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# T is the set of all numbers that can be written as the following sum

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T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 09:41
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: t-is-the-set-of-all-numbers-that-can-be-written-as-the-follo-164005.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 15 Apr 2015, 03:54, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 09:49
apologies, I could not get the question
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 09:54
hi0parag wrote:
apologies, I could not get the question

A mistype which I now corrected in Edit: "written as sum" => "written as a sum".
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 09:59
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 10:06
Max Value - When a,b,c,d is positive = 1+2+3+4+5 = 15

Min Value - When we have 3 negatives and 1 positive (since if we had all 4 positive, then abcd would also be positive and the expression 5*(|abcd|/abcd) would have a positive value

So to get the lowest possible value for T: b,c,d - negative, a - positive

which yields 1-2-3-4-5 = -13

Range = 28

The only thing I am confused about is whether 0 will be counted as part of the range

So, is the range 28 or 29?
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 10:09
yellowjacket wrote:
Max Value - When a,b,c,d is positive = 1+2+3+4+5 = 15

Min Value - When we have 3 negatives and 1 positive (since if we had all 4 positive, then abcd would also be positive and the expression 5*(|abcd|/abcd) would have a positive value

So to get the lowest possible value for T: b,c,d - negative, a - positive

which yields 1-2-3-4-5 = -13

Range = 28

The only thing I am confused about is whether 0 will be counted as part of the range

So, is the range 28 or 29?

Last edited by nirimblf on 07 Jul 2008, 10:10, edited 1 time in total.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 10:10
My first thought is this:

|a|/a is going to be +/- 1...either positive or negative 1. The absolute value of any number divided by that number will be -1 or 1 because, for example if a = -4, then that is |-4|/-4, or 4/-4 = -1. But |4|/4 = +1.

The only thing I'm not sure about is the |abcd|/abcd. I think this should be the same...I came up with a range of 30.

On a number line, the farthest left it could be is -1 - 2 -3 -4 -5. And the most positive that it could be is 1 + 2 + 3 + 4 + 5. These equal -15 and +15 respectively. 15 - (-15) = 30.

nirimblf wrote:
T is the set of all numbers that can be written as a sum involving distinct real numbers 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

It is an 800score question. Initially I guessed the solution. Then, reviewing it, it took me a few minutes and still I got it wrong. The solution isn't that complicated (try), but I really doubt if I could do it in 2 minutes. What do you think?

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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 10:15
jallenmorris wrote:
My first thought is this:

|a|/a is going to be +/- 1...either positive or negative 1. The absolute value of any number divided by that number will be -1 or 1 because, for example if a = -4, then that is |-4|/-4, or 4/-4 = -1. But |4|/4 = +1.

The only thing I'm not sure about is the |abcd|/abcd. I think this should be the same...I came up with a range of 30.

On a number line, the farthest left it could be is -1 - 2 -3 -4 -5. And the most positive that it could be is 1 + 2 + 3 + 4 + 5. These equal -15 and +15 respectively. 15 - (-15) = 30.

That's what I initially thought. But note that abcd is positive if a,b,c,d are all negative, therefore |abcd|/abcd will be positive as well.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 10:34
nirimblf wrote:
yellowjacket wrote:
Max Value - When a,b,c,d is positive = 1+2+3+4+5 = 15

Min Value - When we have 3 negatives and 1 positive (since if we had all 4 positive, then abcd would also be positive and the expression 5*(|abcd|/abcd) would have a positive value

So to get the lowest possible value for T: b,c,d - negative, a - positive

which yields 1-2-3-4-5 = -13

Range = 28

The only thing I am confused about is whether 0 will be counted as part of the range

So, is the range 28 or 29?

I think 0 should be considered as the range is over the number line.
Hence ans: D 29
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 11:09
nirimblf wrote:
T is the set of all numbers that can be written as a sum involving distinct real numbers 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

It is an 800score question. Initially I guessed the solution. Then, reviewing it, it took me a few minutes and still I got it wrong. The solution isn't that complicated (try), but I really doubt if I could do it in 2 minutes. What do you think?

10 sec solution. |a|/a = sign(a) ... highest value if +,+,+,+ -> 15, lowest value if +,-,-,- -> -13 => 15-(-13) = 28 -> C
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 11:24
Expert's post
hi0parag wrote:
I think 0 should be considered as the range is over the number line.
Hence ans: D 29

The range is defined to be the largest value minus the smallest value in the set. 15 - (-13) = 28, and it makes no difference whether 0 is in the set.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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07 Jul 2008, 12:09
Another vote for C : 28
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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08 Jul 2008, 09:56
Hi All

|a|/a is going to be +/- 1...either positive or negative 1. The absolute value of any number divided by that number will be -1 or 1 because, for example if a = -4, then that is |-4|/-4, or 4/-4 = -1.

There is a gmat catch @ |abcd|/abcd,
if all the numbers are negative, then minimum value will be -1-2-3-4 +5 (since multiplication of 4 -ves will become positive and 5 will remain positive)
so min value = -5
max value = 15
Range = 20 in this case.

but if we take a = 1, b=-1, c = -1 and d = -1,
then we will have min value = 1-2-3-4 -5 = -14
and max value = 15

so Range = 29
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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08 Jul 2008, 10:07
selvae wrote:
Hi All

|a|/a is going to be +/- 1...either positive or negative 1. The absolute value of any number divided by that number will be -1 or 1 because, for example if a = -4, then that is |-4|/-4, or 4/-4 = -1.

There is a gmat catch @ |abcd|/abcd,
if all the numbers are negative, then minimum value will be -1-2-3-4 +5 (since multiplication of 4 -ves will become positive and 5 will remain positive)
so min value = -5
max value = 15
Range = 20 in this case.

but if we take a = 1, b=-1, c = -1 and d = -1,
then we will have min value = 1-2-3-4 -5 = -14
and max value = 15

so Range = 29

Hey I think min value should be -13 so Range comes out to be again 28
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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08 Jul 2008, 10:23
nirimblf wrote:
T is the set of all numbers that can be written as a sum involving distinct real numbers 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

It is an 800score question. Initially I guessed the solution. Then, reviewing it, it took me a few minutes and still I got it wrong. The solution isn't that complicated (try), but I really doubt if I could do it in 2 minutes. What do you think?

I think the quickest way to answer this question has been posted by maratikus:
maratikus wrote:
10 sec solution. |a|/a = sign(a) ... highest value if +,+,+,+ -> 15, lowest value if +,-,-,- -> -13 => 15-(-13) = 28 -> C

To build on that and just if you want to answer a more difficult question based on the same problem:

How many different elements does the set T contain ?
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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08 Jul 2008, 13:55
maratikus wrote:
nirimblf wrote:
T is the set of all numbers that can be written as a sum involving distinct real numbers 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

It is an 800score question. Initially I guessed the solution. Then, reviewing it, it took me a few minutes and still I got it wrong. The solution isn't that complicated (try), but I really doubt if I could do it in 2 minutes. What do you think?

10 sec solution. |a|/a = sign(a) ... highest value if +,+,+,+ -> 15, lowest value if +,-,-,- -> -13 => 15-(-13) = 28 -> C

best way to do it in less than 30 sec.

if it can be seen in one sought, it even doesnot take more than few seconds. i do not say its a 5 or 10 second question but definitely it should not be taking more than 30 sec. its basically a one sought question.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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08 Jul 2008, 17:20
Expert's post
Oski wrote:
To build on that and just if you want to answer a more difficult question based on the same problem:

How many different elements does the set T contain ?

Nice question. Thirteen- curious if you have a quick way to get there. Clearly all the values in T are odd, which leaves at most 15 possible values (using the range), and 13 and 11 are clearly impossible. That leaves at most 13 elements in T, but verifying the rest of the possibilities are indeed in T seems to take a bit more work. I noted that if |abcd|/abcd = 1, then 15 and -5 are in T (for all positive and all negative signs), and if two of {a,b,c,d} are positive, |a|/a + 2(|b|/b) + 3(|c|/c) + 4(|d|/d) can take on any even value from -4 to 4, which means all positive odd integers from 1 to 9 are in T. Doing something similar for when |abcd|/abcd = -1 got me the rest of the odd negative values.
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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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Re: T is the set of all numbers that can be written as the following sum [#permalink]

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15 Apr 2015, 03:54
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Expert's post
nirimblf 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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: t-is-the-set-of-all-numbers-that-can-be-written-as-the-follo-164005.html
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Re: T is the set of all numbers that can be written as the following sum   [#permalink] 15 Apr 2015, 03:54
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