\(\displaystyle Mean=\frac{Sum\ of\ Options}{Number\ of\ Options}\)

\(\displaystyle Mean=\frac{\left ( 20+15+30+40+30+40\right )}{6}=\frac{175}{6}=$29.17\)

The average of the first 5 students' scores is 90, which means that the total score for these 5 students is 450. To find the total score of all 6 students, you simply add the sixth student's score of 84 to the first 5 students' total score of 450, which gives you a total score of 534 for all 6 students. To determine the average score, you must divide the total score of 534 by 6, which gives you an average score of 89.

Since they had an average score of 90, we know that their total score must have been 270 (90 multiplied by 3). We know that 2 of the scores are 94 and 92, so the third score must be \(\displaystyle 270-92-94\), which is 84.

To find mean, we add up the values in a set and divide by the number of terms in that set. We begin this problem with the knowledge that the mean of the first set is 20. Since that set contains five numbers, we know that its total sum must be 100 (since 100 divided by 5 is 20).

 \(\displaystyle 27+11+14+23 = 75\)

so \(\displaystyle x\) must be 25.

Now, the only step left is to find the mean of \(\displaystyle \left \{ 25,50,11,8,31\right \}\).

These values add up to 125, and when we divide by 5, we are left with a final answer of 25.

If we substitute 3 in for \(\displaystyle x\), we find that our original set is equal to \(\displaystyle \left \{ 3,7,10,11,4\right \}\). To find the mean, add the values and divide by the number of terms in the set (in this case, 5). Here, the sum of the values is 35. Dividing by 5 yields a final answer of 7 for our mean.

To find the average score, you must total the scores of everyone in the class and divide it by the number of students in the class. 8 students received a score of 90, 6 students received a score of 80, and 6 students received a score of 70. The total score of the class is \(\displaystyle (8*90)+(6*80)+(6*70)=720+480+420=1620\). There are 20 students in the class, so the average score must be \(\displaystyle \frac{1620}{20}=81\).

Reese's average score is calculated by dividing the sum of his test scores by the number of tests he took. This semester, Reese took 4 tests and his average score was 88, which means the sum of his test scores must be 362 (88 multiplied by 4). On the first three tests, he scored an 84, 85, and 90. This means that his fourth test score must be \(\displaystyle 352-84-85-90=93\)

Add, then divide by 5:

\(\displaystyle \frac{ 45 + 69 +78+ N+ \left (2N + 3 \right) }{5}\)

\(\displaystyle = \frac{ 45 + 69 +78+3+ N+2N}{5}\)

\(\displaystyle =\frac{ 3N + 195}{5}\)

\(\displaystyle =\frac{3}{5} N +\frac{195}{5}\)

\(\displaystyle =0.6N +39\)

The prime numbers between 75 and 100 are 79, 83, 89, and 97. To find their mean, add them and divide by 4:

\(\displaystyle \frac{79+83+89+97}{4} = 87\)

The sum of the first \(\displaystyle N\) natural numbers is \(\displaystyle \frac{N (N+1)}{2}\), so their mean is \(\displaystyle \frac{N (N+1)}{2} \div N = \frac{ (N+1)}{2}\) 

The mean of the first 99 natural numbers is \(\displaystyle \frac{ 99+1}{2} = 50\).