Counterexample - 0/1 Knapsack

$0$/$1$ knapsack: items have weight and value, capacity is limited, items are indivisible. Maximize value. Greedy by value/weight ratio seems natural. But it fails.

Items: (weight= P0, value= P1), (weight= P2, value= P3), (weight= P4, value= P5). Capacity = $5$. Ratios: $5$, $4$, $4$. Greedy takes item $1$ (ratio P11), weight $2$ used. Remaining capacity $3$. Item $3$ does not fit. Item $2$ fits. Total value: $10$ + $12$ = $22$. But taking just item $3$ gives value $20$ with weight $5$. Wait, that is less. Let me try again..