Kayla drove from Bayside to Chatham at a constant speed of 21 miles per hour and then returned along the same route from Chatham to Bayside. If her average speed for the entire journey was 26.25 miles per hour, at what average speed, in miles per hour, did Kayla return from Chatham to Bayside ?

Options

A. 28
B. 31
C. `31.5`
D. 35

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Define Variables
Let the distance from Bayside to Chatham be \( d \) miles.

### Step 2: Calculate Time for Each Leg of the Journey
- The time taken to travel from Bayside to Chatham at a speed of 21 miles per hour is given by:
\[ \text{Time}_{B \to C} = \frac{d}{21} \]
- Let the speed on the return trip from Chatham to Bayside be \( r \) miles per hour. The time taken for this leg is:
\[ \text{Time}_{C \to B} = \frac{d}{r} \]

### Step 3: Calculate Total Distance and Total Time
- The total distance for the round trip is:
\[ \text{Total Distance} = d + d = 2d \]
- The total time for the round trip is:
\[ \text{Total Time} = \frac{d}{21} + \frac{d}{r} \]

### Step 4: Use the Average Speed Formula
The average speed for the entire journey is given by:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ 26.25 = \frac{2d}{\frac{d}{21} + \frac{d}{r}} \]

### Step 5: Simplify the Equation
We can simplify the equation:
\[ 26.25 = \frac{2d}{\frac{d}{21} + \frac{d}{r}} \]
Cancelling \( d \) from the numerator and denominator (assuming \( d \neq 0 \)):
\[ 26.25 = \frac{2}{\frac{1}{21} + \frac{1}{r}} \]

### Step 6: Find a Common Denominator
The common denominator for the fractions in the denominator is \( 21r \):
\[ \frac{1}{21} + \frac{1}{r} = \frac{r + 21}{21r} \]
Thus, we can rewrite the equation as:
\[ 26.25 = \frac{2 \cdot 21r}{r + 21} \]

### Step 7: Cross-Multiply
Cross-multiplying gives:
\[ 26.25(r + 21) = 42r \]
Expanding this:
\[ 26.25r + 551.25 = 42r \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 42r - 26.25r = 551.25 \]
This simplifies to:
\[ 15.75r = 551.25 \]

### Step 9: Solve for \( r \)
Dividing both sides by 15.75:
\[ r = \frac{551.25}{15.75} = 35 \]

### Conclusion
Thus, the average speed at which Kayla returned from Chatham to Bayside is:
\[ \boxed{35} \text{ miles per hour} \]