Set S consists of five consecutive integers, and set T consi

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Set S consists of five consecutive integers, and set T consists of seven consecutive integers. Is the median of the numbers in set S equal to the median of the numbers in set T?

(1) The median of the numbers in Set S is 0
(2) The sum of the numbers in set S is equal to the sum of the numbers in set T

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by ceilidh.erickson » Sat May 25, 2019 11:31 am

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Set S consists of five consecutive integers, and set T consists of seven consecutive integers. Is the median of the numbers in set S equal to the median of the numbers in set T?

Let's first look at 2 examples of consecutive sets that do have the same median to try to contextualize the question:
S = [3, 4, 5, 6, 7]
T = [2, 3, 4, 5, 6, 7, 8]

These sets will have the same median if the middle 5 terms of the 7-term set are the same as the 5-term set. So we could phrase that as follows:
Is the first term of set T 1 less than the first term of set S?

(1) The median of the numbers in Set S is 0
This only tells us about the 1st set, but not the 2nd one. Insufficient.

(2) The sum of the numbers in set S is equal to the sum of the numbers in set T
By definition, the average (and thereby the median) of a consecutive set is (sum)/(# of terms)

These two sets have the same sum; let's call it x.
The median of set S will be $$\frac{x}{5}$$
The median of set T will be $$\frac{x}{7}$$

We cannot know whether these values are the same.

To give concrete examples, imagine:
S = [5, 6, 7, 8, 9] --> sum = 35, median = 7
T = [2, 3, 4, 5, 6, 7, 8] --> sum = 35, median = 5
or
S = [-2, -1, 0, 1, 2] --> sum = 0, median = 0
T = [-3, -2, -1, 0, 1, 2, 3] --> sum = 0, median = 0

Because we can get a "yes" or a "no" answer to "are the medians the same?", this statements is insufficient.

(1) and (2) together
If the median of set S is 0, set S must be:
S = [-2, -1, 0, 1, 2]

And if both sets have the same sum, both sums must be 0, and set T must be:
T = [-3, -2, -1, 0, 1, 2, 3]

Thus, we know that the median of both sets is 0. This is sufficient.

The answer is [spoiler]C.
[/spoiler]
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education