If it takes six people four days to dig a ditch that's 10 feet long how long will it take three people to dig a ditch 5 feet long .

First, we need to make two assumptions:

1).  Assume that all diggers dig at the same rate.

2).  Assume that the time required is directly proportional to the
length of the ditch.  In other words, no matter how long or short
a ditch is, each inch of it takes the same amount of time to dig.



6 people take 4 days to dig 10 feet.

Half as many people do half as much work in the same length of time.

3 people take 4 days to dig 5 feet.

Wow !  That's the question you asked, so there's no more to calculate.