To solve the problem using minutes without converting to hours, we follow a similar reasoning but use the unit of minutes throughout.

Restating the problem in minutes:

• Brad's mother drives with him for either 15 minutes or 30 minutes each day.
• Over 35 days, the total driving time is capped at 15 hours, which is equivalent to 900 minutes.
• We need to show that during some consecutive period of days, the total driving time will be exactly 8.75 hours, which is equivalent to 525 minutes.

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Solution in minutes:

1. Define cumulative sums in minutes: Let $S_n$ represent the cumulative driving time from day 1 to day $n$ (in minutes): $S_n = t_1 + t_2 + \dots + t_n$ where each $t_k \in \{15, 30\}$ (the driving time on the $k$-th day).

   The problem asks us to find two days $i$ and $j$ (where $i < j$) such that: $S_j - S_i = 525 \quad \text{(minutes)}$

   Equivalently, this means we are looking for two cumulative sums $S_j$ and $S_i$ such that: $S_j \equiv S_i \pmod{525}$

2. Apply the Pigeonhole Principle:

   • The possible remainders when dividing a number by $525$ are integers in the range $[0, 524]$, so there are $525$ possible remainders.
   • However, there are $35$ days, and thus $36$ cumulative sums ($S_0 = 0, S_1, S_2, \dots, S_{35}$).
   • If any two cumulative sums $S_i$ and $S_j$ (with $i < j$) have the same remainder modulo $525$, then: S_j - S_i = k \times 525 \quad \text{(where k is an integer)}. Since $S_j - S_i$ represents driving time and cannot exceed $900$ minutes, the only possible value of $k$ is $1$. Thus: $S_j - S_i = 525.$

3. Ensure total time constraint: The total driving time is at most $900$ minutes. If there is a remainder of $0$ (i.e., $S_i \equiv 0 \pmod{525}$), this means $S_i = 525$, satisfying the condition immediately.

4. Conclude the proof: Using the Pigeonhole Principle and modulo arithmetic, we have shown that either:

   • Two cumulative sums $S_i$ and $S_j$ will have the same remainder modulo $525$, or
   • One cumulative sum $S_i$ will be exactly $525$, ensuring the existence of a consecutive driving period totaling 525 minutes.

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Key Intuition:

By working in minutes and ensuring that the modulo condition aligns with the given time constraints, the solution directly parallels the logic in hours. The choice of $525$ (instead of $8.75$) simplifies the arithmetic by eliminating fractional hours.