This is a rate problem. You need to use the information given to find the rates at which things happen, then combine them to get the answer.

In this case, we assume that a single painter takes some fixed amount of time to paint a house. We can then say that that the rate of painting houses is X houses per hour per painter.

[; R^{paint}_{hourly} = X \frac{house}{hour \cdot painter} ;]

If the painting happens at a rate of Y hours per day, then we can get the daily rate from

[; R^{paint}_{daily} = X \frac{house}{hour \cdot painter} * Y \frac{hour}{day} = X * Y \frac{house}{day \cdot painter} ;]

Note how the units cancelled out. Now if there are N painters, then the total number of houses painted per day is

[; R^{total}_{day} = N painter * \left( X * Y \frac{house}{day \cdot painter} \right) = X * Y * N \frac{house}{day};]

Noting again the cancellation of the unit painters.

So, applying it to this problem. There are 10 painters, 5 houses painted per day, and 6 hours worked per day. That means the base painting rate is 1/12 houses per painter per hour. Thus, if there are 15 painters and 8 hours worked per day, then 10 houses will be painted per day. There are 600 houses, so it will take 60 days.