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## It Was Sum Picnic

Suppose your school PTA sponsored a picnic and covered most of the costs. The high school students had to pay 50 cents, the middle school students had to pay 30 cents, and in order to get teachers to attend, they were charged only 1 cent. It proved to be both a great day and picnic, with a total of 100 people attending. The sum of money collected was \$10.

How many high school students attended the picnic?
How many middle school students attended the picnic?
How many teachers attended the picnic?

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.