Apply the distance, rate, and time problem-solving strategy to find two unknown speeds when a single traveler covers two consecutive segments at different uniform rates, with the segment distances adding up to a known total and the faster rate expressed as a multiple of the slower rate.

Problem: Hamilton drives from his home in Orange County to Las Vegas, a total distance of $255$ miles. He departs at 2:00 PM and drives on congested city freeways. At 4:00 PM the congestion clears and he drives through the desert at a speed $1.75$ times his city speed. He arrives in Las Vegas at 6:30 PM. What was his speed during each part of the trip?

1. Read and draw: Sketch the route from Home to Las Vegas with two arrows — one labeled "city driving" (2:00 PM to 4:00 PM) and one labeled "desert driving" (4:00 PM to 6:30 PM). The total distance between the two endpoints is $255$ miles. Create a rate–time–distance table.

2. Identify: The driving speed for the city segment and the desert segment.

3. Name: Let $r$ = the city driving speed in mph. Because the desert speed is $1.75$ times as fast, it equals $1.75r$. Convert the clock times to elapsed times: city driving lasts from 2:00 PM to 4:00 PM, which is $2$ hours; desert driving lasts from 4:00 PM to 6:30 PM, which is $2.5$ hours. Multiply rate by time to fill in the distance column:

	Rate (mph)	Time (hrs)	Distance (miles)
City	$r$	$2$	$2r$
Desert	$1.75r$	$2.5$	$2.5(1.75r)$
			$255$

4. Translate: The city distance plus the desert distance equals the total of $255$ miles:

$2r + 2.5(1.75r) = 255$

5. Solve: Multiply $2.5 \times 1.75 = 4.375$:

$2r + 4.375r = 255$

Combine like terms: $6.375r = 255$. Divide both sides by $6.375$:

$r = 40$

The city speed is $40$ mph. The desert speed is $1.75 \times 40 = 70$ mph.

6. Check: City: $40 \times 2 = 80$ miles. Desert: $70 \times 2.5 = 175$ miles. Total: $80 + 175 = 255$ miles. $\checkmark$

7. Answer: Hamilton drove $40$ mph in the city and $70$ mph in the desert.

This example demonstrates the consecutive-segment scenario: a single traveler covers two parts of a trip at different uniform speeds, and the individual segment distances add up to a known total. Unlike the equal-distance and opposite-direction examples, here the two speeds are related by a multiplicative factor ($1.75r$ rather than $r + 12$), the elapsed times for each segment are determined by converting separate clock-time intervals to durations, and multiplying the decimal coefficients produces the term $4.375r$. Combining like terms with decimal arithmetic yields $6.375r = 255$, which is solved by a single division.