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T is the set of all numbers that can be written as the follo

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honchos
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T is the set of all numbers that can be written as the follo [#permalink] New post   01 Dec 2013, 17:34
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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
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C
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Bunuel
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   02 Dec 2013, 01:30
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honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   13 Jul 2017, 21:53
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honchos wrote:
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


How to think in such a problem:

"T is the set of all numbers that can be written as the following sum"
Makes me think that T can take a limited number of values. Else the set would have infinite elements and we will not be able to define the range.

|a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) + 5(|abcd|/abcd)

Seeing this, I recall that |x|/x will be 1 or -1 depending on whether x is positive or negative. It can take no other value.

So to maximise the sum (to get range), we should make all such expressions 1. This happens when all a, b, c and d are positive. The sum will be 1 + 2 + 3 + 4 + 5 = 15

To minimise the sum we should try to make as many terms negative as possible. But abcd will become positive if all a, b, c and d are negative. So we should keep 'a' positive and make all rest negative.
1 - 2 -3 - 4 -5 = -13

Note that it doesn't matter what the actual values of a, b, c and d are. |a|/a will always be only 1 or -1.

Range = 15 - (-13) = 28
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   29 Jan 2014, 03:33
Am I missing sth?
1-2-3-4-5 is -14 and not -13, right? so the correct answer would be D
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   29 Jan 2014, 05:51
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FlaCarv wrote:
Am I missing sth?
1-2-3-4-5 is -14 and not -13, right? so the correct answer would be D


1 - 2 - 3 - 4 - 5 = -13 not -14.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   07 Jul 2014, 10:32
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.


Hey Bunuel..there are many values in between that we can never achieve..say 0..so how is range defined in such cases...is it just the max-min..
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   07 Jul 2014, 10:35
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JusTLucK04 wrote:
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.


Hey Bunuel..there are many values in between that we can never achieve..say 0..so how is range defined in such cases...is it just the max-min..


Yes, the range is always the difference between the largest and smallest.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   12 Jul 2017, 18:49
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.


I took abcd as a 4 digit number.
Bunuel , could you please edit the question and replace abcd with a*b*c*d , for clarity. :)
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   12 Jul 2017, 21:24
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rekhabishop wrote:
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.


I took abcd as a 4 digit number.
Bunuel , could you please edit the question and replace abcd with a*b*c*d , for clarity. :)


If abcd were a 4-digit number if would have been mentioned explicitly. Without that, abcd can only be a*b*c*d since only multiplication sign (*) is usually omitted.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   14 Jul 2017, 19:22
I am wondering why no one asked why we are considering "a" positive, why not b or c or d ?
well the answer is: as we are looking for minimum value we should add as less as possible, thus á is considered +ve value.
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T is the set of all numbers that can be written as the follo [#permalink] New post   08 Sep 2020, 09:30
I am confused absolute regarding values. Could some clarify a bit?
As per theory |x| is +ve if x>0 and -ve when x<0
And also as per theory and solutions in multiple questions |-3| is taken as 3.

Aren't the two statements contradictory?
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   08 Sep 2020, 09:41
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realslimsiddy wrote:
I am confused absolute regarding values. Could some clarify a bit?
As per theory |x| is +ve if x>0 and -ve when x<0
And also as per theory and solutions in multiple questions |-3| is taken as 3.

Aren't the two statements contradictory?


The part I've highlighted in red is not correct. |x| is never negative; |x| can be zero, when x is zero, and in all other cases |x| is positive.

What is true is the following:

• When x > 0, |x| = x
• When x < 0, |x| = -x

But in this second case, when x < 0, notice that -x is a positive number (because if x is negative, -x will be positive), so in both cases, |x| is positive.

Sometimes test takers find this confusing at first, because "-x" looks like a negative number. But it only sometimes is -- it all depends on whether x itself is positive or negative.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   08 Sep 2020, 09:48
IanStewart wrote:
realslimsiddy wrote:
I am confused absolute regarding values. Could some clarify a bit?
As per theory |x| is +ve if x>0 and -ve when x<0
And also as per theory and solutions in multiple questions |-3| is taken as 3.

Aren't the two statements contradictory?


The part I've highlighted in red is not correct. |x| is never negative; |x| can be zero, when x is zero, and in all other cases |x| is positive.

What is true is the following:

• When x > 0, |x| = x
• When x < 0, |x| = -x

But in this second case, when x < 0, notice that -x is a positive number (because if x is negative, -x will be positive), so in both cases, |x| is positive.

Sometimes test takers find this confusing at first, because "-x" looks like a negative number. But it only sometimes is -- it all depends on whether x itself is positive or negative.



Thank you so much. This was very helpful.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   25 Sep 2020, 10:37
honchos wrote:
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

I fell again on this type of question. Here's how i analysed.
So, from the given condition 15 can never be the range. Eliminate A.
As we can take a,b,c and d negative the lowest TRAP value we get is -5(-1 - 2 - 3 - 4 + 5). This gives us one of the trap answers 20.

Hence our next step should be to get the least value of the T. This can be done by taking the lowest values of components whose coefficient are larger so that sum is least(largest negative). Only way to do this is taking value of 'a' positive and rest negative so that 5(|abcd|/abcd) give -5('a*b*c*d' should be negative).
Hence the least value if 1 - 2 - 3 - 4 - 5 = -13.
Range = 15 - (-13) = 28.

Answer C.
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   04 Mar 2021, 03:16
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.



Why are we not including '0' in the range?
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Re: T is the set of all numbers that can be written as the follo [#permalink] New post   04 Mar 2021, 03:19
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pranshum1 wrote:
Bunuel wrote:
honchos wrote:
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


\(\frac{|x|}{x}\) is 1 when \(x>0\) and -1 when \(x<0\). For example, if \(x=2\), then \(\frac{|x|}{x}=1\) and if \(x=-2\), then \(\frac{|x|}{x}=-1\).

Thus the maximum value of \(\frac{|a|}{a} + 2(\frac{|b|}{b}) + 3(\frac{|c|}{c}) + 4(\frac{|d|}{d}) + 5(\frac{|abcd|}{abcd})\) is obtained when each of a, b, c and d are positive: 1+2+3+4+5=15.

As for the minimum value: notice that all a, b, c, d and abcd cannot simultaneously be negative. For example if a, b, c and d are negative then abcd will be positive. Thus the minimum value is obtained when a is positive and b, c and d are negative: 1-2-3-4-5=-13.

The range = 15 - (-13) = 28.

Answer: C.



Why are we not including '0' in the range?


What do you mean by "Why are we not including '0' in the range"?

The range of a set is the difference between the largest and smallest elements of the set. The largest element = 15 and the smallest = -13. The range = 15 - (-13) = 28.
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