A merchant uses a faulty scale to measure goods, that is, a scale whose arms are not of equal length. Two customers request goods of equal weight. The merchant places the goods for the first customer in the pan labeled $A$, and the weights in the pan labeled $B$. For the second customer, the merchant places the goods in the pan labeled $B$, and the weights in the other pan. Question: Under this procedure, does the merchant gain, or lose?
If the arms of the balance are $l_{1}$ and $l_{2}$, the weights of the goods are $p_{1}$ and $p_{2}$, and the weight used for measuring the goods is $q$, then

$$
p_{1} l_{1}=q l_{2} \text { and } p_{2} l_{2}=q l_{1} \text {, }
$$

from which

$$
p_{1}+p_{2}=q\left(\frac{l_{2}}{l_{1}}+\frac{l_{1}}{l_{2}}\right)
$$

But

$$
\frac{l_{2}}{l_{1}}+\frac{l_{1}}{l_{2}}=\frac{l_{1}^{2}+l_{2}^{2}}{l_{1} l_{2}}=\frac{\left(l_{1}-l_{2}\right)^{2}+2 l_{1} l_{2}}{l_{1} l_{2}}=2+\frac{\left(l_{1}-l_{2}\right)^{2}}{l_{1} l_{2}}>2
$$

Thus

$$
p_{1}+p_{2}>2 q
$$

so the trader loses. $\Phi$