There are more than one way to do this:

Here's one way:

The least common multiple of 4 people and 3 people is 12 people.

12 people is 3 times as many people as 4 people, and
12 people is 4 times as many as 3 people.

Since 4 people take 6 days, 3 times as many people take only
one-third as long, or only 2 days. (since 2 is one-third of 6)

Since 3 people is only one-fourth as many people as 12, it will take
them 4 times as long as it would take 12 people, or 8 days. (since
4 times 2 is 8).

Answer: 8 days

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Or you can do it this way:

Since 4 people take 6 days, their combined rate is 1/6 of a job per day.
So one person's rate is 1/4 of that rate or 1/4 of 1/6 of a job per day,
or 1/24 of a job per day.  Then 3 people's rate will be 3 time 1/24 of a
job per day, or 3/24 of a job per day.  And 3/24 reduces to 1/8 of a
job per day, so it will take them 8 days to complete the job.

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Another way is: 

Workers times days = Workers times days
         4 times 6 = 3 times X
                24 = 3x
                 8 = x

Edwin