Problem Solving

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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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.

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

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