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Share-based Payments - Share-based Compensation Expense Recognized (Details) - USD ($)
$ in Millions
12 Months Ended
Aug. 31, 2023
Aug. 31, 2022
Aug. 31, 2021
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense $ 42.0 $ 37.4 $ 32.5
Stock Incentive Plan      
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense 42.0 37.4 32.5
Restricted stock awards and units | Stock Incentive Plan      
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense 19.6 17.2 15.1
Performance stock units | Stock Incentive Plan      
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense 15.2 9.9 6.8
Stock options | Stock Incentive Plan      
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense 5.7 8.8 9.2
Director stock units | Stock Incentive Plan      
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]      
Share-based payment expense $ 1.5 $ 1.5 $ 1.4