It may be true that the average home holds 125 ft of 3/4-inch piping, although no source was provided for this factoid. But when you turn on the tap, the water doesn't run through all 125 feet. The water runs the most direct course from the water heater to your tap. Also, I would think that half of that pipe is dedicated to cold water. In my house the distance from the water heater to the farther faucet is less than 50 feet. Using their assumptions, the amount of water in the pipe does not come out to 3.14 gallons, but rather 2.8687 gallons.

The next assumption is that you draw water ten times per day. This assumes that the water in the pipes cools completely between each draw. In most households however, there are two periods of the day when hot water is being drawn; for the morning shower and the evening dishes. During these periods the water in the pipes would probably not cool down very much so you would realistically only need to wait for hot water two or three times per day.

Using the assumptions and calculations from our source, we can confirm that 11,461 gallons would be wasted each year. Using my corrected assumptions, I would put that number closer to 838 gallons. Of course some homes are occupied all day, have a more spread out floorplan, and have a lot more hands to wash. Still, the 11,461 gallons saved are very optimistic. How much money does the pump save on the water bill? Using their savings and California's high water prices you would save around $50 per year, but the reality is probably closer to $4.