Mean Balance
Given information about the mean, can you work out the missing numbers?
Problem
The mean of the set of numbers 8, 4, 5, 4,
,
and 6 is 6.
If another
is added to the set, then the mean is still 6.
What numbers do
and
represent?
This problem is adapted from the World Mathematics Championships
Student Solutions
If
is less than 6, then adding another
to the set would make the mean smaller.
If
is more than 6, then adding another
to the set would make the mean larger.
But adding another
to the set does not change the mean, so
must be exactly 6.
Alternatively:
If the mean of the seven numbers is 6, they must add up to 42.
If the mean of eight numbers is 6, they must add up to 48.
Therefore the eighth number is 48 - 42 = 6
So the mean of the set of numbers 8, 4, 5, 4,
, 6, 6 is 6.
That means that ( 8 + 4 + 5 + 4 +
+ 6 + 6 ) $\div$ 7 = 6,
so 8 + 4 + 5 + 4 +
+ 6 + 6 = 42,
so 33 +
= 42, so
= 9.
So
= 9 and
= 6.