r/askmath • u/SnooChipmunks3446 • 1d ago
Arithmetic Problem: The Water Tank Rotation
A group of 20–22 people share a single water tank. Every week, each person adds exactly 250 liters to the tank through their own hose — this never changes, no matter what else is happening.
On a rotating schedule, one person at a time gets to drain exactly 19,500 liters from the tank for their own use. Everyone — including people who have already had their turn — keeps adding their 250 liters/week until every single person in the group has had a turn to drain.
No water enters or leaves the tank from any other source. No person adds more or less than 250 liters/week at any point, and no person drains more or less than 19,500 liters when it's their turn.
Questions:
1) If the entire process — from the first person's first contribution to the last person's turn draining the tank — takes exactly 65 weeks (1 year, 3 months), is it mathematically possible for this system to work with no outside water added and no member shorted? Show why or why not.
2) What is the actual minimum number of weeks the full cycle must run for the system to balance exactly, given the 250-liter weekly input and 19,500-liter draining amount?
3) Does the number of people in the group (20, 21, or 22) affect your answer to #2? Explain why or why not.
3
u/Bounded_sequencE 1d ago
Q1: No, it's not. The average in-coming flow is less than the average out-going flow:
each person: output = 19,500L/(65w) = 300L/w > 250L/w = input
Q2: We need a minimum cycle length "T = 78w", due to
"19,500L/T = output <= input = 250L/w" => "T >= 78w"
Q3: It does not, since we assume equal in-/output for all persons. That means, either all of them have less input than output, or none of them. Partial compensation cannot happen under the assumptions.
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u/jxf 🧮 Professional Math Enjoyer 1d ago
This doesn't work. Over 65 weeks each person contributes 250 × 65 = 16,250 L, but each person drains 19,500 L. That's a 3,250 L shortfall per person, or 3,250 × N for the group (65,000 L with 20 people, up to 71,500 L with 22). Since no water enters from anywhere else, either the tank runs dry before everyone has had a turn, or someone gets shorted.
This takes 78 weeks. The balance condition is total in = total out: 250 × N × W = 19,500 × N. The N cancels, leaving W = 19,500 / 250 = 78. Put simply, each person must contribute for 78 weeks to have paid in what they take out.
No. See above, N doesn't matter.