a certain experimental mathematics program was tried out in 2 classes in each of 22 elementary schools and involved 31 teachers . each of the classes had 1 teacher and each of the teachers taught at least 1 , but not more than 3 , of the classes . if the number of teachers who taught 3 classes is n , then the least and greatest possible values of n , respectively , are
"one may notice that greatest possible values differ in each answer choice in contrast to the least values , which repeat . to find out the greatest value you should count the total classes ( 22 * 2 = 44 ) , then subtract the total # of teachers since we know from the question that each teacher taught at least one class ( 44 - 31 = 13 ) . thus we get a number of the available extra - classes for teachers , and all that we need is just to count how many teachers could take 2 more classes , which is 13 / 2 = 6.5 . so the greatest possible value of the # of teachers who had 3 classes is 6 . only answer d has this option ."
divide(subtract(multiply(22, 2), 31), 2)