1. **Problem (2a):** Find the ratio of First Class seats to Premium seats to Economy seats given 14, 70, and 168 seats respectively. 2. **Formula and rule:** To find the simplest ratio, divide each number by their greatest common divisor (GCD). 3. **Calculate GCD:** - Factors of 14: 1, 2, 7, 14 - Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70 - Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168 - Common factors: 1, 2, 7, 14 - Greatest common divisor is 14. 4. **Divide each by 14:** - First Class: $\frac{14}{14} = 1$ - Premium: $\frac{70}{14} = 5$ - Economy: $\frac{168}{14} = 12$ 5. **Answer (2a):** Ratio is $1 : 5 : 12$. --- 6. **Problem (2b)(i):** Given ticket cost ratio First Class : Premium : Economy = 14 : 6 : 5 and Premium ticket cost is 114, find First Class and Economy ticket costs. 7. **Formula:** Use ratio to find cost per unit: Cost per unit = $\frac{114}{6} = 19$. 8. **Calculate costs:** - First Class: $14 \times 19 = 266$ - Economy: $5 \times 19 = 95$ 9. **Answer (2b)(i):** First Class $266$, Economy $95$. --- 10. **Problem (2b)(ii):** Premium ticket cost reduced from 114 to 96.90. Find percentage reduction. 11. **Formula:** Percentage reduction = $\frac{\text{Original} - \text{New}}{\text{Original}} \times 100$. 12. **Calculate:** - Reduction = $114 - 96.90 = 17.10$ - Percentage reduction = $\frac{17.10}{114} \times 100 = 15$% 13. **Answer (2b)(ii):** 15% reduction. --- 14. **Problem (Flight time):** Athens time 13:15 departure, Berlin arrival 15:05 local time. Athens is 1 hour ahead of Berlin. 15. **Calculate flight time:** - Convert Berlin arrival to Athens time: $15:05 + 1:00 = 16:05$ - Flight time = $16:05 - 13:15 = 2$ hours $50$ minutes. 16. **Answer (Flight time):** 2 h 50 min. --- 17. **Problem (Average speed):** Distance = 1802 km, time = 2 h 50 min = $2 + \frac{50}{60} = 2.8333$ hours. 18. **Formula:** Average speed = $\frac{\text{Distance}}{\text{Time}}$. 19. **Calculate:** - Speed = $\frac{1802}{2.8333} \approx 635.9$ km/h. 20. **Answer (Average speed):** 636 km/h (rounded). --- 21. **Problem (3a)(i):** Beth invests 2000 at 2% compound interest for 5 years. Find value. 22. **Formula:** Compound interest formula: $A = P(1 + r)^t$ where $P=2000$, $r=0.02$, $t=5$. 23. **Calculate:** - $A = 2000 \times (1.02)^5 = 2000 \times 1.10408 = 2208.16$ 24. **Answer (3a)(i):** 2208.16. --- 25. **Problem (3a)(ii):** Find overall percentage increase. 26. **Formula:** Percentage increase = $\frac{A - P}{P} \times 100$. 27. **Calculate:** - $\frac{2208.16 - 2000}{2000} \times 100 = 10.41$%. 28. **Answer (3a)(ii):** 10.41%. --- 29. **Problem (3a)(iii):** Find minimum years for investment to exceed 2500. 30. **Formula:** Solve $2000(1.02)^t > 2500$. 31. **Calculate:** - $(1.02)^t > \frac{2500}{2000} = 1.25$ - Take natural log: $t \ln(1.02) > \ln(1.25)$ - $t > \frac{\ln(1.25)}{\ln(1.02)} = \frac{0.2231}{0.0198} = 11.27$ 32. **Answer (3a)(iii):** Minimum complete years = 12. --- 33. **Problem (3b):** Population decreases exponentially at 4% per year, current population 255. Find population 16 years ago. 34. **Formula:** $P = P_0 (1 - r)^t$ where $P=255$, $r=0.04$, $t=16$. 35. **Calculate original population $P_0$:** - $P_0 = \frac{P}{(1 - r)^t} = \frac{255}{(0.96)^{16}}$ - Calculate denominator: $(0.96)^{16} \approx 0.5273$ - $P_0 = \frac{255}{0.5273} \approx 483.5$ 36. **Answer (3b):** Approximately 484 people 16 years ago.