Official Solution:


\(\{x, x, x, x, 8, 8, 8, 8, 12, 12, 11,y\}\)

The twelve numbers shown represent, the ages, in years, of the twelve children in a school bus. What is the median age, in years, of the twelve children in the bus?

The median of a list with even number of terms is the average of the two middle terms (after the numbers have been arranged in ascending/descending order).

(1) \(x = 10\)

The list in ascending order is \(\{8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12, y\}\).

Notice that irrespective the value of \(y\), the two middle terms will always be 10 and 10, making the median 10. For example, if \(y < 10\), then the list is {8, 8, 8, 8, \(y\), 10, 10, 10, 10, 11, 12, 12} (the two middle terms are 10 and 10) and if \(y \geq 10\), then the list is {8, 8, 8, 8, 10, 10, 10, 10, \(y\), 11, 12, 12} (the two middle terms are 10 and 10). Sufficient.

(2) \(y = 13\)

The list in ascending order is \(\{x, x, x, x, 8, 8, 8, 8, 11, 12, 12, 13\}\). The median can take more than one value. For example:

If \(x=10\), then the list would be \(\{8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12, 13\}\), making the median equal to 10.

If \(x=12\), then the list would be \(\{8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 12, 13\}\), making the median equal to 12.

Not sufficient.


Answer: A
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Median is the middlemost number when a list is arranged in ascending order, for even number of elements the median is the mean of two middlemost numbers/
Statement 1:

Y 8 8 8 8 Y 10 10 10 10 Y 11 Y 12 Y; Y represent the positions where we can put y (Y age in years so integer)

When Y<8 median is 10
When Y >12 median is 10
When y=10/11 median is still 10
When y= 9, median still 10. So sufficient.

Statement 2.
y=13

x, x, x, x, 8, 8, 8, 8, 11, 12, 12, 13; median= 8.5 for x<=8
But for x=10 we saw that median is 10. So B statement is insufficient. Only A.

This a perfect question that we can use the concept of maximization and minimization. There are three things to keep in mind while solving questions related to the median :

1. By definition, the median is the middle value in a list of numbers arranged in ascending or descending order.
2. The arrangement is always key to solving questions related to the median.
3. On questions where we have a list of elements with many variables, maximization and minimization is the best way to proceed.

Now lets jump into the solving of the question.

We have the list x, x, x, x, 8, 8, 8, 8, 12, 12, 11,y. Since the data here are of ages, x and y have to be integers.

Statement 1 : x = 10

Arranging the elements in ascending order we get, 8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12, y.

Now let us think about the max and min values of y. If y >= 12, then the max median will be 10. If y <= 12, then the list now becomes y, 8, 8, 8, 8, 10, 10, 10, 10, 11, 12, 12. Again, the min median will be 10. Sufficient.

Statement 2 : y = 13

Arranging the elements in ascending order we get, x, x, x, x, 10, 10, 10, 10, 11, 12, 12, 13.

If x<= 10, the list will be as above and then the minimum median is 10, but if x >=13, then the list will be 10, 10, 10, 10, 11, 12, 12, 13, x, x, x, x. In this case the max median will be 12. So the median here can be any value from 10 to 12. Insufficient.

Hope this helps!
Aditya
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Hello Aditya, nice to see you on the gmatclub & thank you for the precise explanation.

AdityaCrackVerbal