When 2 numbers are removed from 1, 3, 5, 7, 9, 11, 13, 15, 1

[Max@Math Revolution]

When 2 numbers are removed from 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19, is the median of new 8 numbers equal to the average (arithmetic mean) of new 8 numbers?

1) The median of new 8 numbers is 10
2) The average (arithmetic mean) of new 8 numbers is 10


* A solution will be posted in two days.

[Dario@VinciaPrep]

This is a set of 10 consecutive odd numbers.
We know that for this set, the average is 10 and the median is 10.
You can see it as two small sets of 5 consecutive integers:
1, 3, 5, 7, 9 and 11, 13, 15, 17, and 19

Statement 1 alone states that the median of the new 8-numbers set is 10. It means that you have to take out one number for each subset, which means that the median will be 10 as in the original set.
Think about this example...
If you remove 1 and 19, the average will still be 10.
If you remove 1 and another number from the second subset, the average will not be 10.
This means that STATEMENT 1 ALONE IS NOT SUFFICIENT.

Statement 2 alone states that the average of the new 8-numbers set is 10. It means that you have to take out two numbers, and that the sum of the two numbers needs to be 20.
You can take out:
- 1 and 19
- 3 and 17
- 5 and 15
- 7 and 13
- 9 and 11
In each of this couples, one element is taken from the first subset, and the other is taken from the second subset. This means that the new median will still be 10, as in the original set - the same value of the new average.
This means that STATEMENT 2 ALONE IS SUFFICIENT.

The answer is (B).

[Max@Math Revolution]

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

When 2 numbers are removed from 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19, is the median of new 8 numbers equal to the average (arithmetic mean) of new 8 numbers?

1) The median of new 8 numbers is 10
2) The average (arithmetic mean) of new 8 numbers is 10


Modify the original condition and the question. In order to make mean=median happen, there should be one pair amongst numbers (1,19),(3,17),(5,15),(7,13),(9,11) which should be removed. For 2), it is always yes and sufficient. On the other hand, for 1), if (1,19) is removed, it is yes, but if (1,17) is removed, it is no. Therefore, the answer is B.