The Fractional Knapsack problem permits items to be broken down fractionally. Given items of weights and values, fill a knapsack of capacity W to maximize value.

Always pick the item with the highest value-per-unit-weight density (value / weight) next. If the capacity is filled before exhausting the item, take a fraction.

An optimal solution to fractional knapsack contains the highest density item plus the optimal solution to the remaining subproblem with decreased capacity.

Proof by contradiction: if we select a lower-density item over a higher-density one, swapping the lower-density fraction with the higher-density one yields a strictly higher profit.