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Problem statement

A dishonest shopkeeper uses a false weight of 900gm instead of 1kg, if he promises to sell his goods at cost price, what is his overall profit?

I approached this problem like this,

when a customer buys 1 kg, he gets 0.9 kg.(profit of 0.1kg for shopkeeper)

when a customer buys 2 kg, he gets 1.8 kg.(profit of 0.2kg for shopkeeper)

So, if the shopkeeper has x kg originally, he will be able to sell (10x/9)kg.

It gives me the correct answer but I still can't visualize this.

For e.g. Let's say the shopkeeper has 9 kg originally, and sells all of it, in this case he will sell 8.1 kg(there is 900gm left). So, in total he will sell, 9.9kg(9kg+900gm leftover). The earlier equation gives 10kg.

What is wrong with reasoning?

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    $\begingroup$ If the shopkeeper sells all of $9$ kg, he will have sold $9$ kg for the price of $10$ kg. $\endgroup$ Commented Jun 6, 2023 at 11:43
  • $\begingroup$ @N.F.Taussig Can you point out what is wrong with my reasoning, which gives me 9.9. $\endgroup$
    – Max
    Commented Jun 6, 2023 at 11:54
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    $\begingroup$ The last comment was a bit loose so I deleted it. The last $900$ gm will be treated by the customer as $1$ kg, so the customer will pay for it as if it's worth $1$ kg. That means even though, from the shopkeeper's point of view, the last $900$ gm is actually $900$ gm in weight, what he earns out of it is what $1$ kg is worth. So the leftover $900$ gm is to be treated as $1$ kg when sold. That's what gives $10$kg instead of $9.9$ at the end. $\endgroup$ Commented Jun 6, 2023 at 12:02
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    $\begingroup$ Other (more difficult) way to think about it: now shopkeeper has 900g left, he sells "900g", but actually he sells 810g, and has 90g left. Repeating it, he will sell total of 9kg + 900g + 90g + 9g + ... = 10kg. $\endgroup$
    – mihaild
    Commented Jun 6, 2023 at 12:34
  • $\begingroup$ @mihaild I considered it, but the problem with this reasoning is that "he uses 900gm weight instead of 1kg, but it is possible that for weights below 1kg, he has the correct weights." What is wrong with my analysis? The question mention only one crooked weight, right? 1kg => 900gm $\endgroup$
    – Max
    Commented Jun 6, 2023 at 13:09

1 Answer 1

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It's really a wording problem, as you state in the title, here's why:
"when a customer buys 1 kg, he gets 0.9 kg"
and
"when a customer buys 2 kg, he gets 1.8 kg"
are both from the customer's point of view. And your next statement
"So, if the shopkeeper has x kg originally, he will be able to sell (10x/9)kg."
is from the shopkeeper's point of view. From the numbers & the formula we can see, that x, the smaller amount, is the "he gets" from the customer's point of view - which means whatever the customer gets is the amount the shopkeeper has originally, and 10x/9, the bigger amount, is "buys" from the customer's point of view - which has a word equivalent "sell" from the shopkeeper's point of view.
Visalizing these:
\begin{array}{c|c|c|c} \text{Math notation} & \text{Relative amount} & \text{Customer} & \text{Shopkeeper} \\\hline x & \text{smaller} & \text{he gets} & \text{has originally} \\\hline 10x/9 & \text{bigger} & \text{buys} & \text{sell} \\\hline \end{array} In your last example, you use different logic when you say
"the shopkeeper has 9 kg originally, and sells all of it, in this case he will sell 8.1 kg"
because here the word "sell" is present two times and refers to two different amounts, and you used the bigger one for x, however the above table shows that the smaller should be used, and then the result is 9 not 10.
Additionally, you introduced two (or one and a half) new words, "left" and "leftover", which were "profit" in the two original examples, and this amount is not sold so it shouldn't be added.

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