Willie has an average (mean) of 84.8 on the first five exams. What is the least he can score in order to have an average of 85 for all six exams?

Question

Willie has an average (mean) of 84.8 on the first five exams. What is the least he can score in order to have an average of 85 for all six exams?

Answer 1

The mean, or average, of a list of numbers is found by finding the sum of the numbers and then dividing the result by how many numbers there are.

mean=(x₁+x₂+...+x_n-1+x_n)/n

So, (mean)•n=(x₁+x₂+...+x_n-1+x_n), multiplying both sides of the equation by n.

In other words, the mean times the number of items equals the sum of the value of the items.

Therefore, the mean of 5 scores being 84.8 indicates that the mean of five scores, 84.8, multiplied by the number of scores, 5, equals the sum of the five scores;

(mean₅)•5=(Sum of 5 scores)

84.8•5=(Sum of 5 scores)

and the mean of 6 scores, 85, multiplied by the number of scores, 6, equals the sum of the 6 scores.

(mean₆)•6=(Sum of 6 scores)

85•6=(Sum of 6 scores)

Since the sum of the 6 scores is the sum of the five scores plus the additional score, x, then:

85•6=(84.8•5+x)

Solve for x

Answer 2

If the mean of five scores is 84.8, then their sum is 5 * 84.8 = 424.

If the mean of six scores is 85, then their sum is 6 * 85 = 510.

So Willie has to score at least 510 - 424 = 86.