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This kind of question has been asked on the forum before as well, but the doubts I have regarding it are not being resolved by the answers provided in those threads hence posting this as a new question.

The solution given in the book does as follows :- $1000*CP=800*SP$, where CP is the cost price of 1 gram and SP is the selling price of 1 gram from that the author did $SP/CP=1000/800$ which is 25% profit

My doubts :-

  1. Why are we writing $1000*CP$ and not $800*CP$ to calculate his total cost. If he is selling 800 grams, shouldn't his cost also be measured for 800 grams only?

  2. Secondly, even if I take that $1000*CP$ as given in the solution to be the total cost and $800*SP$ as the total selling price. In that case why are we equating these 2 things, since as per the question it is given that the shopkeeper claims that the total selling price is same as total cost price but at the same time we know that he is dishonest, so obviously they won't be equal.

I am really confused on these 2 statements

Fin27
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  • I get the gist of your question, but what do you mean by 'doubts'? They look like questions you have, or things you want to understand further. They are not doubts. – fabspro Mar 19 '23 at 04:54

4 Answers4

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The shopkeeper buys $1$ kg for some amount of money. He sells $800$ g for that same amount of money. Money-wise the shopkeeper has broken even. But he still has $200$ g of unsold item. This $200$ g that can still be sold represents his profit.

The shopkeeper will sell the last $200$ g at the same rate (presumably) as the first $800$ g, thus bringing $25\%$ of what he brought in for the initial $800$ g sold.

paw88789
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  • How do we know he has broken even ? since he is dishonest, shouldn't the claim of him being broken even also be wrong ? – Fin27 Mar 19 '23 at 08:45
  • We should not go so far , @Fin27 , otherwise we can question the honesty about even the Input Cost , which may be 0 when he had robbed the Item. We might go even further questioning whether he was honest about being a Shopkeeper. We should stick to the Mathematical Calculations. – Prem Mar 19 '23 at 12:28
  • @Fin27 I'm assuming the information provided in the problem is accurate--if not, there isn't a meaningful way to solve the problem. – paw88789 Mar 19 '23 at 12:32
  • @Prem yeah you are right that would be going too deep, what is your interpretation on the equation of $1000CP=800SP$, do you mean to say that we are kind of using his false claim of "Selling at cost" in this equation? What essence do you get from this equation ? If you could also present an answer, that would be greatful – Fin27 Mar 19 '23 at 18:41
  • Thanks @paw88789 for that answer, I am trying to understand more about how that equation turned out – Fin27 Mar 19 '23 at 18:42
  • @Fin27 Perhaps it would help to think of the stated selling price per unit (which is the cost price per unit) vs. the effective selling price per unit which is $\frac{1000}{800}$ times the cost price per unit, due to the short scale. – paw88789 Mar 19 '23 at 21:22
  • this is an effective way to avoid usage of the $CP$ and the ambiguity around it. – D S Mar 20 '23 at 12:31
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Let $c$ be the CP per gram.
The shopkeeper tells you that he sells at the rate of $c$ per gram (which is his cost price). You are buying $1000$g. So, you pay $1000c$, unaware of his dishonesty. However, in reality, he has just given you $800$g. He had bought this $800$g from a dealer for a cost of $800c$, but is now selling you the same thing at $1000c$. His profit becomes $200c$, hence profit per cent is $25\%$.

Coming to the method described in the question: In the solution given, SP is the net selling price for him per gram, if you consider his fraud. What does he do? Gets $1000CP$ from you, but since he has sold only $800$g, that's $800SP$ for him. So, both should be equal.

Answering your questions:

  1. $1000CP$ is not his total cost. It's what you (the buyer) have paid. He has bought from a dealer at $CP$ per gram. He lies and tells he sells at the same rate. You become happy and buy $1000$g from him, not realizing he's only given you $800$g.

  2. (I think you meant $800*SP$.) We are considering $SP$ to be the price he would've sold you the thing if he would have told the truth (that is, the price he would have sold if he used proper weights but wanted to maintain the same profit %). "so obviously they won't be equal" yes, $CP \neq SP$, however, $1000CP = 800SP$. This is because you give him $1000CP$, and that's $800SP$ for him (since only he knows that he's selling $800$g, and not $1000$g).

D S
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  • We are writing the equation $1000CP = 800SP$ as per shopkeeper's actual transaction, right ? If he is selling me 800 gram, the actual amount transferred from him to me is 800 gram only, so while calculating his cost involved in the transaction, why are we not considering 800CP , why are we taking 1000CP as cost if he is selling only 800 grams, don't we calculate on the quantity that was "actually" transacted ? Why do we care how much grams did he buy from the dealer and also can't it be anything why just limited to 1000 g ? – Fin27 Mar 19 '23 at 08:42
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    no, you've misunderstood. I am referring to the transaction that happens between you and the shopkeeper. He tells you the cost of 1 gram is $CP$, and because of fraudulent weights, you think you're buying $1000$g. So you pay $1000CP$. Here, $CP$ is the cost price for the shopkeeper. $SP$ is the selling price for him per gram. He has actually sold $800$g (only he knows this). So, he receives $800SP$ from you. Since he receives whatever you give him, they must be equal. If this is confusing, refer to the first part of my answer. – D S Mar 19 '23 at 09:24
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    Dealer is just the one who sells the thing to the dishonest shopkeeper. If the shopkeeper lies to you that he sells at cost, meaning his selling price is same as cost price. Which means he had bought the thing from somewhere, the dealer is the "somewhere". – D S Mar 19 '23 at 09:26
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    And no $1000$g isn't necessary, although it makes understanding a bit easier. For example, you want to buy $2000$g, but you get $1600$g only because of the fraudulent weights. So, again the equation is $2000CP = 1600SP \iff 1000CP = 800SP$, the same equation. – D S Mar 19 '23 at 09:29
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    I feel like the following statement "Since he receives whatever you give him, they must be equal." seems to finally remove the confusions, please correct me if I am wrong :- $1000CP=800SP$ is purely written in point of view of the seller and his interaction with the customer. The dealer was only used as a reference by the shopkeeper to inform the customer like heh!, I bought these 1000 grams at CP from that guy and I am selling you these 1000 grams just like the dealer sold it to me-this covers the sells at cost part given; (Continued in next comment due to lack of characters) – Fin27 Mar 19 '23 at 19:02
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    Now though the the quantities being exchanged are different, but we are equating the "worth of the quantities being sold" in 2 different ways , one in terms of SP and one in terms of CP, what he actually received in terms of CP is $1000CP$ from the customer , and $800SP$ is also the "actual" money received only but this $800*SP$ is known by the shopkeeper only, customer doesn't have a clue about what this SP is . I know this is a way too long comment, but please let me know if I am again getting mistaken somewhere, or you feel like I am missing on some point somewhere. Thanks a lot :) – Fin27 Mar 19 '23 at 19:11
  • Let us continue this discussion in chat. – Fin27 Mar 19 '23 at 22:07
  • can you please help me in https://math.stackexchange.com/questions/4662434/profit-loss-question-sales-gimmick , this is almost the same question, but since here conditions are getting varied, I am not able to completely understand it. I request you to please have a look and the comment to this question as well to check if I have understood properly . Thanks again – Fin27 Mar 19 '23 at 22:24
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    @Fin27 you got it right! – D S Mar 20 '23 at 10:38
  • Phew ! thank you so much @D S ,I had a follow up question :- Can we say that the claim of shopkeeper selling the product at cost is actually true? What I mean is $1000*CP$ would be some amount, let's say (X dollars) which he gave to the dealer to buy goods, and shopkeeper is indeed selling (X dollars) only worth of goods to the customer as well, but this time using 800 grams as weight and SP as the price per gram (Continued in next comment due to lack of characters) – Fin27 Mar 20 '23 at 11:37
  • So can we say that the shopkeeper is being dishonest only on using the faulty weights and rather not on making the claim of selling at cost as it is the same $X only being exchanged between the dealer-shopkeeper and then afterwards between shopkeeper-customer – Fin27 Mar 20 '23 at 11:37
  • That depends on how you think. If you think that the shopkeeper is dishonest in the weights, well, he is honest in selling at cost, but he sells you $800$g at the cost price of $1000$g. If you think that the shopkeeper is dishonest in claiming that he sells at cost, then you forget about the weights and think that he sells $800$g for $1000CP$ to earn his profit.
    However, according to me, the dishonesty is claiming that he sells at cost, which attracts customers. And weights a dishonest method to extract profit. But at the end of the day, this followup question really depends on perspective     – D S Mar 20 '23 at 11:58
  • Yes you are right, but still if we are equating those 2 prices ($1000CP$ and $800SP$) , it is actually on the assumption ("sells at cost") that the total price of buying and selling the goods is remaining constant , those $X ; Had he claimed that he (sells at cost per gram) then it would be a dishonest claim, isn't it , because the SP and CP turn out to be different – Fin27 Mar 20 '23 at 12:32
  • I may have misunderstood your last comment. The $SP$ in the equation is virtual, that is, the shopkeeper is selling at cost only, but he sells you $800$g instead of $1000$g. The $SP$ is the selling price he would've adopted if he wanted the same profit percent in an honest way. – D S Mar 20 '23 at 12:41
  • Alternatively, this $SP$ is the rate at which he sold his full stock. Take an example. Let $CP$ be $1$ dollars per gram. Say the shopkeeper has $4$kg in stock. He tells you he is selling you $1000$g, charges $1000$ dollars, but gives you only $800$g. He does this with $5$ customers including you, before his stock is exhausted. How much has he received? $1000 \times 5 = 5000$ dollars. So, he effectively sells $4000$g in $5000$ dollars, or his $SP$ per gram is $5000/4000 = 5/4 = 1.25$. Hence, his profit is $0.25$ dollars, so profit percent is $25%$. – D S Mar 20 '23 at 12:47
  • "...wanted the same profit percent in an honest way. " , Do you mean SP is the price which would fetch him no profit no loss condition in an honest way ? – Fin27 Mar 22 '23 at 08:49
  • I'd say that meaning is ambiguous. Refer to the "full stock" meaning. It's more clear there. – D S Mar 22 '23 at 12:24
  • @Fin27 "...wanted the same profit percent in an honest way. " means he is getting some profit percent now due to wrong weights. If he corrected the weights, then he would have to increase the 'price tag'. The value on the 'price tag' is the meaning of $SP$. But again, this may be confusing. – D S Mar 22 '23 at 12:26
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The shopkeeper claims to be selling $1$ kg at cost, which means that his sales price is $1000 \cdot CP$, where $CP$ is the cost per gram. Since he is actually selling $800$ g at the cost of $1000$ g, $1000 \cdot CP = 800 \cdot SP$, where $SP$ is the actual sales price per gram. The dishonesty comes from selling $800$ g at the cost of $1000$ g.

N. F. Taussig
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  • If he is selling 800 gram, so why shouldn't we take the cost of 800 gram as well only ? why are we considering the cost of 1000 grams now, how does it matter how many grams he bought from the dealer ? – Fin27 Mar 19 '23 at 08:43
  • The shopkeeper claims to be selling $1000$ g, which is why he sets the price at the cost of $1000$ g rather than $800$ g. – N. F. Taussig Mar 19 '23 at 09:37
  • please correct me if I am wrong, as per your solution, we are equating the dishonest selling price to the actual price ? If yes, I just want to know why would these 2 things be equal ? that is what is confusing me – Fin27 Mar 19 '23 at 18:44
  • The dishonest selling price for the $800$ g the shopkeeper is actually selling is the actual cost of $1$ kg since the shopkeeper claims to be selling $1$ kg at cost. It is dishonest since the shopkeeper is actually selling $800$ g for the price of $1000$ g, thereby selling at a $25%$ profit rather than at cost, as he claims. – N. F. Taussig Mar 19 '23 at 19:13
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. . . dishonest shopkeeper claims he sells at cost

There is a bag of stuff on the shelf. It claims to be a bag of $1000$ grams and claims to be sold at cost price. The cost of a bag of $1000$ grams at cost price is $1000 \, *\, CP$. So this is the pricetag on the bag.

In reality the bag only weight $800$ grams. So the price is also $800 * SP$.

These two things both equal the price tag. So they equal each other.

Daron
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  • Thanks a lot @Daron , please correct me if I am wrong, $1000CP$ is the price that both customer and seller knows as obviously it is on the price tag as well, however the price of $800SP$ is only known to the seller? I just want a bit of clarification on the $800*SP$ part , how will this be equal to the price tag amount ? – Fin27 Mar 19 '23 at 19:18
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    The customer and seller both know the number $CP$. But only the shopkeeper knows $SP$. They both know $1000CP$ and $800SP$ because they are both the pricetag. – Daron Mar 19 '23 at 19:22
  • Thank you so much @Daron , could you please look into the other question as well which I had posted on similar lines but I am not able to formulate one of equation over there using the same logic as discussed here in your post :- https://math.stackexchange.com/questions/4662434/profit-loss-question-sales-gimmick – Fin27 Mar 19 '23 at 22:12
  • @Fin27 I have answered your other question too. – Daron Mar 20 '23 at 15:08