**Hint:** Focus on the number of teachers. For example, the number of teachers must be a multiple of what number?

Also, find some constraints by determining the minimim and maximum number of each of the types of attendees.

**Solution Commentary:** First, let T = # teachers, M = # middle school students, and H = # high school students.

Now, there are no "odd" cents in $10.00 (e.g. $10.89 would have an extra 9 cents). Thus, the number of teachers T must be a multiple of 10 (i.e. T = 0,10, 20,..., 1000). Note: You should be able to quickly see why the problem dictates that this big set collapses to T = 0, 10, ..., 100.

In turn, just in case it becomes valuable later, find the minimum and maximum numbers for middle school students M and high school students H.

Suppose the number of teachers was T = 60. Then, A = 60(0.01) + 40(0.30) is the minimum sum collected for this group of attendees. Be sure to stop and think about where the 40 comes from...and why 30 cents is used?

But A > $10. You chould be able to reach this same conclusion for all possible values of T < 60. Which means we need to focus only on T = 70, 80, 90, or 100...and why can we drop 100 as well?

Next, try T = 90....what are the ranges of amounts that could be collected?

By now you should be left with only T = 70 or T = 80. Use trial and error to finish off the problem by finding the true value of T as well as the values of M and H.