A set of 5 numbers has an average of 50. The largest element

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Anaira Mitch: Master | Next Rank: 500 Posts; Posts: 235; Joined: Wed Oct 26, 2016 9:21 pmFollowed by: 5 members; Thanked: 3 times

by Anaira Mitch » Mon Mar 06, 2017 9:47 pm

A set of 5 numbers has an average of 50. The largest element in the set is 5 greater than 3 times the smallest element in the set. If the median of the set equals the mean, what is the largest possible value in the set?
(A) 85
(B) 86
(C) 88
(D) 91
(E) 92

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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 » Mon Mar 06, 2017 10:14 pm

Anaira Mitch wrote:A set of 5 numbers has an average of 50. The largest element in the set is 5 greater than 3 times the smallest element in the set. If the median of the set equals the mean, what is the largest possible value in the set?
(A) 85
(B) 86
(C) 88
(D) 91
(E) 92

The sum of the 5 numbers = 5*50 = 250.

Let the smallest number = x.
Since the greatest number is 5 more than 3 times the smallest number, the greatest number = 3x+5.
Median = 50.
Let the remaining numbers be a and b.

Here are the 5 numbers, in ascending order:
x, a, 50, b, 3x+5.

To MAXIMIZE the value of 3x+5, we must MINIMIZE the values of a and b.
The least possible value for a is x.
The least possible value for b is 50.
Here are the 5 numbers:
x, x, 50, 50, 3x+5.

Since the sum of the 5 numbers is 250, we get:
x + x + 50 + 50 + 3x+5 = 250
5x + 105 = 250
5x = 145
x = 29.

Thus:
Greatest possible value for the largest integer = 3x+5 = 3*29 + 5 = 92.

The correct answer is E.

Last edited by GMATGuruNY on Tue Mar 07, 2017 4:28 am, edited 1 time in total.

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Mon Mar 06, 2017 10:30 pm

Anaira Mitch wrote:A set of 5 numbers has an average of 50. The largest element in the set is 5 greater than 3 times the smallest element in the set. If the median of the set equals the mean, what is the largest possible value in the set?

(A) 85
(B) 86
(C) 88
(D) 91
(E) 92

Hi Anaira,

We have 5 numbers in the set whose mean = median = 50

Thus, the sum of the numbers in the set = 5*50 = 250

Say, the smallest number is x, thus the largest number = 3x + 5

Since there are 5 numbers (Odd numbers of terms in the set), median must be the third term = 50.

So we have three terms ready for the set.

1. The fifth and the largest term = 3x + 5
2. The third term = 50
3. The first and the smallest term = x

We have no clue about the second and the fourth term.

We are given that we have to find out the largest possible value in the set, i.e., the value of (3x + 5).

Since the sum of 5 terms is 250, in order to maximize the value of (3x + 5), we must ensure that the second and the fourth term is as least as possible.

Since the third term = median = 50, let's keep the fourth term = 50 (This is the least value it can get; it cannot be less than 50 since the terms are arranged in the ascending order.)

Since the first term = x, let's keep the second term = x

So, the five terms are: (3x + 5), 50, 50, x, x

=> Sum = 250 = 3x + 5 + 50 + 50 + x + x

=> 250 - 105 = 5x

=> x = 29

=> Largest term = (3x + 5) = 3*29 + 5 = 92

The correct answer: E

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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Anaira Mitch: Master | Next Rank: 500 Posts; Posts: 235; Joined: Wed Oct 26, 2016 9:21 pmFollowed by: 5 members; Thanked: 3 times

by Anaira Mitch » Tue Mar 07, 2017 1:32 am

Thanks Mitch and Jay for your help.

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Tue Mar 07, 2017 7:36 am

Anaira Mitch wrote:A set of 5 numbers has an average of 50. The largest element in the set is 5 greater than 3 times the smallest element in the set. If the median of the set equals the mean, what is the largest possible value in the set?
(A) 85
(B) 86
(C) 88
(D) 91
(E) 92

We could also back-solve this question. It makes the most sense to start with E - the largest- and work your way up. The first answer choice that works will have to be correct.

E) 92

If 92 is 5 more than 3 times the smallest value, then we can find the smallest value by subtracting 5 from 92 and dividing by 3. 92-5 = 87; 87/3 = 29.
We already know that the median is 50, so now we have the following set: 29, A, 50, B, 92.

We also know that the sum of all the terms in the set is 5*50 = 250. So 29 + A + 50 + B + 92 = 250

If we wish to maximize the largest value, we want to minimize everything else. The smallest value A can assume is 29 and the smallest value B could assume would be 50 (any smaller and we'd alter the median.) Our final set looks like this: 29, 29, 50, 50, 92. Low and behold 29+29+50+50+92 = 250. Huzzah! E is our answer.

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by [email protected] » Tue Mar 07, 2017 10:51 am

Hi Anaira Mitch,

David brings up a great option for this question - TESTing THE ANSWERS. If you were unsure of which answer to TEST first though - then there are some built-in Number Properties in this prompt that can help you to quickly eliminate a number of the answer choices.

We're asked to find the LARGEST POSSIBLE value in this set of 5 values. Although the question doesn't explicitly state it, based on the information given (and the answer choices) it's likely that the 5 values are all integers. The question tells us that the largest value is 5 GREATER than 3 TIMES the smallest value....

If you take 3 times an integer, then you get a multiple of 3.
If you add 5 to a multiple of 3, then you get a total that is NOT a multiple of 3.

Thus, the largest possible value will NOT be a multiple of 3 AND - when you subtract 5 from it - you WILL get a multiple of 3... Now let's consider how all of this fits with the given answer choices....

Answer A: 85 - 5 = 80... NOT a multiple of 3
Answer B: 86 - 5 = 81... IS a multiple of 3
Answer C: 88 - 5 = 83... NOT a multiple of 3
Answer D: 91 - 5 = 86... NOT a multiple of 3
Answer E: 92 - 5 = 87... IS a multiple of 3

It's highly likely that the correct answer is either Answer B or Answer E. From here, it's just a matter of doing the necessary arithmetic to prove which one is correct.

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Rich

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Thu Mar 23, 2017 3:23 pm

Anaira Mitch wrote:A set of 5 numbers has an average of 50. The largest element in the set is 5 greater than 3 times the smallest element in the set. If the median of the set equals the mean, what is the largest possible value in the set?
(A) 85
(B) 86
(C) 88
(D) 91
(E) 92

We are given that a set of 5 numbers has an average of 50. Thus, the set has a sum of 250.
We are also given that the largest element in the set is 5 greater than 3 times the smallest element in the set. If we let the smallest element = x, then the largest element is 3x + 5. Finally since the median = average, the median = 50. To determine the largest possible value in the set, let's minimize the first four values.

1st value = x

2nd value = x

3rd value = median = 50

4th value = 50

5th value = 3x + 5

Thus:

x + x + 50 + 50 + 3x + 5 = 250

5x + 105 = 250

5x = 145

x = 29

Thus, the largest value is 3 x 29 + 5 = 92.

Answer: E

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