# Digging a ditch

Here is a curious fact that happened to me the other day. My wife sent me to the garden to dig a ditch to plant azaleas, but since I am not a very hardworking person, but a good businessman, I hired a sick old man who promised to dig the ditch for two euros.

The old man decided to ask for help from his grandson, a healthy and strong boy who agreed to help him in exchange for distributing money according to his abilities.

We know that the old man can use the beak to dig at the same speed that his grandson is able to take out the earth with the shovel. Instead, the grandson is able to use the beak to dig the earth at a speed four times faster than the old man is able to extract it with the shovel.

How should the money be distributed?

#### Solution

The old man should receive a third of the two euros and the grandson the remaining two thirds.

The reasoning is as follows: Suppose the grandson is able to dig the entire hole with the beak in two hours and extract the earth with the shovel in 4 hours. To make that same ditch, the old man would take 4 hours using the beak (the same speed as his grandson with the shovel) and 8 hours in extracting all the earth (four times what it takes the grandson to dig with the beak).

From here we deduce that the chop ratio is 2 to 4 and the ratio of extracting the earth with the shovel is 4 to 8, that is, the same in both cases (one to two). Thus, the old man can chop at the same time that his grandson takes to extract the earth (4 hours) while the grandson can chop the entire ditch in a quarter of the time it would take the old to extract the earth.

Other example figures would give us the same ratios, so we deduce that the old man takes a third of the profits and the grandson twice, that is, two thirds.