Writing a number as sum of ....

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.

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.