1. Audit the Logic (Deconstruct the Problem)
- The Scenario: Two people are moving directly toward each other on a straight road.
- The Key Clue: We need to find the distance between them $\(\mathbf{1}\)$ hour before they meet.
2. Execute the Combined Rate Protocol
- The "Combined Rate" Insight: When two objects are moving towards each other, you can treat them as a single entity closing the gap. You find their combined speed by adding their individual rates.
- Combined Speed:
- Atu's Rate = 56 mph
- Brek $\(^{\prime}$ s Rate $=49\)$ mph
- Combined Rate $\(=56+49=105\)$ mph
- The "One Hour Before" Calculation: The question asks for the distance apart $\(\mathbf{1}\)$ hour before they meet. Since their combined speed is $\(105 m p h\)$, this means that every single hour, the distance between them shrinks by exactly 105 miles.
- If they are 105 miles apart, it will take them exactly one more hour to close that remaining gap.

3. Final Logic Check

The entire 420 -mile distance is irrelevant to the specific question asked. The logic remains the same whether they started 100 miles apart or 10,000 miles apart.

Because Atu and Brek are closing the gap at a combined rate of $\(\mathbf{1 0 5 ~ m p h}\)$, exactly one hour before their meeting point, they will be 105 miles apart.

Correct Answer: 105 miles