### Video Transcript

Michael has the following data:
Six, eight, 𝑘, eight, eight, and nine. If the range is three, which
number could 𝑘 be? Is it (A) three, (B) four, (C)
five, (D) six, or (E) 13?

We recall that we can calculate
the range by subtracting the smallest value from the largest value. In this question, we will
consider what the largest value and smallest values are when 𝑘 takes each of
the five options. One of these options will have
a range of three which will be the correct answer. When 𝑘 is equal to three, our
list of values in ascending order are three, six, eight, eight, eight, and
nine. As the largest value is nine
and the smallest value is three, the range will be equal to nine minus
three. As this is equal to six, option
(A) is not correct.

When 𝑘 is equal to four, the
smallest value is four and the largest value is nine. This time the range would be
equal to nine minus four, which is equal to five. Once again, this is not
correct. When 𝑘 is equal to five the
smallest number is five. The largest number is still
nine. The range in this case is equal
to four, which once again is not correct. When 𝑘 is equal to six, we
have two sixes. We could write these in either
order. Our list is now six, six,
eight, eight, eight, and nine. As the smallest number in this
set of date of is six and the largest is nine, the range is equal to nine minus
six. This is equal to three, which
suggests that option (D) is correct.

We will check option (E) just
to make sure. This time, 𝑘 is equal to
13. This means that the smallest
number is six and the largest number is 13. The range is the difference
between these values. 13 minus six is equal to
seven. So this answer is also
incorrect. This means that the correct
answer is option (D). If the range is three, the
number from the list that 𝑘 could be is six.

There are a few other numbers
that were not one of the options that 𝑘 could be. As long as six remains the
smallest number and nine remains the largest number, the range will always be
three. This means that 𝑘 could be any
one of the four integers, six, seven, eight, or nine. In this question, the only one
of those that was listed as an option was six, which is why this is the only
correct answer.