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Earnings Per Share - Basic and Diluted Earnings per Share, Treasury Stock Method (Details) - USD ($) $ / shares in Units, shares in Millions, $ in Millions	3 Months Ended
	Nov. 30, 2020	Nov. 30, 2019
Earnings Per Share [Abstract]
Net income	$ 59.6	$ 57.0
Basic weighted average shares outstanding (in shares)	37.6	39.5
Common stock equivalents (in shares)	0.2	0.1
Diluted weighted average shares outstanding (in shares)	37.8	39.6
Basic earnings per share (in dollars per share)	$ 1.58	$ 1.44
Diluted earnings per share (in dollars per share)	$ 1.57	$ 1.44

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