Video Transcript
Given that the median of the values
𝑎 minus four, 𝑎 plus six, 𝑎 minus three, 𝑎 plus seven, and 𝑎 plus nine is 30,
determine the value of 𝑎.
In this problem, we have five
different values: 𝑎 minus four, 𝑎 plus six, 𝑎 minus three, 𝑎 plus seven, and 𝑎
plus nine. Now because 𝑎 will not change,
then the values will change only because of the value that is added or subtracted
from it. We are given information relating
to the median, and we can recall that the median of a data set is the middle value
when the data is ordered, either in ascending or descending order.
So, let’s put the values we were
given in ascending order. We’ll need to consider the values
that are added or subtracted with values that have a larger number subtracted from
it being the smallest overall value, which means that the smallest value must be 𝑎
minus four. The next smallest value would be 𝑎
minus three, then 𝑎 plus six, and finally 𝑎 plus seven and 𝑎 plus nine. These values would still be in
ascending order even if the value of 𝑎 is a negative number.
So now we need to find the median,
the middle value. There are five different values, so
when we have an odd number of values, as we have here, the middle value is the
actual middle number. If there were an even number of
values, we would have to find the midpoint of the two values in the middle. So, the middle value is 𝑎 plus
six.
Now, we were given that the median
is 30. So, putting both these pieces of
information together, we know that 𝑎 plus six must equal 30. And subtracting six from both sides
of the equation, we have 𝑎 equals 24. And so, we can give this as our
answer.
As a check, we can substitute the
value of 𝑎 equals 24 into each of the original values, which means the first value
is 20. And the remaining values are 30,
21, 31, and 33. In order, we have 20, 21, 30, 31,
and 33. The median of this ordered set is
30, which confirms our answer that 𝑎 equals 24.
