To find the average, add up all of the scores, then divide by how many scores there were: 

\(\displaystyle \frac{68+78+80+99+75+83}{6}=\frac{483}{6}=80.5\).

Set \(\displaystyle x\) equal to the sum of the tests of the first \(\displaystyle 14\) students.

\(\displaystyle \frac{x}{14}=82, x=1148.\)

Now set up an equation to solve for Sarah's test score:

\(\displaystyle \frac{1148+y}{15}=83, y=97\)

In order to determine whether Sally will qualify for the last time trial, set up an equation to find the mean.  Out of the five scores, one of the scores is unknown, and Sally's final score must be \(\displaystyle 9.5\) or higher.  Let the unknown score be \(\displaystyle x\).  

Write the equation to solve for the mean.

\(\displaystyle \frac{9.3+9.6+8.7+9.1+x}{5}=9.5\)

\(\displaystyle 9.3+9.6+8.7+9.1+x=47.5\)

\(\displaystyle 36.7+x=47.5\)

\(\displaystyle x=10.8\)

Unfortunately, Sally cannot make the gymnastics team because she needs to earn a 10.8, which is not possible on the score limit.

The correct answer is:

\(\displaystyle \textup{Sally will not make the team.}\)

The mean is also the average of a set of numbers. To find the mean, add all of the numbers in the set:  \(\displaystyle 3\), \(\displaystyle 6\), \(\displaystyle 9\), \(\displaystyle 34\), \(\displaystyle 12\), and \(\displaystyle 2\). Then, divide your answer by the number of terms present within the set, in this case \(\displaystyle 6\).

\(\displaystyle 3+6+9+34+12+2=66\)

In this case, the sum of all the numbers is \(\displaystyle 66\). Divide by the size of the set: \(\displaystyle 6\).

\(\displaystyle \frac{66}{6}=11\)

This will give you an average or mean of \(\displaystyle 11\).

To get the total combined score of everyone in the class, multiply the average of the class by the number of students.

\(\displaystyle 84.1*10=841\). Since the teacher decided to drop the lowest 2 scores, let's subtract those 2 scores from the total.

\(\displaystyle 841-71-73=697\).

Now the new total is 697. To find the new average, divide this new total by the new number of grades to be averaged in, which is 8. Remember that the teacher dropped 2 grades so there are less grades to take into account.

\(\displaystyle \frac{697}{8}=87.1\)

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are ten numbers so we add them up and divide by ten:

\(\displaystyle \frac{10+10+11+11+11+12+12+15+27+39}{10}=15.8\)

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are 16 numbers in the set, so we add the numbers together and divide by 16 to find the mean:

\(\displaystyle 76+78+82+83+84+84+88+89+89+89+91+91+93+97+100+100\)

\(\displaystyle =1414\rightarrow \frac{1414}{16}=88.4\)

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are 11 numbers in the set, so we add the numbers together and divide by 11 to find the mean:

\(\displaystyle \frac{32+33+33+34+35+37+38+41+42+45+47}{11}=37.9\)

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are 16 numbers in the set, so we add the numbers together and divide by 16 to find the mean:

\(\displaystyle 7+10+16+16+16+16+17+23+25+27+30+31+41+57+57+58\)

\(\displaystyle =447\rightarrow \frac{447}{16}=27.9\)

To find the mean, there are two steps:

1. Add all of the numbers in the set together.

2. Divide that number by the amount of numbers are in the set.

For this problem there are 12 numbers in the set, so we add the numbers together and divide by 12 to find the mean:

\(\displaystyle \frac{11+13+17+22+22+22+23+23+26+28+47+49}{12}=25.3\)