# How to prove that this man has a profit of 20 and not 10?

This is not actually a puzzle per se, but I'm asking for an explanation about the puzzle's answer.

The puzzle goes like this:

A man bought a goat for 60. He then sold it for 70. Later, he bought that goat back for 80, and then sold it again for 90. How much profit did he make?

I know the answer is 20 since the man will be left with 20 extra units of money on his hands after the last sale. But somebody insists that his profit is just 10 because he had used his first 10 profit to buy back the goat for 80. I know there is a logic error here, but I can't see it. Can you explain, using this thought process, why his profit is not 10, but 20?

• To all the answerers - I am confused by the spoiler warning - the answer is given in the question - are the explanations considered spoilers? I didn't think so and didn't mark it as such (I dislike spoilers when not needed because they make things harder to read) but given five answers before mine did I wondered if I missed something or if there is some convention that all answers, whether spoilers or not should be hidden... – Chris Feb 24 '18 at 18:30
• @Chris I agree, I think we're all too used to using spoilers that we're not really thinking about it. – greenturtle3141 Feb 25 '18 at 17:16
• if he bought a goat for \$60 and eventually sold it for \$90, didn't he make \$30 profit? – JMP Feb 28 '19 at 15:35 • The fact that the goat appreciated while it was out of his hands does not factor into his net profit. At the end of the day, he made two separate smart goat sales. – Aww_Geez May 26 at 21:03 • @Aww_Geez I think that's the root of the error, focusing too much on the 'loss' from 70 to 80. If you look at it as two separate transactions, the answer becomes clear. – Matthew Jensen May 26 at 23:59 ## 14 Answers Based on his thought process: "But somebody insists that his profit is just 10 because he had used his first 10 profit to buy back the goat for 80. " Well, that 10 profit that was used was returned again when the second buyer bought the goat for 80 too, plus another 10 to add to the profits. But honestly, a better way to see it is that: He bought twice, for 60 and 80. He sells twice, for 70 and 90. Money out of pocket was 140, money in was 160. Profit = 20. • I like your second answer - nice and simple :) – puzzledPig Feb 24 '18 at 22:25 • Oh I see! The first answer do it for me. I was wondering too where the first profit gone. I forgot that the profit returns when he sold it again. Thanks! – Chen Li Yong Feb 25 '18 at 15:45 • Your answer convinces me that his profit is indeed 20, but when I work it out myself I still get stuck on 10 and 30! He bought the goat for 60 and sold it for 70. (10 Profit). He bought it back for 80 (-10 Loss), and sold it for 90 (10 Profit). Conversely, he started off with a goat for 60 and ended up with 90, so he should have 30 Profit. – Grace Jul 17 '18 at 3:30 • @Grace You counted -10 loss on the second buy, but where did that come from? The second transaction (buy 80 sell 90) is completely separate from the first. When he bought for 60 and sold for 70, the transaction already "finished" with 10 profit. The second purchase is a fresh transaction, where he lost 80 to buy, then gained 90 when selling, for another net of 10 profit. That's that. – votbear Jul 17 '18 at 4:13 • @Grace Or rather, just imagine that you have 100 money in the start. So you go from 100 -> 40 (buy 60) -> 110 (sell 70) -> 30 (buy 80) -> 120 (sell 90). – votbear Jul 17 '18 at 7:21 Assuming you have already explained the maths as others have suggested (70-60)+(90-80)=20 then here's a few other ideas... Method 1 - switching the numbers up A man buys a goat for £1 and then sells it for £1,000,000. He then buys another goat for £1,000,000 and sells it for £1,000,001. In this case the first sell and new buy price are deliberately the same. So in this example he uses his profit to buy the second goat but hopefully it is obvious to all that he started with £1 and ended with £1,000,001 and thus his profit is £1,000,000 and not just the £1 from the second sale. You could then extend this to more akin to the original example with buying for £1 and selling for £1,000,000 and then buying for £1,000,001 and selling for £1,000,002. Again I hope it would be obvious that the seller has ended with a lot more than £1 profit. Method 2 - Actually do it Offer your friend an experiment. You both start with £100 (you can go for pennies if you prefer to be a bit less rich) and then a random item and perform the transactions as described. A buys a goat from B for £60. A now has £40 and B has £160. A sells the goat to B for £70. A now has £110 and B has £90. A buys the goat from B for £80. A now has £30 and B has £170. A sells the goat to B for £90. A now has £120 and B has £80. Method 3 - separate stacks of money Consider the first man to have two separate stacks of money, one with 60 in and one with 80 in. He uses these two stacks for his two purchases. Each stack will be £10 bigger than when he started, thus £20 profit. The separate stacks ensures that there is no possibility that you could consider the profits of the first set of transactions to have been consumed by the second set of transactions. Method 4 - Reorder the transactions A man buys two goats, one for 60 and one for 80, thus two for 140. He then sells them both for 70 and 90, or both for 160. Thus 20 profit. This technique means that the profit doesn't even exist at the time the second goat is purchased and thus there is no way that it could have been made to go away by the second goat purchase. If this still fails to convince him then offer to actually perform these transactions til this other person has no money left and you have all their money. :) • The method 2, 3, and 4 are still my thought process, which he rejects / can't understand. The first method though, is thought provoking. I never thought that I can switch the number to eliminate the profit issue. Another way to solve this is similar to method 4, I can reorder the transaction so that the man buy goat for 80 and sell for 90, and then buy back the goat for 60 and then sell for 70. Using this, the loss 10 is not even exists, because he sell for 90 and buy for 60. That way, it can show him that the "loss" that he mentioned between sell 70 and buy 80 is not relevant at all. – Chen Li Yong Feb 25 '18 at 15:51 • Surely he can't reject the actually do it thing. If you do all the transaction and person A has 20 more money at the end then he can't deny that result. Likewise the separate stacks of money approach you cannot apply the "he used his profit to buy it" because that money is clearly not touched by the second set of transactions. Hell, you could claim it is two different people each making a 10 money profit which then combines to an overall 20 money profit. – Chris Feb 25 '18 at 15:56 • The fact is his statement is wrong. The above is four demonstrations of why and if he still insists that his logic is correct then there isn't much more to be done. In the same way that if somebody said 2+2=5 you could easily demonstrate this to be wrong but if they kept insisting that it was then you just have to shrug and accept that they are wrong and move on with your life. Once something has been demonstrated to be wrong (in this case that the first ten profit is somehow zeroes out by using it) and the person keeps insisting it is right then there is no point attempting more logic. – Chris Feb 25 '18 at 15:59 • Yeah, basically I just need to nailed the explanation of "where did the 10 dollars profit go?" Once I told him that when he sold it again for 90 dollars, the original 10 dollars profit is returned, plus some 10 more, finally he gets it lol. It's silly me to not notice it in the first place. I just made this question because I know his logic is wrong but I cannot pinpoint where. It's like I know that apple falls from the tree, but I cannot explain why apple falls from the tree. This bothers me a lot lol. – Chen Li Yong Feb 26 '18 at 3:53 Let's say the guy had 70 dollars at the beginning, he bought the goat for 60 dollars and 10 dollars in his pocket. He sold the goat for 70 dollars and together with his 10 dollars he bought a goat for 80 dollars and later he sold it for 90 dollars. He has 90 dollars in his pocket now which is 20 dollars more than his original money. • This is kinda still doesn't explicitly explain where the 10 dollars go. Through other answers, I have got explanation that the 10 dollars is returned when he sold it for 90 dollars. Thanks! – Chen Li Yong Feb 25 '18 at 15:54 A nice way of thinking about it: Let's say the goat is worth 60 dollars. If you buy it for 60, nothing changed, sell it for 70, you gain 10 dollars. When you buy it for 80, you lose 20 dollars, but when you sell it for 90, you gain 30 dollars. So the profit is (-)0+10-20+30=20 dollars. • Yes, this is the mathematics of the riddle. But he rejects it because in his logic he still clinging to the "disappearance of 10 dollars profit". The explanation need to explicitly explain where the 10 dollars profit gone? Through other answers I get logical explanation that the 10 dollars profit is returned, along with additional 10 dollars profit, when he sold the goat for 90 dollars. – Chen Li Yong Feb 25 '18 at 16:03 Here is my explanation. He buys the goat for 60. He sells it for [sealed box of money who knows how much is in there.] Then he buys it again for [sealed box of money who knows how much is in there] plus ten. He has spent a total of 70. He sells it for 90 so his profit is 20. This doesn't rely on adding two separate profits and then wondering if one of them was cancelled when he bought it in the middle. It may feel more intuitive as a result. • Yea, in the end, I used this logic to explain it to him instead of trying to follow his cancelled profit logic to explain from there. – Chen Li Yong May 27 at 6:36 The reasoning behind this may be, How about if we pretend that he bought two different goats. The first one he bought for 60 dollars and then sold it for 70 dollars, therefore making 10 dollars. Then the man bought another goat for 80 dollars and then sold it for 90 dollars therefore making another 10 dollars. This then adds up to a total of 20 dollars. • This is still my thought process, which he rejects, even though he can't deny that he ended up with extra 20. – Chen Li Yong Feb 25 '18 at 15:52 The logic error here is that coming up with a profit of only 10 subtly double-counts the two middle steps. As others have pointed out, the basic math is $$-60+70-80+90=20$$. The faulty logic is 10 profit, 10 loss, and 10 profit summing to 10 profit overall. However, the first 10 profit comes from $$-60+70$$, the loss comes from $$70-80$$, and the second profit comes from $$-80+90$$. So to get a profit of 10, they're actually doing $$-60+70+70-80-80+90=-60+2*70-2*80+90$$. So it's not valid to combine the intermediate loss from $$70-80=-10$$ with the incremental gains of 10. It is, however, valid to compare the intermediate loss to the overall gain of $$-60+90=30$$, as combining those two calculations does not cause any value to be used twice. Say you have \$100 to invest. The sequence is as follows:

• Buy goat for \$60, so you have a goat + \$40 in the bank
• Sell goat for \$70, so you have \$110 in the bank
• Buy a goat for \$80, so you have a goat + \$30 in the bank
• Sell the goat for \$90, you have \$120 in the bank

Your guy is reasoning that he bought the first goat for \$60, and sold for \$70, and now the vendor has \$70+\$10=\$80, which he buys the second goat (or same goat again), and then sells for \$90, plus the \$10 profit =\$100, so he is counting the profit twice, but only uses the trick once.

The absolute simplest way to prove this to your friend is to offer to actually perform these transactions with you - and to make it worth his time, you'll pay him $15 to do so. Ask him to sell you something (a goat, or any suitable substitute) for \$60, then buy it back for \$70, then sell it to you for \$80, then buy it back for \$90. According to him, he'll have only lost \$10 in this series of transactions. Since you paid him \$15 for his trouble, he'll have made a profit. Tell him you're happy to do this over and over for as long as he likes, to be sure you both eventually arrive at the same conclusion. The goat is worth$60 because it is the same goat and it's value never changes. Therefore buying the goat back for 80 means he lost 20, he then made 20 in the exchanges he broke even no way around it

The man earned no profit. His dollars declined in value over the timeframe of the scenario. His buying power at the start was one goat. His buying power at the end was still one goat. The goat is a metaphor for all “real goods”. On paper he has 20 dollars more than he started with, but unfortunately the dollars lost 33% of their buying power in real terms and he is no better off.

• And since he owes capital gains taxes on $20, by April he will be even worse off than when he started. – Bart Simpson May 25 at 10:02 • Well, profit is still$20 by any economic standard. The time-value of money is obviously beyond the scope of the question, but I like your tangent. Indeed, if the two transactions were close together, one might be very concerned about hyper-inflation. – Earlien May 25 at 10:27
• This is obviously a joke answer, but even so it has its own issues. This makes assumptions about the state of the goat in all transactions, the perceived value of the buyer, and the skill of the salesperson. (re: state) consider hairdressing or painting horns as "added value". (re: perceived value) the only person to remain the same is the single salesperson and other people may value the goat differently. (re: salesperson ability) a silver-tongued salesperson can sell anything for more than its worth. – Ian MacDonald May 25 at 13:47
• It’s just a matter of evaluating the profit in goats rather than Dollars. You are looking at this with the assumption of a stable US Dollar. By the end of 2021 you’ll start to see this question my way. – Bart Simpson May 25 at 22:43

He uses 60 dollars to buy a goat. He sells it for 70, gaining 10 dollars and his original 60 back. He buys the goat back for 80 and he has 70 at hand and spends another 10 to get 80. This puts him at a profit of -70 because 10 wasn't his base money. He then sells it for 90 so he profits 20. Unless after you earn the 10 you count it as your own money from then on. So after you gain 10 dollars pretend it is a new problem and you have 70 dollars to buy a goat and you get another 10. You used 80 dollars to buy a goat then sell for 90. You profit 10.

• −½ for being wrong and −½ for being unclear. – Peregrine Rook Jul 15 '18 at 17:51

You neither can prove that the answer is 20 nor 10. Because both 20 and 10 are wrong answers. The man makes 0 profit at the end of these two transactions. Mathematically yes with straight equation, the answer should be 20. But with financial operation factored-in, we need consider that he had to borrow 10 to buy the second time and leaving him with nothing left in his pocket. When he made the 10 in his second sale by selling at 90, he needs to use that 10 that he made to repay the 10 he borrowed in order to buy the goat at 80. In the end, he makes nothing.

• Your explanation assumes he starts with \$60, buys the goat (leaving him with \$0), sells it for \$70 (he has \$70 in hand), borrows \$10 (he has \$80 in hand), buys the goat for \$80 (leaving him with \$0 again), sells it for \$90 (he has \$90 in hand), and then repays the \$10 he borrowed (leaving him with \$80 in hand). At the end he has \$80, when he had \$60 at the start, even factoring in the loan you invented. He's still up \$20. – Rubio Apr 2 '18 at 3:26 • And what if he started with$100,  and didn’t need to borrow any money at all?  He ends up with \\$120. – Peregrine Rook Jul 15 '18 at 17:49