Two Travelers Meeting at Same Time: Solve for Speed

Question

PROBLEM

Whitney left school and drove 7 miles to meet her mother at a restaurant. Her mother left at the same time and drove 8 miles, traveling 5 mph faster than her daughter. If the two arrived at the restaurant at the same time, what was Whitney's speed?

What This Problem Teaches

Setting up equations from distance-rate-time relationships
Working with relative speeds (one person faster than another)
Recognizing when equal times create solvable equations
Cross-multiplying to eliminate fractions efficiently
Checking answers by verifying both travelers' times match

Visualizing the Journey

Whitney left school and drove 7 miles to meet her mother at a restaurant. Her mother left at the same time and drove...

Solution: Method 1 — The Equal Times Approach

Step 1 — Define the variable

Let r = Whitney's speed in mph. Since her mother travels 5 mph faster, the mother's speed is (r + 5) mph.

Step 2 — Set up time expressions

Both travelers take the same amount of time to reach the restaurant. Using the formula Time = Distance ÷ Rate:

Whitney's time = 7/r hours
Mother's time = 8/(r + 5) hours

Step 3 — Create the equation

Since they arrive at the same time, their travel times are equal:

7/r = 8/(r + 5)

Step 4 — Cross-multiply to eliminate fractions

Cross-multiplication gives us:

7(r + 5) = 8r
7r + 35 = 8r

Step 5 — Solve for r

Subtract 7r from both sides:

35 = 8r - 7r
35 = r

Whitney's speed is 35 mph

Solution: Method 2 — The Algebraic Substitution Method

Step 1 — Set up distance and rate variables

Let t = the travel time for both people (in hours). Since Distance = Rate × Time, we can write:

Whitney: 7 = r × t, so r = 7/t
Mother: 8 = (r + 5) × t, so r + 5 = 8/t

Step 2 — Use the speed relationship

Since the mother travels 5 mph faster than Whitney:

8/t = 7/t + 5

Step 3 — Solve for t

Subtract 7/t from both sides:

8/t - 7/t = 5
1/t = 5
t = 1/5 = 0.2 hours

Step 4 — Find Whitney's speed

Using r = 7/t:

r = 7/0.2 = 35 mph

Whitney's speed is 35 mph

Verification

Let's check that both travelers indeed take the same time:

Whitney's travel time

Time = Distance ÷ Speed = 7 ÷ 35 = 0.2 hours

Mother's travel time

Mother's speed = 35 + 5 = 40 mph

Time = Distance ÷ Speed = 8 ÷ 40 = 0.2 hours ✓

Both times equal 0.2 hours (12 minutes), confirming our answer is correct.

Watch Out For These Mistakes

Setting up the speed relationship backwards

Wrong approach: Letting Whitney's speed be (r + 5) and mother's speed be r.

Why it's wrong: The problem states the mother travels faster than Whitney, not the other way around. Always read carefully to see who is faster.

Adding the distances instead of using equal times

Wrong approach: Setting up 7 + 8 = 15 miles and trying to work from there.

Why it's wrong: This is a meeting problem, not a total distance problem. The key insight is that both people travel for the same amount of time, not that they cover the same total distance.

Forgetting to cross-multiply correctly

Wrong approach: From 7/r = 8/(r+5), writing 7(r+5) = 8 instead of 7(r+5) = 8r.

Why it's wrong: When cross-multiplying a/b = c/d, you get ad = bc. Here, that means 7(r+5) = 8r, not 7(r+5) = 8.

Recognizing This Problem Type

Distance-rate-time meeting problems have these telltale signs:

"At the same time" - indicates simultaneous departure or arrival
"X mph faster/slower" - creates a relative speed relationship
Different distances - each person travels a different distance to the same destination
"Find the speed" - asks for rate, not time or distance

Key insight: When two people start simultaneously and arrive simultaneously, they travel for exactly the same amount of time, even if they cover different distances at different speeds. This equal-time condition is what makes the problem solvable.

The Pattern Behind This

This problem follows the general pattern for simultaneous travel problems:

If person A travels distance d₁ at speed s
and person B travels distance d₂ at speed (s + k)
for the same time t, then:

d₁/s = d₂/(s + k)

Cross-multiplying gives us: d₁(s + k) = d₂s, which simplifies to d₁k = s(d₂ - d₁).

In our problem: d₁ = 7, d₂ = 8, k = 5, so 7 × 5 = s(8 - 7), giving us s = 35.

This formula works whenever you have two people traveling different distances at related speeds for the same duration.

Does This Seem Reasonable?

Let's do a sanity check on our answer of 35 mph:

Whitney: 35 mph for 7 miles takes 7/35 = 0.2 hours = 12 minutes

Mother: 40 mph for 8 miles takes 8/40 = 0.2 hours = 12 minutes

Both speeds are reasonable for city driving, and it makes sense that someone traveling a longer distance (8 vs 7 miles) would need to go faster (40 vs 35 mph) to arrive at the same time. The 5 mph speed difference is modest and realistic.

Reality check: If both people traveled the same speed, the person going 8 miles would take longer. Since they arrive simultaneously, the person going farther must travel faster - which matches our solution.

Four "What-If?" Problems

1

Different Distance Ratio

Whitney drives 9 miles while her mother drives 12 miles to the same restaurant. Her mother still travels 5 mph faster, and they arrive simultaneously. What is Whitney's speed now?

Step 1 — Set up variables

Let r = Whitney's speed, so mother's speed = r + 5

Step 2 — Equal time equation

9/r = 12/(r + 5)

Step 3 — Cross-multiply

9(r + 5) = 12r
9r + 45 = 12r

Step 4 — Solve

45 = 12r - 9r = 3r
r = 15

Step 5 — Verify

Whitney: 9/15 = 0.6 hours
Mother: 12/20 = 0.6 hours ✓

Answer: Whitney's speed is 15 mph

2

Reverse the Unknown

Whitney drives 7 miles at exactly 28 mph. Her mother drives 8 miles and arrives at the restaurant at the same time. How much faster (in mph) is the mother traveling?

Step 1 — Find Whitney's travel time

Time = 7 miles ÷ 28 mph = 0.25 hours

Step 2 — Find mother's speed

Mother travels 8 miles in 0.25 hours
Speed = 8 ÷ 0.25 = 32 mph

Step 3 — Calculate speed difference

32 - 28 = 4 mph faster

Step 4 — Verify

Whitney: 7/28 = 0.25 hours
Mother: 8/32 = 0.25 hours ✓

Answer: The mother travels 4 mph faster

3

Time Delay Challenge

Whitney drives 7 miles at 21 mph. Her mother drives 8 miles but leaves 4 minutes later. What speed must the mother maintain to arrive at the restaurant at the same time as Whitney?

Step 1 — Find Whitney's travel time

Time = 7 ÷ 21 = 1/3 hour = 20 minutes

Step 2 — Calculate mother's available time

Mother leaves 4 minutes later, so has 20 - 4 = 16 minutes = 16/60 hours = 4/15 hours

Step 3 — Find required speed

Speed = Distance ÷ Time = 8 ÷ (4/15) = 8 × 15/4 = 30 mph

Step 4 — Verify timing

Whitney: travels 20 minutes
Mother: leaves 4 minutes late, travels 8 miles at 30 mph in 16 minutes
Both arrive after 20 minutes total ✓

Answer: The mother must travel at 30 mph

4

Three Travelers Extension

Whitney (7 miles), her mother (8 miles), and her father (10 miles) all head to the same restaurant. Whitney's speed is r mph, her mother is 5 mph faster, and her father is 8 mph faster than Whitney. All three arrive simultaneously. Find all three speeds.

Step 1 — Set up speed variables

Whitney: r mph, Mother: r + 5 mph, Father: r + 8 mph

Step 2 — Equal time equations

Whitney = Mother: 7/r = 8/(r + 5)
Whitney = Father: 7/r = 10/(r + 8)

Step 3 — Solve first equation

7(r + 5) = 8r
7r + 35 = 8r
r = 35

Step 4 — Verify with second equation

7/35 = 10/(35 + 8)
0.2 = 10/43
This gives 0.233..., not 0.2, so no solution exists with these constraints.

Step 5 — Adjust father's speed difference

For a valid solution, father must be exactly 10 mph faster than Whitney.
Then: Whitney = 35 mph, Mother = 40 mph, Father = 45 mph

Answer: This system has no solution as stated. For simultaneous arrival, the father would need to be exactly 10 mph faster than Whitney.

Frequently Asked Questions

How do you solve distance-rate-time problems when two people arrive at the same time? +

Set up equations using Time = Distance ÷ Rate, then equate the times since they travel for the same duration. In this problem, Whitney's time (7/r) equals her mother's time (8/(r+5)), giving us one equation to solve for Whitney's speed r.

What does it mean when one person travels faster than another in word problems? +

When someone travels '5 mph faster,' add 5 to the base speed. Here, if Whitney travels at r mph, then her mother travels at (r + 5) mph. This relationship becomes part of your equation setup.

Why do you cross-multiply when solving distance-rate-time equations? +

Cross-multiplication eliminates fractions when you have an equation like 7/r = 8/(r+5). Multiplying both sides by r(r+5) gives 7(r+5) = 8r, which is much easier to solve than working with the original fractions.

DN

Dr. Neven Jurkovic

PhD in Mathematics · 15+ years teaching experience · Specializes in making algebra accessible and building genuine mathematical understanding in students of all levels

NJ

Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

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This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-08-04