Researchers showed a new way to write knapsack problems without slack variables. By turning the weight inequality into an equality over a restricted weight range, the number of extra variables drops from W to wₘₐₓ, a major savings when solving with quantum algorithms. They also derived a tighter penalty constant, vₛ, using greedy guesses instead of the conventional vₘₐₓ. Tests on both liquid‑state, ten‑layer QAOA circuits and on a D‑Wave annealer revealed that vₛ consistently improves median success probabilities across problem sizes n=4 to 20. The reduced slack range further raises success rates for most instances, though some show only marginal gains. Both quantum solvers exhibit a 2⁻ⁿ fall‑off in success probability; the D‑Wave device decays slightly faster because embedding adds qubits. However, the slackless approach still needs to solve wₘₐₓ separate QUBO problems, making its total run‑time O(wₘₐₓ·2ⁿ). The penalty‑bound guarantees hold only when the target weight equals the optimal knapsack weight, so they may not apply to other weight choices. Finally, counting all equally optimal solutions can inflate measured success rates, especially when multiple optima exist.