You have 100 kilograms of purely mathematical and hypothetical potatoes (meaning they do not need to conform to the laws of basic physics and other stuff). These potatoes are special because they're 99% water by weight. Overnight, they dehydrate until they become 98% water. You go to weigh them the next morning, and you're shocked by the result.
How many kilos do the potatoes weight the next day?
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On the first day, the 100 kg potatoes is 99% by weight. This means that the dry potato material weighs 1 kg or its dry weight w d r y = 1 . Let the weight of the potatoes be w the next day. Then we have:
w w − w d r y w w − 1 w − 1 ⟹ 0 . 0 2 w w = 0 . 9 8 = 0 . 9 8 = 0 . 9 8 w = 1 = 5 0
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There are two methods of solving this problem.
Method 1: The explanation states that the non-water weight is 1 kilo (1% of 100 kilos). Once the potatoes have dehydrated, the question becomes "1 kilo is 2% of how many kilos?" (there was no change in the non-water weight, only the composition), and in order for the percentage to be twice as big, the weight must be half as heavy.
Method 2: The initial ratio of potato stuff to water is 1:99. If the water is reduced to 98%, the potato stuff accounts for 2% of the weight, meaning the ratio becomes 2:98, or 1:49. Since the potato stuff still weighs 1 kilo, the water must weigh 49 kilos totalling 50 kilos.