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 pool 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 during the day before 5pm, and about a quarter of the visits, I think the three of us will go during the day before 5pm. My guess is that, given your work schedule, all three of us 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 1 YMCA POOL MEMBERSHIP PRICING

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

Table 2 shows the expected pool usage.

TABLE 2 EXPECTED POOL USAGE BY THE SWANSONS

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

Using Tables 1 and 2 for guidance, how many times would the Swansons need to use the pool to break-even? In answer, specify the total number of times the Swansons would need to go to break-even, how many times would just Carol and Anna need to go before 5pm, how many times would John, Carol, and Anna need to go during the day before 5pm, and how many times would John, Carol, and Anna need to go after 5pm.

# of times Carol & Anna visit pool = 0.5x

# of times Carol, Anna, & John visit the pool before 1700 = 0.25x

# of times Carol, Anna & John visit the pool after 1700 = 0.25x

Cost of 1 visit = ticket cost * # of members visiting

Cost of Carol, Anna, & John visiting = (0.5x)(10)(2) = 10x

Cost of Carol, Anna, & John visiting before 1700 = (0.25x)(10)(3) = 7.5x

Cost of Carol, Anna, & John visiting after 1700 = (0.25x)(5)(3) = 3.75x

340 = 10x + 7.5x + 3.75x

340 = 21.25x

16 = 340 / 21.25

The Break Even Point would be 16 times.

After calculating the break-even point, the Swansons 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.

TABLE 3 EXPECTED POOL USAGE BY THE SWANSONS AFTER ANALYZING THE FAMILY CALENDAR

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

Family members	Frequency of usage
Carol and Anna before 5pm	8 times = 0.5 * 16 = 0.5x
Carol, Anna, and John before 5pm	4 times = 0.25 * 16 = 0.25x
Carol, Anna, and John after 5pm	4 times = 0.25 * 16 = 0.25x

Given the usage distribution above in Table 3, 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? Answer this question in an Excel spreadsheet using appropriate formatting and range names.

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.

Cost of Carol & Anna visiting 8 times = 8 *10 * 2 = 160

Cost of Carol, Anna & John visiting before 1700 for 5 times = 5 *10 *3 = 150

Cost of Carol, Anna & John visiting after 1700 for 3 times= 3 * 5 *3 = 45

Total cost per estimated frequency of visits = 160 + 150 + 45 = 355

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. In fact, our neighbors, the Hendersons, Paul and Kristin, have two kids, Robbie and Lexie. If they use a similar usage distribution as we do, I wonder how many times they would need to go?

Yes, Carols reasoning is more effective per the estimated frequency of visits.

QUESTION: Using the usage distribution in Table 4 and the same pricing information in Table 1, calculate the number of times the Hendersons would need to use the pool for it to make sense to purchase the YMCA pool pass. Similar to question 1, specify the total number of times the Hendersons would need to go, how many times would just Kristin, Robbie and Lexie would need to go before 5pm, how many times would Paul, Kristin, Lexie, and Robbie need to go during the day before 5pm, and how many times would Paul, Kristin, Lexie, and Robbie would need to go after 5pm. The Hendersons have four people and the Swansons have three. Explain what happens to the break-even point as the number of people in the family increases. Answer this question in an Excel spreadsheet using appropriate formatting and range names. WHAT IS THE BREAKEVEN GRAPH?

TABLE 4: EXPECTED POOL USAGE BY THE HENDERSONS

Family members	Frequency of usage
Kristin, Robbie, and Lexie before 5pm	50% of the time
Kristin, Robbie, Lexie, and Paul before 5pm	25% of the time
Kristin, Robbie, Lexie, and Paul after 5pm	25% of the time