MIA-20260531-FA1024E5

Here is the step-by-step solution. ### 1. Establish the initial conditions * **Train A** (from Chicago): Leaves at 10:00 AM traveling south at **60 mph**. * **Train B** (from St. Louis): Leaves at 11:00 AM traveling north at **80 mph**. * **Distance between cities:** 300 miles. ### 2. Calculate the situation at 11:00 AM In the first hour (from 10:00 AM to 11:00 AM), only Train A is moving. * Distance Train A covers by 11:00 AM: $$60 \text{ mph} \times 1 \text{ hour} = 60 \text{ miles}$$ At 11:00 AM: * Train A is 60 miles south of Chicago. * Train B is just leaving St. Louis, 300 miles south of Chicago. Therefore, the **remaining distance between the two trains** at 11:00 AM is: $$300 \text{ miles} - 60 \text{ miles} = \mathbf{240 \text{ miles}}$$ ### 3. Calculate the time to meet after 11:00 AM The trains are now moving toward each other, so their speeds add up. * **Combined speed:** $60 \text{ mph} + 80 \text{ mph} = \mathbf{140 \text{ mph}}$ They need to close a 240-mile gap at 140 mph. * **Time to meet:** $$t = \frac{\text{Distance}}{\text{Speed}} = \frac{240}{140} = \frac{12}{7} \text{ hours}$$ Convert $\frac{12}{7}$ hours to hours and minutes: $$\frac{12}{7} = 1\frac{5}{7} \text{ hours} = 1 \text{ hour} + \frac{5}{7} \text{ of an hour}$$ $$\frac{5}{7} \times 60 \text{ minutes} = \frac{300}{7} \text{ minutes} = 42\frac{6}{7} \text{ minutes}$$ So they meet **$1$ hour and $42\frac{6}{7}$ minutes after 11:00 AM**. ### 4. Determine the meeting time $$11:00 \text{ AM} + 1 \text{ hour } 42\frac{6}{7} \text{ minutes} = \mathbf{12:42\frac{6}{7} \text{ PM}}$$ This is approximately **12:42:51 PM**. ### 5. Verification * **Train A** travels for $2\frac{5}{7}$ hours at 60 mph: $$60 \times \frac{19}{7} = \frac{1140}{7} \approx 162.86 \text{ miles south of Chicago}$$ * **Train B** travels for $1\frac{5}{7}$ hours at 80 mph: $$80 \times \frac{12}{7} = \frac{960}{7} \approx 137.14 \text{ miles north from St. Louis}$$ * Position of Train B south of Chicago: $$300 - 137.14 = 162.86 \text{ miles}$$ The positions match. **Answer:** The trains meet at **approximately 12:43 PM** (exactly $12:42\frac{6}{7}$ PM).