1. **State the problem:** Two trains start 342 miles apart and travel toward each other. One train travels at 105 mph and the other at 85 mph. We need to find the time $t$ in hours when they meet. 2. **Formula and concept:** When two objects move toward each other, their relative speed is the sum of their speeds. The distance covered together equals the initial distance between them. 3. **Set up the equation:** Let $t$ be the time in hours until they meet. The total distance covered by both trains together is $$105t + 85t = 342$$ 4. **Simplify the equation:** $$105t + 85t = 190t$$ So, $$190t = 342$$ 5. **Solve for $t$:** $$t = \frac{342}{190}$$ Show canceling common factors: $$t = \frac{\cancel{342}}{\cancel{190}}$$ Since 342 and 190 share a common factor 2: $$t = \frac{171}{95}$$ 6. **Final answer:** The trains will meet after $$\boxed{\frac{171}{95}}$$ hours, which is an exact fraction with no rounding.