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I couldn't find an explanation to this problem that I could understand.

A watermelon consist of 99% water and that water measures 2 litre. After a day in the sun the water melon dries up and now consist of 98% water. How much water is left in the water melon?

I know the answer is ~1 litre, but why is that? I've read a couple of answers but I guess I'm a bit slow because I don't understand why.

I'd like you to assume that I know no maths. Explain it like you would explain it to a 10 year old.

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If 2 litres of water is 99%, then the rest of the watermelon is about 0.02 litres. After drying up, the rest remains about 0.02 litres, but it is now 2% of the watermelon. So the whole watermelon is about 1 litre. – Rahul Sep 25 '12 at 9:31
up vote 4 down vote accepted

At the beginning the solid material is $1\%$ of the total which is a trifle (to be neglected) more than $1\%$ of $99\%$ of the total, or $1\%$ of $2000\ {\rm cm}^3$. Therefore the solid material has volume $\sim20\ {\rm cm}^3$.

After one day in the sun these $20\ {\rm cm}^3$ solid material are still the same, but now they make up $2\%$ of the total. Therefore the total now will be $1000\ {\rm cm}^3$ or $1$ litre. $98\%$ of this volume, or almost all of it, will be water.

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You start with 2 litres of water and $x$ litres (say we measure by volume) of "non water". The percentage of water is

$$ \frac{2}{2 + x} = 99\% = \frac{99}{99+1}$$

You solve this to get that $x : 1 = 2 : 99$ or that $x = 2/99$.

After drying, you have $y$ litres of water and $x$ litres of "non water". Since the non-water bits don't dry, the $x$ is same as before: that is $x = 2/99$. The percentage of water is

$$ \frac{y}{y+x} = \frac{y}{y+ 2/99} = 98\% = \frac{98}{98 + 2} $$

So solving this you get that $y : 98 = 2/99 : 2 = 1 : 99$. Or, in other words, $y = 98 /99 \approx 1$. That's how much water you have left.

To intuitively understand the problem, it is more helpful to think of the proportion of "non-water". The non water started out at 1%. It ended up in 2%. Since the amount of "non water" didn't change, to have its proportion go from 1% to 2% means that the total volume must have decreased by half.

$$ \frac{\text{non water}}{\text{total starting volume}} = 1\% \longrightarrow \frac{\text{non water}}{\text{total final volume}} = 2\% $$

Since the watermelon started out almost all water, for the total volume to decrease by half you must lose at least (and almost exactly) half of the water.

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I really appreciate the effort! But this is too much maths and equations for me. Is there a simpler way to explain it, using metaphors or similar? – Niklas Sep 25 '12 at 9:23

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