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Changes in Stockholders' Equity (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
 3 Months Ended
Nov. 30, 2021
Nov. 30, 2020
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 2,044.5 $ 2,127.5
Net income 87.6 59.6
Other comprehensive income (loss) (10.7) 6.2
Share-based payment amortization, issuances, and cancellations 0.4 4.7
Employee stock purchase plan issuances 0.6 0.3
Cash dividends of $0.13 per share paid on common stock (4.7) (5.0)
Stock options exercised 8.0  
Repurchases of common stock (52.8) (256.1)
Ending balance $ 2,072.9 $ 1,937.0
Cash dividends paid (usd per share) $ 0.13 $ 0.13
Common Stock Outstanding    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 0.5 $ 0.5
Beginning balance (shares) 35.2 38.9
Share-based payment amortization, issuances, and cancellations (shares) 0.1 0.1
Stock options exercised (shares) 0.1  
Repurchases of common stock (shares) (0.3) (2.6)
Ending balance $ 0.5 $ 0.5
Ending balance (shares) 35.1 36.4
Paid-in Capital    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 995.6 $ 963.6
Share-based payment amortization, issuances, and cancellations 0.4 4.7
Employee stock purchase plan issuances 0.6 0.3
Stock options exercised 8.0  
Ending balance 1,004.6 968.6
Retained Earnings    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance 2,810.3 2,523.3
Net income 87.6 59.6
Cash dividends of $0.13 per share paid on common stock (4.7) (5.0)
Ending balance 2,893.2 2,577.7
Accumulated Other Comprehensive Loss    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (98.2) (132.7)
Other comprehensive income (loss) (10.7) 6.2
Ending balance (108.9) (126.5)
Treasury Stock, at cost    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (1,663.7) (1,227.2)
Repurchases of common stock (52.8) (256.1)
Ending balance $ (1,716.5) (1,483.3)
Cumulative effect of adoption of ASC 326    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance   (0.2)
Cumulative effect of adoption of ASC 326 | Retained Earnings    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance   $ (0.2)