1. **Problem statement:** Two vehicles start from towns A and B, 360 km apart. The minibus leaves A at 8:15 am traveling at 90 km/h. The matatu leaves B 2 \frac{1}{3} hours later (which is 2.333 hours) traveling at 110 km/h towards A. We need to find: - The time they meet (12-hour clock). - The distance from A where they meet. - The distance from the motorist's home to A, given he leaves at 10:30 am at 100 km/h and arrives at B at the same time as the minibus. 2. **Step 1: Define variables and times** - Distance between A and B: $d = 360$ km - Minibus speed: $v_m = 90$ km/h - Matatu speed: $v_t = 110$ km/h - Minibus start time: 8:15 am - Matatu start time: $8:15 + 2.333$ hours = $8:15 + 2$ hours 20 minutes = 10:35 am 3. **Step 2: Let $t$ be the time in hours after 8:15 am when they meet.** - Distance minibus travels: $90t$ - Matatu starts after 2.333 hours, so it travels for $(t - 2.333)$ hours (only if $t > 2.333$) - Distance matatu travels: $110(t - 2.333)$ 4. **Step 3: Since they meet, sum of distances equals total distance:** $$90t + 110(t - 2.333) = 360$$ 5. **Step 4: Solve for $t$:** $$90t + 110t - 110 \times 2.333 = 360$$ $$200t - 256.67 = 360$$ $$200t = 360 + 256.67 = 616.67$$ $$t = \frac{616.67}{200} = 3.0833 \text{ hours}$$ 6. **Step 5: Convert $t$ to time:** - $3.0833$ hours = 3 hours + 0.0833 hours - $0.0833 \times 60 = 5$ minutes - Meeting time = 8:15 am + 3 hours 5 minutes = 11:20 am 7. **Step 6: Distance from A where they meet:** $$\text{Distance} = 90 \times 3.0833 = 277.5 \text{ km}$$ 8. **Step 7: Motorist leaves at 10:30 am and arrives at B at the same time as minibus (11:20 am).** - Travel time for motorist: $11:20 - 10:30 = 50$ minutes = $\frac{50}{60} = 0.8333$ hours - Motorist speed: 100 km/h - Distance motorist travels: $100 \times 0.8333 = 83.33$ km 9. **Step 8: Since motorist travels from home to B, and distance B to A is 360 km, the distance from home to A is:** $$360 - 83.33 = 276.67 \text{ km}$$ **Final answers:** - The vehicles meet at **11:20 am**. - They meet **277.5 km** from A. - The motorist's home is **276.67 km** from A.