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19 Nov 2023, 03:10
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E
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Difficulty:
35%
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Question Stats:
80% (02:16) correct
20%
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The average (arithmetic mean) of 11 numbers is 50. When the numbers are arranged in order from least to greatest, the average of the 6 smallest numbers is 44 and the average of the 6 greatest numbers is 57. What is the median of the 11 numbers?
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
2023-12-03_20-07-31.png [ 39.36 KiB | Viewed 11944 times ]
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
Attachment:
2023-12-03_20-07-31.png [ 39.36 KiB | Viewed 11944 times ]
03 Dec 2023, 09:49
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pablovaldesvega
The average (arithmetic mean) of 11 numbers is 50. When the numbers are arranged in order from least to greatest, the average of the 6 smallest numbers is 44 and the average of the 6 greatest numbers is 57. What is the median of the 11 numbers?
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
Attachment:
2023-12-03_20-07-31.png
Firstly, since we are dealing with 11 numbers, which is an odd count, the median will be the middle number when arranged in order. This means the median is the 6th largest number.
Next, given that the average of the 11 numbers is 50, the total sum of these numbers is 11*50 = 550.
We are also given that the average of the 6 smallest numbers is 44 and the average of the 6 greatest numbers is 57. Thus, the sum of the 6 smallest numbers is 6*44, and the sum of the 6 greatest numbers is 6*57. As the 6th largest number is counted in both groups, the combined sum of these two groups is greater than the total sum by exactly the value of the 6th largest number.
Therefore, the 6th largest number equals (6*57 + 6*44) - 550 = 6(57 + 44) - 550 = 6*101 - 550 = 606 - 550 = 56.
Answer: E.
General Discussion
19 Nov 2023, 03:23
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pablovaldesvega
The average (arithmetic mean) of 11 numbers is 50. When the numbers are arranged in order from least to greatest, the average of the 6 smallest numbers is 44 and the average of the 6 greatest numbers is 57. What is the median of the 11 numbers?
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
A) 47
B) 50
C) 50.5
D) 53.5
E) 56
A = {\(x_1, \quad x_2,\quad ...\quad, x_5,\quad x_6,\quad x_7,\quad....\quad x_{10},\quad x_{11}\)}
The median of the set = \(x_6\)
The average (arithmetic mean) of 11 numbers is 50.
\(x_1 +x_2+x_3+...x_{10}+x_{11} = 11*50 = 550\) --- (1)
...the average of the 6 smallest numbers is 44...
\(x_1 +x_2+x_3+...x_{5}+x_{6} = 6*44\) --- (2)
...the average of the 6 greatest numbers is 57...
\(x_6 +x_7+x_8+...x_{10}+x_{11} = 6*57\) --- (3)
Adding equations (2) and (3) we see that \(x_6\) has been added twice as the term is common both in equation (2) and equation (3)
\((x_1 +x_2+x_3+...x_{10}+x_{11}) + x_6 = 6*(44+57)\)
Substituting the value of (1)
\(550 + x_6 = 6*(44+57)\)
\( x_6 =606 - 550 = 56\)
Option E
