He expected to complete the fence in x days at $y per day, so xy=1200.
But he completed the fence (at the same cost) in x+3 days at a cost of y-22.50 dollars a day, so (x+3)(y-22.5)=1200,
But xy=1200, so:
Multiplying through by -x/22.5:
x²-160+3x=0, or x²+3x=160.
Completing the square:
So, x=√649/2-3/2=12.74-1.5=11.24 days approx.
He expected to take about 11.24 days to complete the job at $106.78 per day.