Gecilia leaves home for school on her bike at 9:05 AM, riding at an average speed of 10 miles per hour. Fifteen minutes later, her mother realizes that Cecilia forgot to take her lunch. Cecilia's mother immediately gets into her car and drives after Cecilia at an average speed of 25 miles per hour. At what time will Cecilia mom catch up to her?

Options

A. 9:15A
B. 9:20A
C. 9:26A
D. 9:30A

Explanation

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the time Cecilia rides before her mother starts.
Cecilia leaves home at 9:05 AM, and her mother realizes she forgot the lunch 15 minutes later. Therefore, Cecilia has been riding for:
- **9:05 AM + 15 minutes = 9:20 AM**

### Step 2: Calculate the distance Cecilia travels by the time her mother starts.
Cecilia rides at an average speed of 10 miles per hour. The time she rides before her mother starts is 15 minutes, which we convert to hours:
- **15 minutes = 15/60 hours = 0.25 hours**

Now, we can calculate the distance (d) Cecilia travels:
- **Distance (d) = Speed × Time**
- **d = 10 miles/hour × 0.25 hours = 2.5 miles**

### Step 3: Set up the equations for the distances traveled.
Let \( t \) be the time in hours that Cecilia's mother drives until she catches up to Cecilia. In this time, Cecilia will continue to ride her bike. The distance Cecilia travels during this time is:
- **Distance = Speed × Time = 10 miles/hour × (t + 0.25 hours)**

Cecilia's mother starts driving 15 minutes (or 0.25 hours) later than Cecilia, so the time she drives is \( t \) hours. The distance her mother travels is:
- **Distance = Speed × Time = 25 miles/hour × t**

### Step 4: Set the distances equal to each other.
Since both Cecilia and her mother will have traveled the same distance when they meet:
- **10(t + 0.25) = 25t**

### Step 5: Solve for \( t \).
Expanding the left side:
- **10t + 2.5 = 25t**

Now, rearranging the equation:
- **2.5 = 25t - 10t**
- **2.5 = 15t**
- **t = 2.5 / 15**
- **t = 0.1667 hours (approximately)**

### Step 6: Convert \( t \) to minutes.
To convert hours to minutes, multiply by 60:
- **t = 0.1667 hours × 60 minutes/hour = 10 minutes**

### Step 7: Calculate the time when Cecilia's mother catches up.
Cecilia's mother starts driving at 9:20 AM and drives for approximately 10 minutes:
- **9:20 AM + 10 minutes = 9:30 AM**

### Final Answer:
Cecilia's mother will catch up to her at **9:30 AM**.

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