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Arm of the Crooked Bean

A merchant has a fraudulent balance: He buys in on the short arm, and sells out on the longer, and thereby gains 5.19395 percent. But if the longer arm measured half an inch more, the shorter remaining unaltered, his gain would be 11.8713 percent. Required the length of each arm of the beam.

Solution:

Source: J. Hunterís Text-book of Arithmetic, for the Use of Teachers, 1847