Multi-Leg Journey Problem: Running and Flying Times

What This Problem Teaches

• Setting up systems of equations when you have multiple constraints
• Recognizing that "distance = rate × time" applies to each segment separately
• Using substitution to reduce two equations to one variable
• Converting between different time units (hours to minutes)
• Checking solutions by verifying they satisfy all original conditions

Solution: Method 1 — The Standard System Approach

This is a classic two-leg journey problem where we need to find how time is split between two different speeds.

Step 1 — Define the variables

Let t = time spent running (in hours)
Then (2 - t) = time spent flying (in hours)

We use this relationship because the total time is 2 hours, so the two time periods must add up to 2.

Step 2 — Set up the distance equation

Using distance = rate × time for each segment:

Step 3 — Solve for running time

Expand the equation:

Step 4 — Find flying time

Flying time = 2 - t = 2 - 0.5 = 1.5 hours

Step 5 — Convert to minutes for clarity

Running time: 0.5 hours = 30 minutes
Flying time: 1.5 hours = 90 minutes

Solution: Method 2 — The Weighted Average Approach

Instead of setting up equations formally, we can think about this as finding the right "mix" of two different speeds to achieve the overall constraints.

Step 1 — Calculate what "all flying" would give us

If Eudora flew the entire 2 hours at 76 km/h:

This is 32 km more than her actual 120 km journey.

Step 2 — Find the speed difference per hour

Each hour of running instead of flying reduces the distance by:

Step 3 — Calculate running time needed

To reduce the distance by 32 km:

Step 4 — Find flying time

Flying time = 2 - 0.5 = 1.5 hours = 90 minutes

Verification

Let's check our answer by substituting back into the original constraints:

Time check: 0.5 + 1.5 = 2 hours ✓

Distance check:

• Running distance: 12 km/h × 0.5 h = 6 km
• Flying distance: 76 km/h × 1.5 h = 114 km
• Total distance: 6 + 114 = 120 km ✓

Both constraints are satisfied, confirming our solution is correct.

Common Pitfalls

The Pattern Behind This

This problem follows the general structure for multi-leg journey problems:

The key insight is that you need as many constraints as unknowns. With two unknown times, we need two constraints: one for total time and one for total distance.

This pattern extends to problems with fuel efficiency (mpg), work rates (items/hour), or any scenario where you're combining different rates over different time periods.

How to Spot This Problem Type

Look for these tell-tale signs:

• "Then" or "next" — signals multiple legs of a journey
• Two different speeds/rates — running and flying, walking and driving, etc.
• Total distance and total time both given — provides the two constraints needed
• "How long did she spend...?" — asks for time breakdown, not just total time

Variations include: different vehicles with different fuel efficiency, workers with different productivity rates, or investments with different interest rates. The mathematical structure remains the same.