Pool Membership: A Break-Even Point That Is Not a Whole Number
Riverside Pool sells two options to families. A Family Pass costs $260 for the season plus a one-time $20 processing fee, and then admits the whole family for $6 per visit. Without a pass, each family member pays $9 per visit. The Delgado family has 4 members and expects to visit between 6 and 20 times this summer. Their daughter writes: "Family Pass = 260 + 6v, no pass = 36v. Setting them equal gives 30v = 260, so v ≈ 8.67. Since 8.67 rounds to 9, the pass starts paying off at 9 visits." The parents want a defensible answer before they buy, and they also want to know how much they save at 20 visits, expressed as a percent of the no-pass cost.
1. Write both cost models, find the smallest whole number of visits at which the pass is cheaper, and verify it by checking the neighboring whole numbers.
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Models (v = number of visits, 6 ≤ v ≤ 20):
- Family Pass: P(v) = 280 + 6v (the $20 fee belongs to the pass option)
- No pass: N(v) = 4 × 9v = 36v
Solve 280 + 6v = 36v → 280 = 30v → v = 9.33…
Because visits are whole numbers, test the integers on either side:
- v = 9: P = 280 + 54 = $334; N = $324 → no pass is cheaper by $10
- v = 10: P = 280 + 60 = $340; N = $360 → pass is cheaper by $20
Answer: the pass first wins at 10 visits.
2. Evaluate the daughter's reasoning. Identify every error and explain why each one matters for the family's decision.
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Two separate problems:
She dropped the $20 processing fee. With the fee, the break-even equation is 280 + 6v = 36v, giving v = 9.33, not 8.67. Omitting a fixed cost on only one side of the comparison systematically favors that option.
She rounded a threshold in the wrong direction. A break-even of 9.33 means that at 9 visits the pass has not yet caught up - the pass is $10 more expensive. Rounding down to 9 tells the family to buy at the exact visit count where buying is still a loss. Thresholds derived from an inequality must be rounded up to the next whole number (the smallest integer satisfying P(v) < N(v)).
Check that catches both errors: substituting whole numbers on each side. v = 9 gives $334 vs $324, which immediately contradicts her conclusion.
3. The family ends up visiting 20 times. Compute the savings and express it as a percent of the no-pass cost, then state one reason the percent figure could mislead a neighbor deciding whether to buy.
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Costs at v = 20
- Pass: 280 + 6(20) = 280 + 120 = $400
- No pass: 36(20) = $720
Savings: 720 − 400 = $320 Percent of no-pass cost: 320 ÷ 720 = 0.444… ≈ 44.4%
Why the percent can mislead: it is tied to v = 20. The percent saved is not a property of the pass; it grows with the number of visits and is actually negative below 10 visits. A neighbor who visits 6 times would pay $316 with the pass versus $216 without - a 46% loss. Quoting "44% savings" without the visit count implies a benefit that does not exist at other attendance levels.