1. Problem 13: Number of Ducks and Pigs We are given that the total number of ducks and pigs is 35 and the total number of legs is 98. Each duck has 2 legs and each pig has 4 legs. We need to find how many ducks and pigs there are. 2. Define variables: Let $d$ = number of ducks Let $p$ = number of pigs 3. Write the system of equations based on the problem: $$d + p = 35$$ $$2d + 4p = 98$$ 4. Solve the system: From the first equation, express $d$: $$d = 35 - p$$ Substitute into the second equation: $$2(35 - p) + 4p = 98$$ $$70 - 2p + 4p = 98$$ $$70 + 2p = 98$$ $$2p = 98 - 70$$ $$2p = 28$$ $$p = 14$$ 5. Find $d$: $$d = 35 - 14 = 21$$ 6. Final answer for problem 13: There are 21 ducks and 14 pigs. --- 7. Problem 14: Racing Strategies Carla runs half the time and walks half the time. Allison runs half the distance and walks half the distance. Both walk at the same speed and run at the same speed. We need to determine who arrives home first. 8. Let the total distance be $D$, running speed be $v_r$, walking speed be $v_w$. 9. Carla's time: She runs half the time and walks half the time, so if total time is $T$, then: Running time = $\frac{T}{2}$, Walking time = $\frac{T}{2}$ Distance run by Carla = $v_r \times \frac{T}{2}$ Distance walked by Carla = $v_w \times \frac{T}{2}$ Total distance: $$v_r \frac{T}{2} + v_w \frac{T}{2} = D$$ $$\frac{T}{2}(v_r + v_w) = D$$ $$T = \frac{2D}{v_r + v_w}$$ 10. Allison's time: She runs half the distance and walks half the distance. Distance run = $\frac{D}{2}$, distance walked = $\frac{D}{2}$ Time running = $\frac{D/2}{v_r} = \frac{D}{2v_r}$ Time walking = $\frac{D/2}{v_w} = \frac{D}{2v_w}$ Total time: $$T = \frac{D}{2v_r} + \frac{D}{2v_w} = \frac{D}{2}\left(\frac{1}{v_r} + \frac{1}{v_w}\right)$$ 11. Compare Carla's and Allison's times: Carla's time: $$T_C = \frac{2D}{v_r + v_w}$$ Allison's time: $$T_A = \frac{D}{2}\left(\frac{1}{v_r} + \frac{1}{v_w}\right)$$ 12. To compare, multiply both times by $\frac{v_r v_w}{D}$ (positive, so inequality direction preserved): $$T_C \times \frac{v_r v_w}{D} = 2 \frac{v_r v_w}{v_r + v_w}$$ $$T_A \times \frac{v_r v_w}{D} = \frac{v_r + v_w}{2}$$ 13. Note that by AM-HM inequality: $$\frac{v_r + v_w}{2} \geq \frac{2 v_r v_w}{v_r + v_w}$$ So: $$T_A \times \frac{v_r v_w}{D} \geq T_C \times \frac{v_r v_w}{D}$$ Therefore: $$T_A \geq T_C$$ 14. Conclusion: Carla arrives home first or at the same time as Allison, but since speeds are positive and distinct, Carla arrives first. Final answers: - Problem 13: 21 ducks and 14 pigs - Problem 14: Carla arrives home first.