MIA-20260531-2667B893

To solve this, we first assume the first train is heading **south** from Chicago toward St. Louis. (If it were heading any other direction, it would not meet the second train, which is heading north from St. Louis.) Here is the step-by-step reasoning: ### 1. Account for the head start The first train leaves at 10:00 AM and travels alone for **1 hour** before the second train departs at 11:00 AM. *Distance covered by the first train in that hour:* $$60 \text{ mph} \times 1 \text{ hour} = 60 \text{ miles}$$ So, by 11:00 AM, the gap between the two trains has shrunk from 300 miles to: $$300 - 60 = 240 \text{ miles}$$ ### 2. Determine their combined closing speed After 11:00 AM, the trains are moving toward each other. Their speeds add up: $$60 \text{ mph} + 80 \text{ mph} = 140 \text{ mph}$$ ### 3. Calculate how long it takes to close the remaining gap With 240 miles left and a combined speed of 140 mph, the time required to meet is: $$\text{Time} = \frac{\text{Remaining Distance}}{\text{Combined Speed}} = \frac{240}{140} = \frac{12}{7} \text{ hours}$$ ### 4. Convert the fractional hours to minutes $$\frac{12}{7} \text{ hours} = 1\frac{5}{7} \text{ hours}$$ Convert the fractional part to minutes: $$\frac{5}{7} \times 60 = \frac{300}{7} \approx 42.857 \text{ minutes}$$ So they meet **1 hour and about 42.86 minutes** after 11:00 AM. ### 5. Find the exact meeting time $$11:00 \text{ AM} + 1 \text{ hour} = 12:00 \text{ PM (noon)}$$ $$12:00 \text{ PM} + 42\frac{6}{7} \text{ minutes} = 12:42\frac{6}{7} \text{ PM}$$ The fraction of a minute is: $$\frac{6}{7} \times 60 \approx 51.4 \text{ seconds}$$ --- ## Answer The trains meet at exactly **12:42 and $\frac{6}{7}$ minutes PM**, or approximately **12:42:51 PM**. *(Verification: The first train travels for $1 + \frac{12}{7} = \frac{19}{7}$ hours, covering $60 \times \frac{19}{7} = \frac{1140}{7}$ miles. The second train travels for $\frac{12}{7}$ hours, covering $80 \times \frac{12}{7} = \frac{960}{7}$ miles. Together: $\frac{1140+960}{7} = \frac{2100}{7} = 300$ miles. ✓)*