Problem Solving

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RiyaR: Senior | Next Rank: 100 Posts; Posts: 98; Joined: Sat May 31, 2014 8:29 pmFollowed by: 2 members; Thanked: 1 times

by RiyaR » Wed Sep 10, 2014 10:06 am

The average of a set of five distinct integers is 300. If each number is less than 2,000, and the median of the set is the greatest possible value, what is the sum of the two smallest numbers?

A) -4494
B) -3997
C)-3494
D)-3194
E) The answer cannot be determined from the data given

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Source: Beat The GMAT — Problem Solving | Wed Sep 10, 2014 10:06 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Sep 10, 2014 10:24 am

RiyaR wrote:The average of a set of five distinct integers is 300. If each number is less than 2,000, and the median of the set is the greatest possible value, what is the sum of the two smallest numbers?

A) -4494
B) -3997
C)-3494
D)-3194
E) The answer cannot be determined from the data given

Let the 5 integers, in ascending order, be as follows:
a, b, c, d and e.
The value of c -- the median -- must be as great as possible.
Since the 5 integers must be distinct, and each must be less than 2000, the greatest possible values for c, d and e are 1997, 1998, and 1999.
Sum of the 5 integers = (number)(average) = 5*300 = 1500.
Thus:
a+b = 1500 - 1997 - 1998 - 1999 = 1500 - (2000 - 2000 - 2000) + (3+2+1) = -4500 + 6 = -4494.

The correct answer is A.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Sep 10, 2014 10:30 am

RiyaR wrote:The average of a set of five distinct integers is 300. If each number is less than 2,000, and the median of the set is the greatest possible value, what is the sum of the two smallest numbers?

A) -4494
B) -3997
C)-3494
D)-3194
E) The answer cannot be determined from the data given

Let the 5 integers be a, b, c, d and e, such that a < b < c < d < e

The average of a set of five distinct integers is 300
So, the SUM of all 5 numbers = (5)(300) = 1500

Each number is less than 2,000 AND we want to MAXIMIZE the median (which is c)
So, let e = 1999
d = 1998
and c = 1997
Now that we have MAXIMIZED the median, what is the sum of the two smallest numbers (i.e., a + b)?
Well, we know that a + b + c + d + e = 1500
So, we can write a + b + 1997 + 1998 + 1999 = 1500

IMPORTANT: To make things easy calculations-wise, notice that 1997 + 1998 + 1999 is ALMOST 6000. In fact it's 6 less than 6000.
So, we can write: a + b + (6000 - 6) = 1500
Now subtract 6000 from both sides: a + b - 6 = -4500
Add 6 to both sides: a + b = -4494

Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Wed Sep 10, 2014 12:05 pm

Hi RiyaR,

Sometimes the answer choices to a PS question are written in such a way as to provide a huge shortcut to the Test Taker who is paying attention.

In this prompt, answer E ("cannot be determined") is a relative rarity on PS questions on Test Day. It might show up on some rare Ratio questions (and might be the correct answer), but normally there 5 answers to consider (as opposed to 4 answers and 1 "cannot be determined").

In the event that answer E was actually a value (something greater than -3194, since the answers are arranged in ascending order), here is the shortcut that I hinted at earlier....

We have a set of 5 distinct values that has an average of 300.

Sum/5 = 300

Sum of the 5 terms = 1500.

We're told that each number is less than 2,000, but we want to make the median as high as possible. By extension, we have to make the 2 values that are larger than the median as high as possible.....

These 3 values would each be really close to 2,000; for estimation purposes, we'll just say that they total 6,000.

To figure out the sum of the 2 smallest numbers, we have this equation...

(sum of 2 smallest numbers) + 6,000 = 1,500

(sum of 2 smallest numbers) = -4,500

There's only one answer that's close....

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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RiyaR: Senior | Next Rank: 100 Posts; Posts: 98; Joined: Sat May 31, 2014 8:29 pmFollowed by: 2 members; Thanked: 1 times

by RiyaR » Thu Sep 11, 2014 9:29 am

Thank you all so much! I should have thought of this solution!

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Tue Dec 12, 2017 10:29 am

RiyaR wrote:The average of a set of five distinct integers is 300. If each number is less than 2,000, and the median of the set is the greatest possible value, what is the sum of the two smallest numbers?

A) -4494
B) -3997
C)-3494
D)-3194
E) The answer cannot be determined from the data given

We are given that the average of a set of five distinct integers is 300, and thus, the sum of the 5 integers is (5)(300) = 1,500. Since each number is less than 2,000 and we want the median to be as large as possible, the median should be 1,997, so that the two largest numbers could be 1,998 and 1,999. Therefore, the sum of the three largest numbers in the set is 1,997 + 1,998 + 1,999 = 5,994. Since the sum of the 5 numbers is 1,500, the sum of the two smallest numbers must be 1,500 - 5,994 = -4,494.

Answer: A

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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