1. **State the problem:** We are given two numbers $A$ and $B$ such that their sum is 28, i.e., $$A + B = 28$$ We are also given that half of $A$ plus one-fourth of $B$ equals -2, i.e., $$\frac{1}{2}A + \frac{1}{4}B = -2$$ 2. **Write down the system of equations:** $$\begin{cases} A + B = 28 \\ \frac{1}{2}A + \frac{1}{4}B = -2 \end{cases}$$ 3. **Solve the system:** From the first equation, express $B$ in terms of $A$: $$B = 28 - A$$ 4. Substitute $B$ into the second equation: $$\frac{1}{2}A + \frac{1}{4}(28 - A) = -2$$ 5. Simplify the equation: $$\frac{1}{2}A + 7 - \frac{1}{4}A = -2$$ 6. Combine like terms: $$\left(\frac{1}{2} - \frac{1}{4}\right)A + 7 = -2$$ $$\frac{1}{4}A + 7 = -2$$ 7. Subtract 7 from both sides: $$\frac{1}{4}A = -2 - 7$$ $$\frac{1}{4}A = -9$$ 8. Multiply both sides by 4 to solve for $A$: $$\cancel{4} \times \frac{1}{\cancel{4}}A = -9 \times 4$$ $$A = -36$$ 9. Substitute $A = -36$ back into $B = 28 - A$: $$B = 28 - (-36)$$ $$B = 28 + 36$$ $$B = 64$$ **Final answer:** $$A = -36, \quad B = 64$$