Calculate Ricardo's Average Speed: Doubling Speed and Rest Breaks Included

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

• Step 1: Calculate the total distance traveled.

• Step 2: Determine the time taken for each segment of the journey.

• Step 3: Use these times to calculate the total journey time.

• Step 4: Apply the average speed formula using the total distance and total time.

Let's work through each step:

Step 1: Calculate the total distance traveled. Ricardo travels:

• 18 km in the first segment,

• 12 km in the second segment,

• 10 km in the third segment.

Total distance is $18 + 12 + 10 = 40$ km.

Step 2: Determine the time taken for each segment of the journey.

• First segment: $\frac{18}{X}$ hours.

• Second segment: $\frac{12}{2X} = \frac{12}{2X} = \frac{6}{X}$ hours (since he doubles his speed to $2X$).

• Rest: $\frac{1}{2}$ hour.

• Third segment: $\frac{10}{X}$ hours.

Step 3: Calculate the total journey time by adding all the parts together:

Total time = $\frac{18}{X} + \frac{6}{X} + \frac{1}{2} + \frac{10}{X} = \frac{34}{X} + \frac{1}{2}$.

Convert $\frac{1}{2}$ into a fraction with common denominator $X$:

$\frac{1}{2} = \frac{X}{2X}$.

So, total time becomes $\frac{34}{X} + \frac{X}{2X} = \frac{34 + X}{X}$ hours.

Step 4: Apply the average speed formula:

$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{40}{\frac{34 + X}{X}} = \frac{40 \times X}{34 + X}$ km/h.

Thus, the average speed of Ricardo's journey is $\frac{40X}{34 + X}$ km/h.