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Comprehensive Income - Schedule of Changes in Components of Accumulated Other Comprehensive (Loss) Income (Details) - USD ($)
$ in Millions
 3 Months Ended
Nov. 30, 2024
Nov. 30, 2023
Accumulated Other Comprehensive Income (Loss), Net of Tax [Roll Forward]    
Beginning balance $ 2,378.8 $ 2,015.4
Other comprehensive loss before reclassifications (17.3) (2.1)
Amounts reclassified from accumulated other comprehensive loss 0.5 0.6
Other comprehensive loss items, net of tax (16.8) (1.5)
Ending balance 2,463.4 2,064.1
Accumulated Other Comprehensive Loss    
Accumulated Other Comprehensive Income (Loss), Net of Tax [Roll Forward]    
Beginning balance (114.9) (112.6)
Ending balance (131.7) (114.1)
Foreign Currency Items    
Accumulated Other Comprehensive Income (Loss), Net of Tax [Roll Forward]    
Beginning balance (70.9) (65.0)
Other comprehensive loss before reclassifications (17.3) (2.1)
Amounts reclassified from accumulated other comprehensive loss 0.0 0.0
Other comprehensive loss items, net of tax (17.3) (2.1)
Ending balance (88.2) (67.1)
Defined Benefit Pension Plans    
Accumulated Other Comprehensive Income (Loss), Net of Tax [Roll Forward]    
Beginning balance (44.0) (47.6)
Other comprehensive loss before reclassifications 0.0 0.0
Amounts reclassified from accumulated other comprehensive loss 0.5 0.6
Other comprehensive loss items, net of tax 0.5 0.6
Ending balance $ (43.5) $ (47.0)