1. **Problem statement:** Two vehicles start at the same time from towns A and B, 160 km apart, traveling towards each other. The lorry travels at 45 km/h from A to B, and the car travels at 75 km/h from B to A. They meet at town C. 2. **Formula and rules:** When two objects move towards each other, the sum of distances they cover equals the total distance between them: $$d_1 + d_2 = D$$ where $d_1$ and $d_2$ are distances traveled by the lorry and car respectively, and $D=160$ km. Speed = Distance / Time, so Time = Distance / Speed. Since they start at the same time and meet at the same time, their travel times are equal: $$t = \frac{d_1}{45} = \frac{d_2}{75}$$ 3. **Find the meeting time (a)(i):** Let $t$ be the time in hours after 11:50 am when they meet. From the total distance: $$d_1 + d_2 = 160$$ Using $d_1 = 45t$ and $d_2 = 75t$: $$45t + 75t = 160$$ $$120t = 160$$ $$t = \frac{160}{120} = \frac{4}{3} = 1.3333 \text{ hours}$$ Convert $1.3333$ hours to hours and minutes: $$1.3333 \times 60 = 80 \text{ minutes}$$ So, they meet 1 hour 20 minutes after 11:50 am. Meeting time = 11:50 am + 1 hour 20 minutes = 1:10 pm. 4. **Find distance from A to C (a)(ii):** Distance from A to C is distance traveled by lorry: $$d_1 = 45t = 45 \times \frac{4}{3} = 60 \text{ km}$$ 5. **Find average speed of car from C to A (b):** The car stops at C for 1 hour 40 minutes = $1 + \frac{40}{60} = 1.6667$ hours. Time taken by lorry to travel from C to B: Distance from C to B: $$160 - 60 = 100 \text{ km}$$ Time: $$t_{lorry} = \frac{100}{45} = 2.2222 \text{ hours}$$ Total time for lorry from A to B: $$t_{total} = 1.3333 + 2.2222 = 3.5555 \text{ hours}$$ The car arrives at A at the same time as the lorry arrives at B. Time car travels from B to C: $$t = 1.3333 \text{ hours}$$ Stop time at C: $$1.6667 \text{ hours}$$ Let $v$ be the average speed of car from C to A. Distance from C to A: $$60 \text{ km}$$ Time car travels from C to A: $$t_{car} = \frac{60}{v}$$ Total time for car from B to A: $$1.3333 + 1.6667 + \frac{60}{v} = 3.5555$$ Simplify: $$3 + \frac{60}{v} = 3.5555$$ $$\frac{60}{v} = 0.5555$$ $$v = \frac{60}{0.5555} \approx 108 \text{ km/h}$$ **Final answers:** - (a)(i) The vehicles met at **1:10 pm**. - (a)(ii) Distance from A to C is **60 km**. - (b) Average speed of car from C to A is approximately **108 km/h**.