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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, 2021	Nov. 30, 2020
Earnings Per Share [Abstract]
Net income	$ 87.6	$ 59.6
Basic weighted average shares outstanding (in shares)	35.1	37.6
Common stock equivalents (in shares)	0.4	0.2
Diluted weighted average shares outstanding (in shares)	35.5	37.8
Basic earnings per share (in dollars per share)	$ 2.50	$ 1.58
Diluted earnings per share (in dollars per share)	$ 2.46	$ 1.57

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- Definition

Number of [basic] shares or units, after adjustment for contingently issuable shares or units and other shares or units not deemed outstanding, determined by relating the portion of time within a reporting period that common shares or units have been outstanding to the total time in that period.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 260
-SubTopic 10
-Section 50
-Paragraph 1
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=124432515&loc=d3e3550-109257

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 260
-SubTopic 10
-Section 45
-Paragraph 10
-URI http://asc.fasb.org/extlink&oid=125511455&loc=d3e1448-109256

+ Details

Name:	us-gaap_WeightedAverageNumberOfSharesOutstandingBasic
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration