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

Question

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

A. 15
B. 20
C. 28
D. 29
E. 30

Bunuel · Accepted Answer

\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.

KarishmaB · Answer

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

FlaCarv · Answer

Am I missing sth?
1-2-3-4-5 is -14 and not -13, right? so the correct answer would be D

Bunuel · Answer

1 - 2 - 3 - 4 - 5 = -13 not -14.

JusTLucK04 · Answer

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..

Bunuel · Answer

Yes, the range is always the difference between the largest and smallest.

rekhabishop · Answer

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.

Bunuel · Answer

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.

jyotipes21@gmail.com · Answer

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.

realslimsiddy · Answer

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?

IanStewart · Answer

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.

realslimsiddy · Answer

Thank you so much. This was very helpful.

unraveled · Answer

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.

pranshum1 · Answer

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

Bunuel · Answer

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.

anirchat · Answer

This is a question and a half !! Great one.

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

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