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# QOTD # 15 The greatest of the 21 positive integers in a cert

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QOTD # 15 The greatest of the 21 positive integers in a cert [#permalink]  10 Sep 2016, 02:04
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The greatest of the 21 positive integers in a certain list is 16. The median of the 21 integers is 10. What is the least possible average (arithmetic mean) of the 21 integers?

A) 4
B) 5
C) 6
D) 7
E) 8

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Question: 15
Page: 180
[Reveal] Spoiler: OA

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Re: QOTD # 15 The greatest of the 21 positive integers in a cert [#permalink]  10 Sep 2016, 02:06
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Explanation

You are given that the median of the 21 positive integers is 10 and the greatest of the 21 integers is 16. This means that when the 21 integers are listed in order from least to greatest,

• the 1st through 10th integers are between 1 and 10, inclusive;
• the 11th integer is 10;
• the 12th through 20th integers are between 10 and 16, inclusive; and
• the 21st integer is 16.

The least possible average of the 21 integers would be achieved by using the least possible value of each integer as described above in the reordered list:

• the 1st through 10th integers would each be 1;
• the 11th integer would be 10;
• the 12th through 20th integers would each be 10; and
• the 21st integer would be 16.

For the least possible integers, the sum is (10)(1) + 10 + (9)(10) + 16, or 126. Therefore the least possible average is $$\frac{126}{21}$$ or 6.

The correct answer is Choice C.
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Re: QOTD # 15 The greatest of the 21 positive integers in a cert [#permalink]  19 Sep 2016, 17:00
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Carcass wrote:
The greatest of the 21 positive integers in a certain list is 16. The median of the 21 integers is 10. What is the least possible average (arithmetic mean) of the 21 integers?

A) 4
B) 5
C) 6
D) 7
E) 8

Let the following spaces represent the 21 integers listed in ASCENDING ORDER, with a red space indicating the median:
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

The greatest of the 21 positive integers in a certain list is 16
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 16

The median of the 21 integers is 10
_ _ _ _ _ _ _ _ _ _ 10 _ _ _ _ _ _ _ _ _ 16

What is the least possible average (arithmetic mean) of the 21 integers?
We must MINIMIZE the 19 remaining integers.
So, the first 10 integers must be 1, since 1 is the smallest positive integer.
1 1 1 1 1 1 1 1 1 1 10 _ _ _ _ _ _ _ _ _ 16

The remaining integers must be greater than or equal to 10 and less than or equal to 16
Since, we're trying to MINIMIZE the remaining integers, we'll let each number be 10 to get:
1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 16

Average = (1+1+1+1+1+1+1+1+1+1+10+10+10+10+10+10+10+10+10+10+16)/21
= 126/21
= 6

[Reveal] Spoiler:
C

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Re: QOTD # 15 The greatest of the 21 positive integers in a cert [#permalink]  04 May 2018, 09:21
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Carcass wrote:
The greatest of the 21 positive integers in a certain list is 16. The median of the 21 integers is 10. What is the least possible average (arithmetic mean) of the 21 integers?

A) 4
B) 5
C) 6
D) 7
E) 8

There are 21 numbers in the list, and, if we list the numbers in ascending order, the median is the 11th number.

Thus, we can let each of the first 10 numbers = 1, each of the next 10 numbers = 10, and the last number = 16.

Thus, the average is (10 x 1 + 10 x 10 + 16)/21 = 126/21 = 6.

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Re: QOTD # 15 The greatest of the 21 positive integers in a cert   [#permalink] 04 May 2018, 09:21
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