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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

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  • $\begingroup$ 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. $\endgroup$
    – fabspro
    Mar 19, 2023 at 4:54

4 Answers 4

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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.

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  • $\begingroup$ How do we know he has broken even ? since he is dishonest, shouldn't the claim of him being broken even also be wrong ? $\endgroup$
    – Fin27
    Mar 19, 2023 at 8:45
  • $\begingroup$ 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. $\endgroup$
    – Prem
    Mar 19, 2023 at 12:28
  • $\begingroup$ @Fin27 I'm assuming the information provided in the problem is accurate--if not, there isn't a meaningful way to solve the problem. $\endgroup$
    – paw88789
    Mar 19, 2023 at 12:32
  • $\begingroup$ @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 $\endgroup$
    – Fin27
    Mar 19, 2023 at 18:41
  • $\begingroup$ Thanks @paw88789 for that answer, I am trying to understand more about how that equation turned out $\endgroup$
    – Fin27
    Mar 19, 2023 at 18:42
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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).

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  • 1
    $\begingroup$ 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. $\endgroup$
    – D S
    Mar 19, 2023 at 9:24
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    $\begingroup$ 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". $\endgroup$
    – D S
    Mar 19, 2023 at 9:26
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    $\begingroup$ 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. $\endgroup$
    – D S
    Mar 19, 2023 at 9:29
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    $\begingroup$ 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 :- $1000*CP=800*SP$ 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) $\endgroup$
    – Fin27
    Mar 19, 2023 at 19:02
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    $\begingroup$ 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 $1000*CP$ from the customer , and $800*SP$ 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 :) $\endgroup$
    – Fin27
    Mar 19, 2023 at 19:11
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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.

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  • $\begingroup$ 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 ? $\endgroup$
    – Fin27
    Mar 19, 2023 at 8:43
  • $\begingroup$ 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. $\endgroup$ Mar 19, 2023 at 9:37
  • $\begingroup$ 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 $\endgroup$
    – Fin27
    Mar 19, 2023 at 18:44
  • $\begingroup$ 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. $\endgroup$ Mar 19, 2023 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.

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  • $\begingroup$ Thanks a lot @Daron , please correct me if I am wrong, $1000*CP$ is the price that both customer and seller knows as obviously it is on the price tag as well, however the price of $800*SP$ 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 ? $\endgroup$
    – Fin27
    Mar 19, 2023 at 19:18
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    $\begingroup$ The customer and seller both know the number $CP$. But only the shopkeeper knows $SP$. They both know $1000*CP$ and $800*SP$ because they are both the pricetag. $\endgroup$
    – Daron
    Mar 19, 2023 at 19:22
  • $\begingroup$ 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 :- math.stackexchange.com/questions/4662434/… $\endgroup$
    – Fin27
    Mar 19, 2023 at 22:12
  • $\begingroup$ @Fin27 I have answered your other question too. $\endgroup$
    – Daron
    Mar 20, 2023 at 15:08

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