# Does the Potato Problem assume 1% water = 1 lb of water?

The Universal Book of Mathematics states the problem without saying anything of the mass of water or potatoes:

Fred brings home 100 pounds of potatoes, which (being purely mathematical potatoes) consist of 99 percent water. He then leaves them outside overnight so that they consist of 98 percent water. What is their new weight? The surprising answer is 50 pounds.

Thus I understand Method 1, but not 2 quoted beneath:

100 lb of potatoes, 99% water (by weight), means that there's 99 lb of water, and 1 lb of solids. It's a 1:99 ratio.

If the water decreases to 98%, then the solids account for 2% of the weight. The 2:98 ratio reduces to 1:49. Since the solids still weigh 1 lb, the water must weigh 49 lb for a total of 50 lbs for the answer.

Doesn't Method 2 assume that 99% of water = 99 lb of water? How do we know that 1% water by weight = 1 lb of water?

I have a BA in Economics, and already know elementary algebra.

• What is 1% of 100 lb? – Wojowu Jan 22 at 21:16
• @Wojowu 1 lb, obviously. But how's this relevant? – Pamela Lee Jan 22 at 21:16
• 1% of water by weight means that 1% of weight is water. 1% of weight is 1 lb, hence 1 lb of water. – Wojowu Jan 22 at 21:18
• These percentages are explicitly stated as by weight. That's the answer to your question. – Ethan Bolker Jan 22 at 21:19

## 1 Answer

We have all the information we need, no more assumptions are needed.

As the original potatoes are $$99\%$$ water, the $$100$$ lbs must be split as $$1$$ pound solid potato flesh and $$99$$ pounds water. Now, the solid potato flesh does not evaporate so you'll always have $$1$$ pound of solid potato flesh. Post evaporation we must have $$x$$ pounds of water where $$\frac x{x+1}=.98\implies x =.98x +.98\implies .02x=.98\implies x =49$$ Thus at the end we must have $$49+1=50$$ pounds of combined water and solid potato flesh.

• How does 99% water imply 1 lb potato flesh + 99 lbs water? The question never states 99% of 100 lbs is water. – Pamela Lee Jan 22 at 21:20
• Yes it does. You wrote "100 pounds of potatoes, which (being purely mathematical potatoes) consist of 99 percent water" – lulu Jan 22 at 21:21
• I just quoted your version of the statement of the problem. If you are referring to some other statement of the problem, you should include that statement (not a link) in your post. – lulu Jan 22 at 21:24
• What on earth do you imagine $99\%$ of $100$ pounds might mean? – lulu Jan 22 at 21:31
• The problem statement doesn't say that...it states "100 pounds of potatoes, which (being purely mathematical potatoes) consist of 99 percent water." – Pamela Lee Jan 23 at 0:16