You can't really talks about an english and a french army, at least not before Charles VIII.
Both Plantagenet/Lancaster and Valois relied on huge mercenaries armies that, whatever their own origin, sold themselves to each side and eventually were used to ravage countryside (that did better than any "national" army).
They'll end by capturing their own castles, for their own purposes and dried out the treasuries of many cities to make them left by tribute, bribe, or war.
Concerning david assets, he's in truth when he said the english army was more contractual, but he forgets the social base of a non-mercenary army : the yeomen. These free land-owners formed the social base of archery and, we could almost say, the spine of english army.
The french army relied as well on mercenary, but aslo on feudal levies. It means less direct commendement, more disorganized but reflecting the royal power as it was the first cause to limit the contractual, urban militias that existed in the XII century.
For numbers between English and French side during the battles (as said, mercenaries could change side, being removed, levies downed, etc.)
For Azincourt/Agincourt we have
England : 6000/7000 men with 5 000 bowmen.
France : 12 000/15 000 men with 5 000/8 000 horsemen in center, 2 000 horsemen on side, 5 000 bowmen and crossbowmen in the back
For Verneuil
England : 8 000/10 000
France : 11 000
Scotland : 4 000/5 000
For Formigny
England : 7 000 (essentially bowmen)
France : 5 000 with 2 500 horsemen, 2 500 footmen
Brittany : 1 500 horsemen
For Castillon
England : 9 000
France : 10 000
As you can see it's hard to give an actual ratio of men, or actual number for armies that, I point it, weren't standing before Charles VIII (who put in place the first real standing armies in Europe since Roman Empire)