# Is it possible to determine the profit percentage? If yes, then how?

A shopkeeper who professes to sell his goods at cost price, uses a faulty balance that has one arm 4% longer than the other. Is it possible to determine his profit percentage? If yes, then how?

Thanks,

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For definiteness, assume that the cost to the shopkeeper of $1$ gram of the substance is $50$ dollars, and therefore the selling price is $50$ dollars per (short) "gram." The shopkeeper places a $1$ gram weight at the end of the short arm of the balance. Then (s)he delicately places some of the substance into a pan at the end of the long arm, until the balance balances.
Let $w$ be the actual weight of substance in the pan, and let $a$ be the length of the short arm of the balance. Then the long arm has length $1.04a$, so $$(1.04a)(w)=(a)(1),$$ and therefore $w=\dfrac{1}{1.04}$.
The cost to the shopkeeper of what (s)he sells for $50$ dollars is therefore $\dfrac{50}{1.04}$. The selling price is $50$, so the selling price divided by the cost is $$\frac{50}{\frac{50}{1.04}}.$$ This ratio is $1.04$, so the profit percentage is $4\%$.