calculating

[cplusplus123480]

Remove

[TheJJJunk]

cout<<"Please enter the____ quiz score <0-110>: ";
cin >> quiz__;

while (quiz__ < 0 || quiz__ > 110)
{
    cout << "Score must be between 0 and 110! Enter score: ";
    cin >> quiz__;
}

As for the accuracy, this statement

projAverage=(proj1+proj2+proj3+proj4)/4.0;
quizAverage=(quiz1+quiz2+quiz3)/3.0;
courseAverage=(projAverage+quizAverage)/2.0;

needs to be after the user enters the values.