John and Carol Swanson had one eight year old daughter, Anna, who enjoyed swimming. The local YMCA, a not-for-profit organization, located about two miles from their house, offered a summer pool membership for families. John and Carol were trying to decide whether to purchase a summer pool membership for the family. John said I think we should purchase the n n npool membership for the summer. Anna loves swimming and the cost for the pool membership is $340 for the family for the entire summer. This is a great deal. John continued The cost is $10 per person each time, so, the three of us would need to go at least 12 times in order to make it worth our while to buy the pass. In past summers, I think weve gone at least that many times.

Carol was not sure: I dont think well use it enough to justify the cost. If we pay each time, the cost is $10 per person. Lots of times you have to work and its only me and Anna going to the pool. If we wait until 5pm, the cost is $5 per person.

Carol continued, I think its likely that half of the time, only Anna and I will go to the pool, and about a quarter of the visits, I think the three of us will go during the day. My guess is that, given your work schedule, we will be able to take advantage of the $5 discounted rate by waiting until 5pm to go about a quarter of the time. John considered Carols reasoning, Sounds like youve put a lot of thought into this. Im impressed. How many times would we need to go in order to make it worth our while to buy the season pass? Carol clarified Johns question, I think we need to determine how many times just Anna and I would need to go, how many times the three of us would need to go, and then how many times the three of us would need to go after 5pm to get the $5 per person rate. Carol continued This will be a fun problem! It reminds of me of doing multi-product break-even problems in the Accounting course I took last summer! Table 1 shows the YMCA pricing; Table 2

Pricing Type	Price
Family pool membership	$340
Price per person, before 5pm	$10
Price per person, after 5pm	$5

Family members	Frequency of usage
Carol and Anna	50% of the time
Carol, Anna, and John before 5pm	25% of the time
Carol, Anna, and John after 5pm	25% of the time

Question #1: How many times would the Swansons need to use the pool to break-even? How many times would just Carol and Anna need to go; how many times would John, Carol, and Anna need to go during the day; and how many times would John, Carol, and Anna need to go after 5pm?

After calculating the break-even point, the Swansons all sat down with the summer calendar to confirm Carols expected usage. After carefully considering their vacations and other commitments, they changed their expected usage slightly and determined that Carol and Anna could go to the pool 8 times, all three family members could go 5 times before 5pm and 3 times after 5pm. Table 3 shows the revised expected pool usage.

Family members	Frequency of usage
Carol and Anna	8 times
Carol, Anna, and John before 5pm	5 times
Carol, Anna, and John after 5pm	3 times

Question #2: Given the usage distribution above, calculate the total cost of going to the pool each time rather than buying the pool pass. Should the Swansons buy the pass? Why or why not?

After finishing the analysis, the Swansons determined they should buy the pool pass. One night, after John got home from work, Carol and Anna really wanted to go the pool. John said Honestly, Id rather do something else tonight. Maybe we could watch a movie together? Carol responded: John, I think we should go to the pool. I mean, weve paid for the pool pass and we need to take advantage of it.

Question #3: Do you agree with Carols reasoning that they need to use the pool since theyve paid for the pass? Why or why not?

A week later after making the decision to buy the pool pass, the Swansons had the opportunity to buy season passes to a local amusement park. The amusement park priced their season passes differently than the YMCA pool pass. John said I think we should look into buying a season pass. The cost is $100 per adult for the summer and $75 per child for the summer. If we dont buy the pass, we would pay $65 per adult and $50 for Anna each time we went to the park. The pass pays for itself in two times!

Carol responded, Thats a good deal. I agree we should buy the pass because last summer we all went three times. Carol commented: The pricing for the amusement park makes more sense to me. Unlike the YMCA, they price the season pass per person, rather than per family. I think the YMCA could learn a thing or two from the amusement park; the YMCA pool membership pricing seems to incentivize large families to buy the pass but not smaller families.

Question #4: Do you agree with Carols statement that the YMCA pool membership pricing policy incentivizes larger families to join the pool, but not smaller families? Why or why not?

Question #5: Why do you think the YMCA would choose the price their pool passes per family, rather than per person?

Question #6: Why do you think the amusement park provides such a strong incentive to purchase the season pass?