Step 1 :Let's denote the water output rate of the Gray family's sprinkler as G and the water output rate of the Wright family's sprinkler as W.
Step 2 :We have two equations based on the problem: \(G + W = 40\) (the sum of the two rates was 40 L per hour) and \(25G + 15W = 850\) (the combined total output of water).
Step 3 :We can solve this system of equations to find the values of G and W.
Step 4 :The solution to the system of equations is {G: 25, W: 15}.
Step 5 :Final Answer: The water output rate for the Gray family's sprinkler was \(\boxed{25 \mathrm{~L/hour}}\) and for the Wright family's sprinkler was \(\boxed{15 \mathrm{~L/hour}}\).