This is a distance, rate, and time problem involving two trains moving toward each other. Let's break it down and solve step by step:

Problem Details:

• Distance between two towns: 1026 miles
• One train is slower than the other by 13 mph
• The two trains meet after 6 hours

Unknowns:

• Let the rate of the faster train be $x$ mph.
• Then the rate of the slower train is $x - 13$ mph.

Formula:

For both trains traveling toward each other, their combined distance covered after 6 hours must be 1026 miles. Using the distance formula: $\text{Distance} = \text{Rate} \times \text{Time}$

The combined distance covered by both trains in 6 hours is: $6(x) + 6(x - 13) = 1026$

Solving the equation:

$6x + 6(x - 13) = 1026$ $6x + 6x - 78 = 1026$ $12x - 78 = 1026$ $12x = 1026 + 78$ $12x = 1104$ $x = \frac{1104}{12} = 92$

Final Answer:

• The faster train travels at 92 mph.
• The slower train travels at $92 - 13 = 79$ mph.

Would you like more details, or have any questions?

Related Questions:

1. How would the solution change if the trains met after 8 hours instead of 6?
2. What if the distance between the towns were 1200 miles but the trains met after 6 hours?
3. How do you calculate the time it would take if both trains traveled the same speed?
4. How can we use a graphing calculator to solve this system of equations?
5. What happens to the rates if one train is 20 mph slower than the other?

Tip: When solving rate problems, always set up your equation using the distance formula $\text{Distance} = \text{Rate} \times \text{Time}$ and keep track of units carefully!