1. **State the problem:** Runners D, E, and F ran a sprint with a combined time of 72 seconds. Runner D's time is half of E's time and one-third of F's time. We need to find how long each runner took. 2. **Define variables:** Let $t_E$ be the time for runner E. Then runner D's time $t_D = \frac{1}{2} t_E$ and runner F's time $t_F = 3 t_D$ (since $t_D = \frac{1}{3} t_F$ implies $t_F = 3 t_D$). 3. **Express all times in terms of $t_E$:** Since $t_D = \frac{1}{2} t_E$, then $t_F = 3 t_D = 3 \times \frac{1}{2} t_E = \frac{3}{2} t_E$. 4. **Write the equation for total time:** $$t_D + t_E + t_F = 72$$ Substitute: $$\frac{1}{2} t_E + t_E + \frac{3}{2} t_E = 72$$ 5. **Simplify the equation:** $$\left(\frac{1}{2} + 1 + \frac{3}{2}\right) t_E = 72$$ $$\left(\frac{1}{2} + \frac{2}{2} + \frac{3}{2}\right) t_E = 72$$ $$\frac{6}{2} t_E = 72$$ $$3 t_E = 72$$ 6. **Solve for $t_E$:** $$t_E = \frac{72}{3} = 24$$ 7. **Find $t_D$ and $t_F$:** $$t_D = \frac{1}{2} t_E = \frac{1}{2} \times 24 = 12$$ $$t_F = \frac{3}{2} t_E = \frac{3}{2} \times 24 = 36$$ **Final answer:** Runner D took 12 seconds, runner E took 24 seconds, and runner F took 36 seconds.