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There are two coins, Coin A and Coin B. You can transfer Coin A for Coin B or Coin B for Coin A only in the two following ways:

Give 100 Coin A for 10 Coin B

Or

Give 10 Coin B for 50 Coin A

In this situation, which of the coins is worth a higher value? A friend of mine presented this problem and we were unable to find a solution. Any thought?

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    $\begingroup$ The exchange ratio of A to B is 10 to 1 in the first transfer and 5 to 1 in the second, so surely both imply that B is more valuable? However, it's a very odd question, focusing only on transfers between coins and saying nothing about the rates at which the coins might exchange for real goods. $\endgroup$ – Adam Bailey Mar 14 '16 at 10:07
  • $\begingroup$ Coin A is more valuable because if those are the only two types of trades allowed, then you can only ever get less Coin A than you started with; Coin B has no power to get you a good deal on Coin A, or you'd rather prefer to work for Coin B rather than A. That's what it looks like to me. $\endgroup$ – Kitsune Cavalry Mar 15 '16 at 2:33
  • $\begingroup$ @KitsuneCavalry that is very true, and that when couple with the other comments is the source of the dilemma. $\endgroup$ – Jamman00 Mar 15 '16 at 2:36
  • $\begingroup$ @KitsuneCavalry But equally you can only get less B than you start with, eg if you start with 20 B you could trade 2 x 10 B for 2 x 50 A = 100 A which you could then trade for just 10 B. $\endgroup$ – Adam Bailey Mar 15 '16 at 10:55
  • $\begingroup$ @AdamBailey So would you say this just looks like a very imperfect money market that hasn't cleared? Because I think you are correct, so that would be my best explanation given that. $\endgroup$ – Kitsune Cavalry Mar 15 '16 at 17:21
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From the first equation, $$100A_{1}=10B_{1}$$ so $A_{1}=0.1B_{1}$.

From the second equation, $$10B_{2}=50A_{2}$$ so $A_{2}=0.2B_{2}$.

So it follows, $$0.2B_{2}\ge A_{2} > A_{1} \ge 0.1B_{1}$$

Now, as $A_{2}>A_{1}$ then the only possible conclusion from the inequality is that for sure $0.2B_{2}>A_{1}$ and thus we deduce that $B$ has the higher value.

Hence if you are forced to choose strictly within the constraints set out in the problem then one would be best served by choosing only the B coin.

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The word 'valuable' is not well defined in this question.

If I am willing to buy an apple from you for \$1 and but I am only willing to sell an apple to you for \$4 which is more valuable: An apple or \$2? You cannot tell because neither option is sufficient to get the other option by trade.

You are faced with the same dilemma in your problem. There are different prices at which to sell and buy so neither one of the options is clearly more valuable.

EDIT:
As @Adam Bailey points out in his excellent answer in this situation even though the exact value ratio is ill defined, it is clear that coin B is more valuable than coin A, because you can exchange one coin B for five coin A, thus you can get more by chosing coin B.

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Since the exchange ratio of A to B is 10 to 1 in the first transfer and 5 to 1 in the second, it is clear that B is more valuable, although as @denesp says the exact value ratio is undefined. It's a very odd question because it focuses only on transfers between the coins and says nothing about the rates at which the coins might exchange for real goods, in other words, the prices of goods in terms of the coins.

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Give 100 Coin A for 10 Coin B

Or

Give 10 Coin B for 50 Coin A

100a >>> 10b (Think: 100 "pennies" buys 10 "dimes")

100b >>> 500a (Think: 100 "dimes" buys 500 "pennies")

Coin "B" is of higher value.

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