# A loss and gain problem

This is a very simple but confusing puzzle.

A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase.

The lady gives him a $1000$ rupee note.

The shopkeeper has no change, so he goes next door to another shopkeeper to get change for the $1000$ rupee note. He keeps $200$ for himself and returns $800$ to the customer.

Later, the second shopkeeper from next door comes back with the $1000$ rupee note with a stamp on it saying "counterfeit" and takes his money back.

How much loss does the first shopkeeper face?

• Indeed that puzzle is phrased confusingly. What does it even mean? Duplicate? – dfeuer May 30 '13 at 18:06
• that is fake currency notes – iostream007 May 30 '13 at 18:08
• I think the lady purchased an item worth 200 with a 1000 bill. The shopkeeper makes the correct change, later to find out that the bill was counterfeit? – Mark May 30 '13 at 18:13
• so at selling lady and shopkeeper are OK but that 1000rs will've to given by shopkeeper and this is lOSS – iostream007 May 30 '13 at 18:17
• Yes. It seems like that. And that question doesnt have any accepted answers! – selvin Jun 13 '13 at 7:44

The shopkeeper hasn't suffered any loss until the next shop's owner comes over and demands his 1000 rupees back. The shopkeeper hasn't gained anything either, since his profit margin is zero.

Thus, after he reimburses the next shop's owner, he's out 1000 rupees.

since shopkeeper selling good with zero profit so $200$rs of good selling is not his profit.so when he return 1000 to his neighbor shopkeeper he is in LOSS of $1000$rs. People thinks its answer will be 800.

• Try to explain this to the IRS :) The good part of this question is: Reading comprehension. I like this question and will upvote it! – imranfat May 30 '13 at 18:11
• IRS? what is this? – iostream007 May 30 '13 at 18:13
• @iostream007: The Internal Revenue Service, the US national tax authority. – Ross Millikan May 30 '13 at 18:15
• This is correct. It is simple if you just focus on what he had at the start of the day, which was the goods +1000 rs, and what he had at the end of the day, which was nothing. Since he received nothing (except a fraudulent note) he is out that much. – Ross Millikan May 30 '13 at 18:18
• ok i didn't get that was joke :P – iostream007 May 30 '13 at 18:18

The shopkeeper loses $1000$ rupees. Here is a flow chart of each transaction:

$$0\xrightarrow{\text{sold goods}} -200\xrightarrow{\text{borrowed from neighbor}} 800\xrightarrow{\text{change to woman}} 0\xrightarrow{\text{paid back neighbor}} -1000$$

Another way to think about this is total gain/loss for all parties involved. The woman gains $200$ from the goods, and $800$ from the change. There is no gain or loss for the neighbor. Since its a zero sum game, it follows that the shopkeeper loses $1000.$

The easiest way to solve this problem is to ignore the counterfeit bill until the end. Instead, assume the shopkeeper's neighbor comes back the next day to make change himself. Only after making change does the shopkeeper notice the counterfeit.

1. Customer gives shopkeeper 1000rs note. Shopkeeper is +1000.
2. Shopkeeper gives neighbor 100rs note. Receives 1000rs change. Shopkeeper is +1000.
3. Shopkeeper gives customer 800rs + 200rs in goods. Shopkeeper is 0.
4. Neighbor comes and asks for change for 1000rs note. Hands shopkeeper same note. Shopkeeper gives change. Shopkeeper is 0.
5. Shopkeeper notices note is counterfeit. Fake 1000rs is worth nothing. Shopkeeper is -1000.

This is the flow of money had the exchange been genuine. Click to expand.

1. Everyone is at zero standing, meaning no net money flows have happened.
2. The shopkeeper gives the item worth 200 to the customer and receives 1000 in return, but hasn't given change yet. Hence he is 800 up, the customer is 800 down. No net change on the creditor.
3. The shopkeeper changes his 1000 for an 800 and 200. No net change on anybody.
4. The shopkeeper gives the 800 worth of change to the customer. All of them have 0 net flow again.

Now, because the 1000 is counterfeit, we just remove the 1000 from the diagrams. We literally just delete the 1000, and then recalculate the standings. Click to expand.

1. Everyone is still at 0 standing.
2. The shopkeeper gives an item worth 200 to the customer and receives nothing, hence he is 200 down and the customer 200 up.
3. The shopkeeper gets 800 and 200 from the creditor in exchange for the fake bill, hence he is 800 up. The creditor is 1000 down since he lost that much but got thing in return.
4. The shopkeeper gives the customer their change of 800, bringing the shopkeeper to 0 and the customer to 1000.
5. The creditor comes back and takes his 800 and 200 again. This brings the creditor back to 0, and the shopkeeper down by 1000.

So the net loss is 1000.

The shopkeeper lost 200 worth of goods, plus 800 worth of money, which totals to 1000.

i) Assuming that the second shop keeper $S_2$ is not doing something fake (saying the bill is fake):

$L=-0+800+x=800+x$

$S_1=+0 -0+1000-800-x+0-1000=-800-x$

$S_2=+0-1000-0+1000$

So shop keeper looses 1000, where $x=200$ and $0$ is the fake bill.

ii) If the the second shop keeper $S_2$ is doing something fake:

$L=-1000+800+x=-200+x=0$

$S_1=+1000 -1000+1000-800-x+0-1000=-800-x=-1000$

$S_2=+1000-1000-0+1000=1000$

In both cases $S_1$ looses $1000$.

He starts with 0. She gives him 1000, he gives her 800. Now he has 200. Now the neighbor takes 1000, so he gives the 200 he has and 800 more, for a net loss of 800.

Another way to see this is to look from the lady's perspective. To her, the two shops form a "black box" where somehow she pays in fake (that is, 0) money and receives 800. So 800 had to be lost from somewhere. The neighbor didn't lose any money, so it had to come from the original person.

However, he also lost 200 in supply. So maybe you want to say he lost 1000, because he does not have those items to sell to someone who will give him real money.

You can see this from the lady's perspective as well: in addition to getting 800 she also got 200 worth of supply, which also had to come from the original person.

• And what has happened to the fake note he got from the lady. That is also his potential loss right? – selvin Jun 13 '13 at 8:26
• So wouldnt the loss be 800 in Cash + 200 in goods + 1000 as fake note which he cant use further? – selvin Jun 13 '13 at 8:28
• The note wasn't actually money, though. If the shopkeeper (original) lost 2000, then that means the neighbor had to gain 1000. But the neighbor clearly made/lost nothing. The "potential loss" from the note was actually part of the real loss from his neighbor demanding the money back. – Eric Stucky Jun 14 '13 at 0:49
• Think about it this way: Pretend the note was real for a minute. Then the shopkeeper gained 200, even if the neighbor thinks it's a fake and has him trade 1000 in change back for the (not fake) 1000 note. Okay, so if it was real, the shopkeeper gained 200. Then the fact that the note wasn't real means that he actually gained 200-1000=-800. It doesn't make sense to remove that note's value twice :) – Eric Stucky Jun 14 '13 at 0:52

His loss is $1000$.

$$-200_{item}+1000_{fake}-1000_{fake} + (800+200) - 800 + 1000_{fake} - (800+200) = -200_{item} - 800 + 1000_{fake}.$$

He lost 1000 rupees.

If the money was genuine the profit would be zero. (No profit for selling the goods.)

But the money was counterfeit, and hence his loss was the value of the counterfeit money: 1000 rupees.

## protected by Willie WongAug 7 '13 at 9:20

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