Writing a number as sum of ....

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

wilderness: Master | Next Rank: 500 Posts; Posts: 102; Joined: Sat Mar 15, 2008 4:03 am; Thanked: 4 times

Writing a number as sum of ....

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 ?

Quote

Source: Beat The GMAT — Problem Solving | Fri Jul 25, 2008 8:49 am

GMAT/MBA Expert

Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

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

ianstewartgmat.com

Quote

wilderness: Master | Next Rank: 500 Posts; Posts: 102; Joined: Sat Mar 15, 2008 4:03 am; Thanked: 4 times

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,

Quote

floravaze: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Wed Jun 25, 2008 5:30 am

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

touch the sky

Quote

GMAT/MBA Expert

Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

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

ianstewartgmat.com

Quote

sudhir3127: Legendary Member; Posts: 829; Joined: Mon Jul 07, 2008 10:09 pm; Location: INDIA; Thanked: 84 times; Followed by:3 members

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?

Quote

reachac: Legendary Member; Posts: 572; Joined: Fri Feb 22, 2008 9:25 am; Thanked: 21 times

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

Quote

sudhir3127: Legendary Member; Posts: 829; Joined: Mon Jul 07, 2008 10:09 pm; Location: INDIA; Thanked: 84 times; Followed by:3 members