We can write the equation to solve this problem as:
(72 + h)/(160 + h) = 0.560
Where h is the number of consecutive hits need.
Solving for h gives:
color(red)((160 + h)) xx (72 + h)/(160 + h) = color(red)((160 + h)) xx 0.560
cancel(color(red)((160 + h))) xx (72 + h)/color(red)(cancel(color(black)(160 + h))) = (color(red)(160) xx 0.560) + (color(red)(h) xx 0.560)
72 + h = 89.6 + 0.560h
-color(red)(72) + 72 + h - color(blue)(0.560h) = -color(red)(72) + 89.6 + 0.560h - color(blue)(0.560h)
0 + (1 - color(blue)(0.560))h = 17.6 + 0
0.44h = 17.6
(0.44h)/color(red)(0.44) = 17.6/color(red)(0.44)
(color(red)(cancel(color(black)(0.44)))h)/cancel(color(red)(0.44)) = 40
h = 40
You will need color(red)(40) consecutive hits to move your batting average from 0.450 to 0.560