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Writing a number as sum of ....

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by wilderness » Fri Jul 25, 2008 8:49 am
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


OA is 28. But I dont have any clue how to even attack this problem, let alone find the answer. Any cluses ?
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Re: Writing a number as sum of ....

by Ian Stewart » Fri Jul 25, 2008 9:39 am
wilderness 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


OA is 28. But I dont have any clue how to even attack this problem, let alone find the answer. Any cluses ?
As you've written the problem, the range is infinite. I think it is supposed to read:

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

You might then notice that |a|/a can only be equal to 1 or -1, and similarly for the other terms, and from there try to work out what the maximum and minimum possible values are for the expression above. Can post the complete solution if you get stuck.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by wilderness » Fri Jul 25, 2008 10:29 am
Thanks, Ian. Now its clear. Actually it was written in exactly the same way in the software as I have posted (wondering if this last sentence of mine would be correct in SC ;-)
Guess the software also missed the division sign.

But I am amazed by the level of your skill. Do you have some magic Genie ;-)

Thanks a lot.

Regrds,
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by floravaze » Fri Jul 25, 2008 10:41 am
Hi Ian;
do post the complete answer.. i am quite stuck.i know that the min value is -1 but how do i get the max value?please
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by Ian Stewart » Fri Jul 25, 2008 11:54 am
Thanks, wilderness :)

The question relies on the fact that:

|x|/x = 1 if x is positive
|x|/x = -1 if x is negative

So, if a, b, c and d are all positive then:

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

will just be equal to 1+2+3+4+5 = 15. So 15 is the maximum possible value of the expression.

To find the minimum value, notice that the last term, 5|abcd|/abcd, has the largest coefficient (5)- this term makes the largest contribution to the sum. To get the minimum value, we'll need to ensure that abcd is negative, and for that to happen, one of a, b, c or d will need to be positive. Notice also that |a|/a makes the smallest contribution to the sum; to get the minimum value, we need to make a positive, and b, c and d negative. With this combination

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

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

So the range is max - min = 15 - (-13) = 28.

I'd add that I don't think I've ever seen a real GMAT question that is much like this one.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by sudhir3127 » Fri Jul 25, 2008 8:46 pm
i was wondering why shud we take mod a/a = +1 even that shud be negative one giving the total as -15

the range would be 15-(-15) = 30..

Any thoughts on it?
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by reachac » Fri Jul 25, 2008 9:00 pm
sudhir3127 wrote:i was wondering why shud we take mod a/a = +1 even that shud be negative one giving the total as -15

the range would be 15-(-15) = 30..

Any thoughts on it?
coz if we take mod a/a = -1 this means a is -ve....so expresion abcd becomes +ve hence we get -1 - 2 - 3 - 4 + 5 = 5
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by sudhir3127 » Fri Jul 25, 2008 9:05 pm
kewl !!!!!!!!!.. got it .. sorrie abt missing that point...
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by raunekk » Sat Jul 26, 2008 6:37 am
Thanks Ian,

I felt as if i am in a classroom.. :D
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