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

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Math Expert
Joined: 02 Sep 2009
Posts: 53063
A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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10 Jun 2015, 03:13
5
19
00:00

Difficulty:

55% (hard)

Question Stats:

69% (02:28) correct 31% (03:06) wrong based on 318 sessions

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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

Kudos for a correct solution.

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Math Expert
Joined: 02 Sep 2009
Posts: 53063
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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15 Jun 2015, 03:29
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3
Bunuel 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

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

Two techniques will help us efficiently interpret the information given in the question. First, we draw a number line with 5 dots representing the 5 numbers in the set. Second, we label these numbers A, B, C, D, and E, with the understanding that A ≤ B ≤ C ≤ D ≤ E.
Attachment:

2015-06-15_1526.png [ 7.99 KiB | Viewed 3132 times ]

We are told that:
A + B + C + D + E = 250 (The set of 5 numbers has an average of 50.)
E = 5 + 3A (The largest element is 5 greater than 3 times the smallest element in the set.)
C = 50 (The median of the set equals the mean.)
We want to maximize E. We should arrange our dots on the number line such that we obey the constraints, yet also note where we have some flexibility.
Attachment:

2015-06-15_1526_001.png [ 8.7 KiB | Viewed 3128 times ]

Point D can be anywhere on the line from Point C to Point E. Since D only appears in one of our formulas above (A + B + C + D + E = 250), we maximize E by minimizing D. Thus, D = C = 50

Similarly, Point B can be anywhere on the line from Point A to Point C. We maximize E by minimizing B, so B = A.
Attachment:

2015-06-15_1527.png [ 6.4 KiB | Viewed 3126 times ]

A + B + C + D + E = 250
A + (A) + 50 + 50 + (5 + 3A) = 250
105 + 5A = 250
5A = 145
A = 29

E = 5 + 3A = 5 + 3(29) = 5 + 87 = 92.
Attachment:

2015-06-15_1528.png [ 7.35 KiB | Viewed 3128 times ]

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Joined: 09 Jun 2015
Posts: 4
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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10 Jun 2015, 04:59
3
Since GMAT typically doesn't require difficult arithmetic, I figured there was at least 1 option that when you subtracted 5, was easily divisible by 3 (Biggest -5 = 3 x smallest).

There were 2 options:
86 and 92.

I quickly calculated what the smallest would have to be in these cases:
86, the smallest would have to be 27
92, the smallest would have to be 29

Using logic, we can see the data set, in order, looking like this:

Smallest, Smallest, 50 (this is the median), 50 (this is the lowest possible after the median to get the biggest remaining value), Biggest

I subbed in the 2 options and 92 was correct:

29, 29, 50, 50, 92-> mean of 50.
Manager
Joined: 10 Jun 2015
Posts: 118
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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10 Jun 2015, 05:24
1
Bunuel 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

Kudos for a correct solution.

The largest possible value is 92. At the first look you can eliminate (A), (C), and (D). 29, 29, 50, 50, 92
Manager
Joined: 13 Mar 2013
Posts: 163
Location: United States
GPA: 3.5
WE: Engineering (Telecommunications)
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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14 Jun 2015, 04:15
1
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

Keeping in mind the condition -
Largest value = 5 + 3*smallest . Option A, C , D can be eliminated .

Option B .
86 = 5 + 3(smallest)
smallest = 27

Similarly , Option E - smallest = 29.
To find Max. Make the other min .

27,27,50,50,86

29,29,50,50,92

Hence option E right .
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Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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09 Dec 2016, 03:46
2
Here is what i did in this Question -->
Given data => mean =50
And number of terms =5
Hence sum=250.

Now let the smallest term be a
The largest term = 3a+5

Mean =50=median

Third term will be 50 (if the set is arranged in increasing or Decreasing Order)

To maximise the largest term we must minimise each term
First Term =a
Second term =a
Third=50
Fourth =50
Fifth = 3a+5

Hence 5a+105=250
Thus,a=29
The largest term = 3*29+5 = 92

Hence E

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Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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23 Jan 2017, 11:05
1
Average of $$5$$ no. is $$50$$, so total is $$5*50=250$$
Let, the first no. is $$x$$. So, 5th no. is $$3x+5$$
We need to maximize the 5th number, so we need to minimize the others number. The minimum value of 1st no. and 4th no. could be $$x$$ and $$50$$ respectively.
So, it is like $$x+x+50+50+3x+5=250 => x=29.$$
So, 5th no. is $$3*29+5=92.$$
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Hasan Mahmud

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Joined: 10 Feb 2017
Posts: 3
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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11 Feb 2017, 11:41
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Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2827
Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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13 Dec 2017, 09:35
Bunuel 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.

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Jeffery Miller

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Re: A set of 5 numbers has an average of 50. The largest element in the  [#permalink]

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13 Dec 2017, 11:13
a + b + c + d + e = 250

since the median = 50 c = 50

a + b + 50 + d + (5 + 3a) = 250

I struggled to realize that to maximize E i have to minimize the other terms.

As the set is in crescent order, b = a and d = c

so:

a + a + 50 + 50 + 5 + 3a = 250
5a = 145
a = 29

so e = 5+ 3(29) = 92
Re: A set of 5 numbers has an average of 50. The largest element in the   [#permalink] 13 Dec 2017, 11:13
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