What is the new average in the following case?
A set of n numbers has an average value of the terms equal to 45. When two new terms are added to the set, the average value increases by 4, what is the change in the average value if terms equal to those added are instead removed from the set?
Let there be n terms in the set. The average value is 45; therefore the sum of the n terms is 45n.
The addition of two terms a and b to the set, increases the total sum of the terms to 45n + a + b and their average value becomes (45n + a + b)/ (n + 2) which is equal to 45 + 4 as given.
So a + b = 49(n + 2) – 45n = 49n + 98 – 45n = 4n + 98
When the terms with the same value as those added are removed from the set the total sum of the terms in the set is given by 45n – 4n – 98. The average is (41n – 98)/ (n - 2).
The change in the average is dependent on the number of terms in the set.
Since the average of n items is 45, the total of the n items = 45n.
When 2 new items are added, the average increases by 4. So the new total of n+2 items is (n+2)(45+4) = 49n+98.
Therefore the terms added has a value 49n+98-45n = 4n+98.
So if the terms equal to the added are removed, then the remaining total = 49n+98-(4n+98) = 45n. The remaining number of terms = n. So the average of the remaining n terms = 45n/n = 45.