Calculating the Break: Driver's Average Speed Conundrum at 54 km/h

To solve this problem, we'll follow these steps:

• Step 1: Convert the driver's travel time from minutes to hours.
• Step 2: Calculate the total distance for each leg of the trip.
• Step 3: Determine the driver's total time using the average speed formula.
• Step 4: Calculate the break time by subtracting travel times from the total time.

Now, let's work through each step:
Step 1: The driver's travel time is given as 45 minutes, which is $\frac{45}{60} = 0.75$ hours.
Step 2: Calculate the distance for each trip. For the forward trip at 90 km/h:
$\text{Distance} = 90 \times 0.75 = 67.5$ km.
The return trip covers the same distance of 67.5 km at 67.5 km/h, taking $\frac{67.5}{67.5} = 1$ hour.
Step 3: Using the given average speed $54$ km/h, set up the equation for the total trip:
$54 = \frac{\text{Total Distance}}{\text{Total Time}}$
The total distance traveled each way is $67.5$, resulting in a round trip of $2 \times 67.5 = 135$ km.
Let $T$ be the total time:
$54 = \frac{135}{T}$
Solving for $T$, we get:
$T = \frac{135}{54} = 2.5$ hours.
Step 4: Calculate the break time. The total traveling time is $0.75 + 1 = 1.75$ hours. Thus, break time is:
$2.5 - 1.75 = 0.75$ hours or $45$ minutes.

Therefore, the break taken by the delivery driver was $45$ minutes.