The Potato Problem!

How well do you think you can estimate something? How good are you at guessing the answers to problems? How fast can you do it? How accurate are your guesses? In this note, we're going to put your abilities to the test. And we're going to do that with a simple math problem. Are you ready?

I want you read the following problem carefully and (this part is really important) not do the calculations in your mind or on a piece of paper. I want you to simply guess. Also note down how fast you're doing the guessing. We're going to take that into consideration as well. Got that? Here comes the problem:

Suppose you have \(100\) pounds of potatoes. And these potatoes consist of \(99\) % water. You decide to leave the potatoes outside and let them dehydrate until they consist of \(98\) % water. Now the potatoes should weigh a little less than they were before. How much do they weigh now?

Okay, I want you to guess ... now! Don't do the math. Just guess [you'll miss out on something fun if you don't]. And you don't have to worry about being correct. You just have to get something close to the correct answer. Have you done that? Write your guess down.

So, how well do you think you did? What was your answer? \(99\) pounds? \(98\) pounds? \(90\) pounds? Prior to writing this note, I did a little survey on this problem with about \(20\) people. And the average answer was \(98.7\) pounds. Was your answer in this ballpark?

Surprisingly, the correct answer is \(50\) pounds! Yes, the weight decreases by \(50\)% because of the \(1\)% decrease of water! People find this answer very non-intuitive.

Here's a short solution of the problem. Since initially you had \(99\)% water, the water weighed \(99\) pounds while the non-water part weighed \(1\) pound. After the potatoes dehydrate, you're left with \(98\)% water and \(2\)% non-water stuff. Remember that the weight of the non-water stuff never decreases. So, \(2\)% of the new weight is equal to \(1\) pound.

\(1\) pound is \(2\)% of how many pounds? \(\boxed{50}\) pounds and that is our answer.

Were you surprised to see this answer? You're not alone. Your brain probably thought, "Hey, the water decreased by \(1\)%. The weight of the potato can't decrease that much." So, you went with an answer that wasn't too far off from \(100\) pounds.

Feel free to post what your guess was in the comments section. And if you have any feedback, I'll be happy to hear it.

The picture was taken from here

100 0 98 x (weight of water) 1(weight of mass) .............................. in pounds..........

100 - 98 1 --------------- = ------- 98-0 x x= 49 pounds.... of water after dehyadration...... add 1 pound of mass............

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I actually knew it because its an exact duplicate from a jelly problem on Brilliant I did a few days ago, level 2 algebra I think. A beautiful, nontrivial problem nonetheless. – Spock Weakhypercharge · 3 years, 6 months ago

@Spock Weakhypercharge – Exactly! I solved the same one! – Finn Hulse · 3 years, 6 months ago

@Spock Weakhypercharge – Oh! I didn't know that there was already a problem on this on Brilliant. Should have posted this earlier. By the way, I didn't make this problem. 'The Potato Paradox' is a simple problem with a very non-intuitive answer. Check out the Wikipedia article on it. – Mursalin Habib · 3 years, 6 months ago

@Mursalin Habib – lol.. everything isn't a paradox brother – Manas Tripathi · 3 years, 6 months ago

@Manas Tripathi – Technically, this problem isn't a paradox in the usual sense . It is a simple problem with a surprising answer. But this problem is commonly referred to as the 'Potato Paradox' [check out the Wikipedia article on it above]. I guess this is because people find the answer non-intuitive and paradoxical. – Mursalin Habib · 3 years, 6 months ago

@Spock Weakhypercharge – yea, saw that also. but if i haven't, my answer would probably be 95. HAHA – Astro Enthusiast · 3 years, 2 months ago

Mind = Blown.

I almost argued you that decreasing 99 to 49 pounds doesn't make sense to say that you're decreasing by 1%.

Let me explained you why this works.

1/2 = 50%

2/3 = 66.66....%

3/4 = 75%

4/5 = 80%

..................

when we keep doing this, you'll see that when you increase by 1 by 1... the percentage increases. But it keeps decreasing the rate of increase and stuffs. So we need to increase the fraction a lottttt to make the percentage change by 1, as we see below.

49/50 = 98%, 99/100 = 99%

Hope I answered your question. You tagged #JustForFun, but I'm freaking serious about this! XD

Oh, I forgot my guess, but I know I can't get this answer, lel. XD – Samuraiwarm Tsunayoshi · 3 years, 6 months ago

@Samuraiwarm Tsunayoshi – Nice explaining on your part! I already knew why this works [I shared this note, remember? :) ]. Thanks nevertheless. Did you enjoy it? – Mursalin Habib · 3 years, 6 months ago

@Mursalin Habib – Yup! My mind's actually stunned for 2 mins. Now I know I can't use common sense any more. XD – Samuraiwarm Tsunayoshi · 3 years, 6 months ago

Did anyone else look at the answer like a million times and still think,"That is witchcraft"? Anyway, i thought this note is very fascinating. Great job! – Victor Song · 3 years, 6 months ago

@Victor Song – Thanks! – Mursalin Habib · 3 years, 6 months ago

@Victor Song – Same here!!!! – Tanya Gupta · 3 years, 6 months ago

Still hard to believe it – Agung Rizka · 3 years, 6 months ago

@Agung Rizka – That is what makes this problem so cool. Even though it's pretty easy to solve, the answer is really non-intuitive and most people wouldn't come close to it if they guessed. – Mursalin Habib · 3 years, 6 months ago

I guessed 98 percent, interesting – Arshiya Sheikh Sheikh · 3 years, 6 months ago

wat my brain hurts – Faraz Masroor · 3 years, 6 months ago

the weight will reduce by 50%.... – Ameya Patil · 3 years, 6 months ago

my guess was 50. – Manish Jha · 3 years, 6 months ago

This problem can be solved in less than a minute.......

by using the concept of mixtures and alligations......

now, understand that the potato is made up of water and the mass.... the weight of mass in going to remain the unchanged.....

so as total weight is 100 pounds..... and 99% is water and 1% is mass..... the weight of water in the potatoes is 99 pounds and mass is of 1 pound.......

So to get the mixture of water and mass to get the potato....... with 98% of water.......

mix the two concentrations.... with weight of the mass is 1 pound after dehydration

concentration of WATER MASS

by alligation rule... solve .....

So, 50 pounds is the weight of the potatoes after dehyadration.............................hahahahahah – Ameya Patil · 2 years, 8 months ago

I guessed 99.2 – Sachin Arora · 2 years, 8 months ago

It's really hard to belive..can someone carry out this experiment and check maths. .... – Tushar Panda · 2 years, 9 months ago

nice trick – Akash Srivastav · 3 years, 6 months ago

me too solve it... – Binil Gopi P G · 3 years, 6 months ago

I was wrong my answer came 97.02 lbs – Vijaypratap Singh · 3 years, 6 months ago

99.9 The tiniest bit possible! Oh well, turns out I was wrong. I had to read it twice to understand, thought. – Pola Forest · 3 years, 6 months ago

This problem made me awake since I have been sleeping while reading.......Yep I could hav done this – Naga Ramsai · 3 years, 6 months ago

owesome! – Chandan Burnwal · 3 years, 6 months ago

99 pound – Faiq Mehmood · 3 years, 6 months ago

first i got 98.7 gm but latter i find out it 50 gm. – Dibyasingh Rautaray · 3 years, 6 months ago

so 1 pound=2%means 50 pounds= 50*2=100% – Sundar K · 3 years, 6 months ago

wow.. yah had 98,. perfect sense by ur solution,. – Letso Jakes Jacobs · 3 years, 6 months ago

a nice trick the brain plays with us – Souradeep Roy Choudhury · 3 years, 6 months ago

ya its 50 pounds only........ – Naga Rohit · 3 years, 6 months ago

100x2%=2 100-2=98.

98.00x1%=.98.00 98.00-.98.00=97.07 97.07 +2=99.07 Ans

Md.Wahidur Rahman Khan. – Khan Rahman · 3 years, 6 months ago

Yeah The Answer was tricky and Correct .. How I calculated was if 1% of 100 is weight of non water (say y) then how 2% of how much will be y ? it is 50 of course .. But there is something in the Question which causes (I think) to guess an answer closer to 100 .. and it works .. if some body reads it they will think it should be closer to 100 . . > " Now the potatoes should weigh a little less than they were before" .. when our mind reads little less it is ought to think a number closer 100 .. – Abhishek Singh · 3 years, 6 months ago

i thought it would be like 52% around – Jagadeep Dadi · 3 years, 6 months ago

Not surprising at all... – Shylett Anthony · 3 years, 6 months ago

98.5% – Gleich Libertia · 3 years, 6 months ago

99 – T.m. Ishtiak Mokut · 3 years, 6 months ago

how can anyone GUESS answer for this? :P indeed you have to calculate! – Vivek Patil · 3 years, 6 months ago

99.5 – Qaiser Imran · 3 years, 6 months ago

49 – Qaiser Imran · 3 years, 6 months ago

99 pounds – Venkataramana Venkataramana · 3 years, 6 months ago

here's simple to understand.
1% =1lb (which is obvious){ the "potato only" content which will not change }
Let total mass after dehydration = y
\(0.98y + 0.02y = y \)
\(0.02 y = 1lb\) { "potato only" content not changed }
So, \(y=50 lb \)
and ofcourse water is \(0.98 \times 50 lb= 49 lb\) .
:)

– Shahbazz Patel · 3 years, 6 months ago

@Shahbazz Patel – thid is what i really require... although I got the answer somehow I need some systematic way to approach this problem... tq:) – Parakkal Dignesh · 3 years, 6 months ago

99.5 – Youssof Rabie · 3 years, 6 months ago

nice one – Archiet Dev · 3 years, 6 months ago

I thought it was 98. – Muhammad N Kamal Aw · 3 years, 6 months ago

i guessed 98% – Aanam Aafzaal · 3 years, 6 months ago

50 indeed is the answer. it's easier to visualize if you plug this in in a table. the trick here is to use percent by weight not percent composition – Vol Vox · 3 years, 6 months ago

interesting one... post some more ques. like this... :) – Venkataramana Gundi · 3 years, 6 months ago

really smart one... – Fahad Shihab · 3 years, 6 months ago

intersting man ..i was also wrong – Malik Suleman Khalid · 3 years, 6 months ago

haha... very interesting question.... but I got the 50 lbs answer just after reading the question... (1/0.01=100lbs and 1/0.02=50 lbs) ... if 1lb potatoes are 2% of the total mass then the total mass ought to be 50 lbs... :D – Chauhdry Sahab · 3 years, 6 months ago

it's still confusing me because this is the first time I join with brilliant.org. but, I enjoy with that. – Khusnul Khotimah · 3 years, 6 months ago

Totally true. I saw a question just like this a while ago and I think I immediately thought 98 – Robert Fritz · 3 years, 6 months ago

Think it this way: There s a giant potato of 100 lb. So initially 99 lb was water. We wanna find the amount of water lost so that finally water will contribute to 98% of the weight. Let X be the weight of the water lost. Reduction in weight of water =99-X Reduction in total weight =100-X Now this must comprise 98% by weight. Putting in equation, ((99-X)/(100-X))*100= 98% Solving for X, we get X= 50 lb Which gives the remaining weight= 100-X = 50 lb – Harisree Krishnamurthy · 3 years, 6 months ago

Why mountains being close to sun than ground remain colder? – Vijaypratap Singh · 3 years, 6 months ago

Potatoes have seldom been a man's problem in this part of the world, I mean India, Pakistan, Bangladesh, Sri Lanka, Nepal and also include Middle East countries. – Irfan Iftekhar · 3 years, 6 months ago

99% – Shabbir Shakir · 3 years, 6 months ago

i guessed 90 lbs – Dustin Jones · 3 years, 6 months ago

Beautiful.. Never able to guess that it would be around 50%. If I would have taken my pen and pencil, then I would have got it correct. Not gud in guessing – Deepak Vijayalu K · 3 years, 6 months ago

i thing 11 potatos....not fully correct i guess....:-p – Tariq Ajeej · 3 years, 6 months ago

98 – Ashwin Upadhyay · 3 years, 6 months ago

ans is 98 is it correct – Aditya Raj · 3 years, 6 months ago

yoyoyo my famalam sweats suck it – Arnnav John · 3 years, 6 months ago

Nah, this solution is wrong I think. If the water decreases maybe because evaportaion,etc. The non-water stuff will not be increased, this is why I say the solution is false. So the ratio of the potato before and now is 100:99, so the answer must be 99 pounds (in my opinion) – Andrew Fanggara · 3 years, 6 months ago

@Andrew Fanggara – The problem with your reasoning is that you consider the new, 98% weight of the water to be based on the initial weight of 100 pounds. It's incorrect; the water is 98% of the new weight. – Ivan Koswara · 3 years, 6 months ago

@Andrew Fanggara – no man. 50 is correct. remember that the non-water part is always fixed and only the water content changes. the water has changed to 98%, which means the non-water is now 2% (regardless of this, its weight is still the same after drying). now, if 1 lb is 2%, then by ratio and proportion, 50 lbs is the answer. – Vol Vox · 3 years, 6 months ago

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The Potato Problem!

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