max value, we need to make all the negative signs become positive
hence a = a, b= -b, c= c, d=-d
max value = 1 - (-2) + 3 - (-4) + 5 = 15

Finding min value, we need to make all positive signs negative
i.e a=-a, b=b, c=-c, d=d
Hovewer with this abcd will become positive and increase it by +5
Hence we need to make a=-a as +1 will make lesser difference than +5
so a=a, b=b, c=-c, d=d
min value = 1 - 2 + (-3) - 4 + (-5) = -13

Hence range = 15 - (-13) = 28

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Bunuel

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$\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$?

Let's say
a = +ve,
b = -ve,
c = +ve,
d = -ve,
abcd = +ve

=> $\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$ = 1+2+3+4+5 = 15

For range we have to find the lowest possible number
Let's say
a = +ve,
b = +ve,
c = -ve,
d = +ve,
abcd = -ve

=> $\frac{|a|}{a} - \frac{2b}{|b|} + \frac{3|c|}{c} - \frac{4d}{|d|} + \frac{5|abcd|}{abcd}$ = 1-2-3-4-5 = -13

Range = 15 - (-13) = 28

Option D

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12 Jul 2025, 01:01

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to maximize, convert all the values into +

for a, it's already positive, then a +, b has - in front of it, to make it positve, b should be negative, like this for c and d
a +, b -, c + and d - , abcd = +

1+2+3+4+5 = 15 Max

For minimum let's make lowest negative number; for this change all signs into negative
a = -, b +, c - and d + , and abcd +

-1-2-3-4+5 = -5

But wait, the highest number abcd is positive, we need to change it to negative, then let's inverse 1 number, we will pick the lowest number to minimize the negative number
a + , then abcd = -

1-2-3-4-5 = - 13.

Range then 15-(-13) = 28

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