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

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.

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.
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.
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.

The correct answer is E.
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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.

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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The largest possible value is 92. At the first look you can eliminate (A), (C), and (D). 29, 29, 50, 50, 92

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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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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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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looking at the answer choices can help you solve the problem quicklier!!! \m/

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

THERE IS AN ANSWER.
THE QUESTION ISN'T FLAWED!

With due respect, Mike showed the outcomes for the same median, not the same mean (average) though.

The smallest number is X while the largest is 3x+5.

To find the largest value of 3x+5, we need to simply put the numbers in this way

x , x , 50 , 50 , 3x+5

Now see, this set gives us the largest value of 3x+5. The conditions are met. Median is 50. We know mean is 50 from the first line of the question.

So (x+x+50+50+ 3x+5) divided by 5 should give the mean value of 50. We solve to get

X=29

so largest value of

3x+5 is 3(29)

=92.

Answer is (E) 92

Hi can someone please explain this. If the mean = median in a set, isn't the set a sequence in AP? This would imply that the largest no is sum of the lowest no and 4 times the common diff. We are also give n that "The largest element in the set is 5 greater than 3 times the smallest element in the set. " We can use this info to get the value of the highest. Is that incorrect ?

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