Problem 15 · AMC 8 Stretch
Core
Arithmetic & Operations
seeking-complementsasking-key-questions
A \(16\)-ounce bottle is full of juice. On day \(1\) you drink \(1\) oz, then refill the bottle to the top with water. On day \(2\) you drink \(2\) oz of the mixture and refill with water. On day \(3\) you drink \(3\) oz and refill, and so on, drinking \(1\) more ounce each day, until the bottle is empty. In total, how many ounces of WATER do you drink?
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Answer: 120 ounces of water
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Hint 1 of 4
Trying to track the juice-to-water ratio each day is a headache. Ask the complement question instead: how much WATER goes INTO the bottle each day?
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Hint 2 of 4
Each day you refill with exactly as much water as you just drank: \(1\) oz on day 1, \(2\) oz on day 2, and so on. Since the bottle ends up empty, every drop of water that ever went in eventually gets drunk.
Still stuck? Show hint 3 →
Hint 3 of 4
Be careful about the last day: when you drink the final \(16\) oz the bottle is empty and you do NOT refill. So water is added only on days \(1\) through \(15\). Add \(1+2+3+\cdots+15\).
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Approach: Seeking complements — count the water in, not the mixture
- Don't track the mixture — ask the complement question: how much water enters the bottle? Each day you replace exactly what you drank, so on day \(k\) you add \(k\) oz of water.
- The drinking amounts climb \(1, 2, 3, \ldots\) and reach \(16\) on day \(16\). On that last day you drink the final \(16\) oz and the bottle is empty, so you do NOT refill — no water is added on day \(16\). Water is added only on days \(1\) through \(15\).
- Because the bottle ends empty, all the water that went in gets drunk: water drunk \(= 1+2+3+\cdots+15 = \frac{15 \times 16}{2} = 120\) oz.
- Check another way: total liquid drunk over all \(16\) days is \(1+2+\cdots+16 = 136\) oz. Of that, only the original \(16\) oz was juice, so the water is \(136 - 16 = 120\) oz. Same answer.
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