Briefly

The inspector's problem

The inspector's problem

Inspector Jones's job is to check the weight and measures of the transactions that are made in the town to ensure that the poor coal man does not give half a ton of extra or that the industrious butcher does not time himself selling more meat for the same price.

The problem of Inspector Jones is that there are scales that are not balanced since the foothold is not centered, an error that can eventually be expensive.

As Benjamin Franklin said, we cannot judge by appearances and I have drawn the scales so as not to give clues as to what their problem is. In the first test, three pyramids are weighed on one side and eight cubes on the other. But when you put a cube, the long arm of the balance is balanced with six pyramids!

If we assume that a pyramid weighs an ounce, How much do the eight cubes weigh?

Solution

Items that are weighed in unbalanced scales will show a difference in weight with their actual weight equal in proportion to the distance at which the support point is. Weigh the items on one side of the scale and then on the other. Multiply the two results and calculate the square root of the result of this multiplication, so you will get the real weight of the product.

In the long arm a pyramid is equivalent to two cubes and 2/3, while in the other arm it is equivalent to 1/6 of a cube.
1/6 × 2 × 2/3 = 4/9
whose square root is 2/3. Therefore a pyramid weighs 2/3 of a cube.

If a pyramid weighs an ounce, a cube will weigh 1.5 ounces and the answer to the question what is the real weight of the 8 cubes? it is 12 ounces.