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

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18
45 00:00

Difficulty:   95% (hard)

Question Stats: 38% (02:20) correct 62% (02:15) wrong based on 627 sessions

### HideShow timer Statistics 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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Re: T is the set of all numbers that can be written as the follo  [#permalink]

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26
18
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.

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

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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]

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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]

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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.

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]

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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.

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]

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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.

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]

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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.

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]

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2
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]

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

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_________________ Re: T is the set of all numbers that can be written as the follo   [#permalink] 24 Sep 2018, 17:15
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