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10 Jul 2024, 07:00
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\(\{a_1, \ a_2, \ a_3, \ ..., \ a_n\}\)
In the list of n numbers shown above \(a_1 < a_2 < a_3 < ... < a_{n}\). Each of the following could change the median of the list EXCEPT:
A. Dividing \(a_n\) by 2
B. Taking the square root of \(a_1\)
C. Squaring \(a_n\)
D. Doubling \(a_n\)
E. Increasing \(a_n\) by 20
In the list of n numbers shown above \(a_1 < a_2 < a_3 < ... < a_{n}\). Each of the following could change the median of the list EXCEPT:
A. Dividing \(a_n\) by 2
B. Taking the square root of \(a_1\)
C. Squaring \(a_n\)
D. Doubling \(a_n\)
E. Increasing \(a_n\) by 20
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10 Jul 2024, 08:15
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Bunuel
GMAT Club Official Explanation:
The median of a list of values is the middle value for an odd number of values and the average of the two middle values for an even number of values, that is when the values are arranged in order. Thus, the question essentially asks for an option that cannot possibly change the order of the values in the list.
Given list: \(\{a_1, \ a_2, \ a_3, \ ..., \ a_n\}\), where \(a_1 < a_2 < a_3 < ... < a_{n}\).
A. Dividing \(a_n\) by 2
This operation could change the order of the values. Example:
Consider the list {1, 2, 3, 4, 5}. The median is 3.
Dividing \(a_n\) (5) by 2, the new list is {1, 2, 3, 4, 2.5}. The new sorted list is {1, 2, 2.5, 3, 4}. The new median is 2.5.
The median has changed.
B. Taking the square root of \(a_1\)
This operation could change the order of the values. Example:
Consider the list {0.25, 0.3, 0.35, 0.4, 0.45}. The median is 0.35.
Taking the square root of \(a_1\) (0.25), the new list is {0.5, 0.3, 0.35, 0.4, 0.45}. The new sorted list is {0.3, 0.35, 0.4, 0.45, 0.5}. The new median is 0.4.
The median has changed.
C. Squaring \(a_n\)
This operation could change the order of the values. Example:
Consider the list {0.1, 0.2, 0.3, 0.4, 0.5}. The median is 0.3.
Squaring \(a_n\) (0.5), the new list is {0.1, 0.2, 0.3, 0.4, 0.25}. The new sorted list is {0.1, 0.2, 0.25, 0.3, 0.4}. The new median is 0.25.
The median has changed.
D. Doubling \(a_n\)
This operation could change the order of the values. Example:
Consider the list {-1.4, -1.3, -1.2, -1.1, -1}. The median is -1.2.
Doubling \(a_n\) (-1), the new list is {-1.4, -1.3, -1.2, -1.1, -2}. The new sorted list is {-2, -1.4, -1.3, -1.2, -1.1}. The new median is -1.3.
The median has changed.
E. Increasing a_n by 20
This operation will increase the largest term by 20, hence whatever \(a_n\) was, its value will increase, and the order of the elements would still remain the same: \(a_1 < a_2 < a_3 < ... < a_{n}\).
Answer: E.
General Discussion
10 Jul 2024, 07:28
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Bunuel
Dividing a_n by 2 will make a_n smaller. Since a_n is the largest element, it will move to a position earlier in the list, potentially altering the median
Quote:
If a_1 is a fraction, the square root is a larger fraction. This can change the position of a_1 in the list, affecting the median
Quote:
If a_n is a fraction, the square is a smaller fraction. This can change the position of a_n in the list, affecting the median
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If a_n is negative, doubling it makes it more negative, potentially affecting the median
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This operation will not change the relative order of elements to affect the median
Quote:
