A dishonest shopkeeper claims he sells at cost but his 1 kg weight actually measures 800 grams when selling. Find his profit/loss %

Question

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

Answer 1

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 $25\%$.

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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.

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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).

Answer 2

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.

Answer 3

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.

Answer 4

. . . 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.