1. **State the problem:** Mark and Tracy's combined ages total 44 years. We need to find Mark's current age given a complex age relationship. 2. **Define variables:** Let $M$ = Mark's current age, $T$ = Tracy's current age. 3. **Given:** - $M + T = 44$ - Mark is twice as old as Tracy was when Mark was half as old as Tracy will be when Tracy is three times as old as Mark was when Mark was three times as old as Tracy. 4. **Break down the complex statement step-by-step:** - Step A: "Mark was three times as old as Tracy" at some time in the past. Let the time elapsed since then be $x$ years ago. Then, $M - x = 3(T - x)$. - Step B: "Tracy is three times as old as Mark was" at that time. So, Tracy's current age $T = 3(M - x)$. - Step C: "Mark was half as old as Tracy will be" when Tracy is three times as old as Mark was (from Step B). Let the time until Tracy is three times as old as Mark was be $y$ years in the future. Then, $M - y = \frac{1}{2}(T + y)$. - Step D: "Mark is twice as old as Tracy was" when Mark was half as old as Tracy will be (from Step C). Let the time elapsed since then be $z$ years ago. Then, $M = 2(T - z)$ and $M - z = \frac{1}{2}(T + y)$. 5. **Solve Step A:** $$M - x = 3(T - x)$$ $$M - x = 3T - 3x$$ $$M - 3T = -2x$$ $$x = \frac{3T - M}{2}$$ 6. **Solve Step B:** $$T = 3(M - x)$$ Substitute $x$: $$T = 3\left(M - \frac{3T - M}{2}\right) = 3\left(\frac{2M - 3T + M}{2}\right) = 3\left(\frac{3M - 3T}{2}\right) = \frac{9M - 9T}{2}$$ Multiply both sides by 2: $$2T = 9M - 9T$$ $$2T + 9T = 9M$$ $$11T = 9M$$ $$M = \frac{11}{9}T$$ 7. **Use $M + T = 44$:** $$\frac{11}{9}T + T = 44$$ $$\frac{11}{9}T + \frac{9}{9}T = 44$$ $$\frac{20}{9}T = 44$$ $$T = 44 \times \frac{9}{20} = 19.8$$ 8. **Find $M$:** $$M = \frac{11}{9} \times 19.8 = 24.2$$ 9. **Final answer:** Mark is approximately $\boxed{24}$ years old. This satisfies the total age and the complex age relationship given.