Paradox

[Steven G]

I have a sack of potatoes that weigh 100 lbs
The ratio of water to solids is 99:1. That is the potatoes are 99 lbs water and 1 lb solid.
I leave the potatoes outside and eventually it is 98% water.
Therefore water:solid is 98:2 or 49:1

I still have the 1 lb of solid which means that I have 49 lbs of water (since water:solid = 98:2 or 49:1)
So if the water drops from 99% to 98% we have 49 lbs of water + 1 lb solid = 50 lbs.
This says that we lost 50 lbs by losing 1% of water.
What is going on here?

 

[khansaheb]

I have a sack of potatoes that weigh 100 lbs
The ratio of water to solids is 99:1. That is the potatoes are 99 lbs water and 1 lb solid.
I leave the potatoes outside and eventually it is 98% water.
Therefore water:solid is 98:2 or 49:1

I still have the 1 lb of solid which means that I have 49 lbs of water (since water:solid = 98:2 or 49:1)
So if the water drops from 99% to 98% we have 49 lbs of water + 1 lb solid = 50 lbs.
This says that we lost 50 lbs by losing 1% of water.
What is going on here?

In this problem, the amount of solid would remain same (1 lb) and you will lose water

In the dried mixture there will be 1 lb of solid and 'w' lb water → total weight = (1 + w) lb

We know

1/(1+w) = 0.02 → 1+ w = 50 → w = 49 lbs

Yes .... you have to lose 50 lbs of water to record a gain of 1% solids.
Total water loss is (99-49 = ) 50 lbs and final weight of the mixture is 50 lbs

 

[Dr.Peterson]

This says that we lost 50 lbs by losing 1% of water.

Surely you know that percentages require you to state the base with respect to which it is calculated. Wording matters.

You didn't lose 1% of the water; it was the percentage of the whole that is water that decreased by 1, from 99 to 98; and that whole changed considerably.

You actually lost about 50% of the water (and 50% of the whole).

Another way to think about it is that the constant amount of solids changed from 1% to 2% of the whole, doubling the percentage; so the whole had to be cut in half, from 100 times the solids to 50 times the solids. Since the solids stay the same, that mean you lost 50 lb of water.