I have a 16-ounce bottle of wine and want to make it last as long as possible, so I establish the following plan: On the first day I’ll drink 1 ounce of wine and refill the bottle with water. On the second day I’ll drink 2 ounces of the mixture and refill the bottle with water. On the third day I’ll drink three ounces of the mixture and again refill the bottle with water. If I continue until the bottle is empty, how many ounces of water will I have drunk?
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It’s easy to get bogged down in calculating proportions, but there’s a simpler solution. If I continue until the bottle is empty, then eventually I will have drunk all the water that I’ve added to the bottle. I add 1 ounce on the first day, 2 on the second, and so on to the 15th day. (I drink off the entire bottle on the 16th day, so no water is added then.) So in all I’ll have drunk 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120 ounces of water.
(Another solution is to calculate that in the end I’ve drunk 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 136 ounces in total, and then deduct the 16 ounces of wine.)
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