1. Problem Q10: You have black and white socks in the ratio 4:3. How many socks must you pull to be certain of having a single pair of the same color? 2. To solve this, use the Pigeonhole Principle: if you have $n$ categories and pick $n+1$ items, at least two items must be in the same category. 3. Here, the categories are colors: black and white (2 categories). 4. To guarantee a pair of the same color, you must pick $2 + 1 = 3$ socks. 5. Explanation: If you pick 2 socks, they could be one black and one white. Picking a third sock ensures at least one color repeats. 6. Problem Q11: Mary drives from Accra to Kumasi at 50 mph. Abena starts 20 minutes later from Kumasi to Accra at 60 mph. The distance between cities is 200 miles. Which car is nearer to Accra when they meet? 7. Let $t$ be the time in hours Mary has driven when they meet. 8. Mary’s distance from Accra at time $t$ is $50t$ miles. 9. Abena starts 20 minutes (1/3 hour) later, so her driving time when they meet is $t - \frac{1}{3}$ hours. 10. Abena’s distance from Accra at meeting time is $200 - 60\left(t - \frac{1}{3}\right)$ miles. 11. At meeting, distances are equal: $50t = 200 - 60\left(t - \frac{1}{3}\right)$. 12. Solve for $t$: $$50t = 200 - 60t + 20$$ $$50t + 60t = 220$$ $$110t = 220$$ $$t = 2$$ 13. Distance from Accra when they meet: $$50 \times 2 = 100$$ miles. 14. Both cars are at the same distance from Accra when they meet. Final answers: - Q10: 3 socks - Q11: (c) Both are at the same distance from Accra