To find the median, order the numbers from smallest to greatest and then find the one in the middle. Thus, the answer is .

The median of five numbers (an odd number) is the number in the middle when they are arranged in ascending order. 

Assume both statements. The orderings 

,

, and

are all consistent with both statements. But in the first scenario,  is the median; in the other two,  is the median. Therefore, it cannot be determined which is the median.

The median of a dataset with an even number of elements is the arithmetic mean of the two elements that fall in the middle when the elements are arranged in ascending order. Since there are 100 elements and , this means the fiftieth-highest and fiftieth-lowest elements.

The mode is the number that comes up most frequently in a set.  Rita goes to the gym 4 times a week for 511 weeks.  She clearly goes 4 times per week far more often than she goes 1, 2, or 3 times per week.  Therefore the mode is 4 days/week.  It is NOT 511 weeks.  That is the frequency with which 4 days/week occurs, but not the mode.

If the known values are ordered from least to greatest, the set looks like this:

Below are each of the choices, followed by the set that results if it is added to the above set, followed by the median - the middle element - and the mode - the most frequently occurring element.

Only the addition of 11 yields a set with median and mode equal to each other.

Of the known elements, 6 occurs the most frequently - three times. Since the unknown  occurs only twice, and it cannot be equal to any of the other elements, its value does not affect the status of 6 as the most frequent element. Therefore, regardless of , 6 is the only mode.

The mode is the number that appears most frequently:

The mode of a set of data is the entry that appears most often within the set. One easy way to determine the mode is by arranging the set in increasing order:

Now that the set is arranged in increasing order, we can see how often each value appears in the set. The value 7 appears three times, which is more than any other entry is repeated, so this is the mode of the set.

The mode of a set of data is the entry that occurs most often within the set. An easy way to determine which entry occurs the most often is by arranging the set in increasing order:

Now we can see that 29 is repeated more often than any other number in the set, so this is the mode.

The mode of a set of data is the entry that appears most frequently within the set. An easy way to determine the mode is by arranging the set in increasing order:

Now we can see that the value of  is repeated more times than any other value, so this is the mode of the set of data.