1. **Stating the problem:** A body of weight $w$ newton is on a rough horizontal plane with a coefficient of static friction $\mu_s = \frac{1}{4}$. A horizontal force acts on the body causing a friction force $f$ such that $0 < f \leq 4$. We need to find the value of $w$. 2. **Understanding friction force:** The maximum static friction force is given by: $$f_{max} = \mu_s N$$ where $N$ is the normal force. On a horizontal plane, $N = w$ (weight). 3. **Given friction force range:** The friction force $f$ lies in the interval $]0,4]$, meaning the maximum friction force is 4 N. 4. **Using the maximum friction force:** Since $f_{max} = 4$ and $f_{max} = \mu_s w$, we have: $$4 = \frac{1}{4} w$$ 5. **Solving for $w$:** Multiply both sides by 4: $$w = 4 \times 4 = 16$$ 6. **Conclusion:** The weight $w$ of the body is 16 N. **Final answer:** $w = 16$ N (option d).