Solve Distance, Rate, and Time Using Two Different Speeds

What This Problem Teaches

• How to use the fundamental relationship Distance = Rate × Time when the distance is constant but rates and times vary
• Setting up equations where the same unknown appears in both rate expressions
• Converting between time units (minutes to hours) to maintain dimensional consistency
• Recognizing that speed differences and time differences work together to determine absolute values
• Verification techniques that check both the mathematical solution and real-world reasonableness

Solution: Method 1 — The Single-Variable Setup

The key insight is that David travels the exact same distance regardless of whether he drives or bikes. This constraint gives us the equation we need.

Step 1 — Define the variable

Let b = David's bike speed in miles per hour. Since his car is 4.5 mph faster than his bike:

Step 2 — Convert times to hours

The speeds are in mph, so we need times in hours for the Distance = Rate × Time formula:

Step 3 — Set up the distance equation

Since the distance to work is the same whether David drives or bikes:

Step 4 — Solve for bike speed

Distribute and collect terms:

Multiply everything by 12 to clear fractions:

Step 5 — Calculate the distance

Now we can find the distance using either mode of travel:

Solution: Method 2 — Speed Ratio from Time Ratio

Since distance is constant, we can use the inverse relationship between speed and time to solve this more directly.

Step 1 — Establish the time ratio

David takes 20 minutes by car and 45 minutes by bike. The time ratio is:

Step 2 — Convert to speed ratio

For constant distance, speeds are inversely proportional to times. So:

Step 3 — Use the speed difference

If the speeds are in ratio 9:4, then car speed is 9k and bike speed is 4k for some constant k. The difference is:

Step 4 — Find actual speeds

Step 5 — Calculate distance

Verification

Let's verify this answer by checking that both methods of travel give the same distance:

We should also verify the speed difference:

All conditions are satisfied.

Common Pitfalls

Does the Answer Make Sense?

A distance of 2.7 miles feels reasonable for a work commute. Let's think about the speeds:

• Bike speed of 3.6 mph: This is quite slow for cycling, but reasonable for city biking with traffic lights, hills, or a cautious rider.
• Car speed of 8.1 mph: This is very slow for highway driving but perfectly reasonable for city driving with traffic, red lights, and congestion.
• Speed difference of 4.5 mph: In stop-and-go city traffic, the advantage of a car over a bike can be surprisingly small.

The relationship also makes intuitive sense: David takes more than twice as long to bike (45 vs 20 minutes), but his car is only 4.5 mph faster. This suggests a short urban commute where traffic limits car speeds significantly.

The Pattern Behind This

This problem belongs to the "same distance, different rates" family. The general structure is:

The constraint can be additive (speed difference), multiplicative (one speed is k times another), or indirect (given cost differences, fuel efficiency, etc.). The key insight is always that distance remains constant.

Where This Shows Up in Real Life

• Urban transportation planning: City planners compare commute times by car, bike, and public transit to design transportation systems and set speed limits.
• Delivery logistics: Companies optimize routes by comparing delivery times for trucks, drones, and foot couriers over the same distances.
• Athletic training: Runners and cyclists track pace differences between different training intensities over identical course distances.

What If?