2025 AMC 8 Problem 19

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To help you master 2025 AMC 8 Problem 19, this page includes:

The original AMC problem to try, plus the correct answer

Clear, step-by-step solutions using different methods

Key ideas, useful formulas, and takeaways you can apply to similar problems

Related problems from past AMC exams, plus new practice problems that build the same skills

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2025 AMC8 Problem 19

Solution

Solution 1

Stage-by-Stage Motion Method

Track the cars stage by stage. We first find out which car enters the middle section first, and then calculate where the other car is at the exact moment the second car enters the middle section.

Step 1: Divide the road into segments

Based on the problem description, we split the 15-mile road into three 5-mile segments and assign the given speed limit to each.

Segment 1 (0 to 5 miles): 25 mph

Segment 2 (5 to 10 miles): 40 mph

Segment 3 (10 to 15 miles): 20 mph

Step 2: Calculate the time to cross the first segment

We calculate the time it takes for each car to cross its starting segment by dividing the distance by its speed. We then convert these times from hours into minutes for easier tracking.

t_{A} = \frac{5}{25} = \frac{1}{5} hr = 12 min

t_{B} = \frac{5}{20} = \frac{1}{4} hr = 15 min

Step 3: Find positions at the 15-minute mark

At exactly 15 minutes, Car B has just reached the 10-mile mark and is entering the middle segment. Car A reached this segment earlier, at the 12-minute mark, so it has already been traveling in the new 40 mph zone for a few minutes. We calculate how much farther Car A traveled during these extra minutes to find its exact position.

Δ t_{A} = 15 - 12 = 3 min = \frac{1}{20} hr

Distance added = 40 \times \frac{1}{20} = 2 miles

x_{A} = 5 + 2 = 7 miles

Step 4: Find the exact meeting point

We now know both cars’ positions at the 15-minute mark. They are moving toward each other in the same segment, meaning they share the same speed limit of 40 mph. Because their speeds are identical, they will cover equal distances and meet exactly halfway between their current positions.

x_{meet} = \frac{7 + 10}{2} = 8.5 miles

Correct answer: D. $8.5$
8.5 (Choice D). To verify, we can check the total travel time to reach the 8.5-mile mark. Car A travels 5 miles at 25 mph (12 mins) and 3.5 miles at 40 mph (5.25 mins), totaling 17.25 minutes. Car B travels 5 miles at 20 mph (15 mins) and 1.5 miles at 40 mph (2.25 mins), also totaling 17.25 minutes. The identical travel times confirm the meeting point.

Solution 2

Algebraic Distance-Time Method

Write an equation for each car’s position as a function of time. By setting the two position equations equal, we can solve for the exact time they meet, and then use that time to find the distance.

Step 1: Define variables and assumptions

Let time be measured in hours and distance in miles from town A. We assume the cars meet in the middle segment (between 5 and 10 miles from town A), which allows us to set up continuous algebraic equations for that specific part of their journey.

Step 2: Write the position equation for Car A

We write an equation for Car A’s position in the middle segment. We know it takes a fraction of an hour to complete the first 5 miles. For any time after that, its position is 5 miles plus the distance it travels at its new speed of 40 mph.

t_{A} delay = \frac{5}{25} = \frac{1}{5} hr

x_{A} (t) = 5 + 40 (t - \frac{1}{5}) = 40 t - 3

Step 3: Write the position equation for Car B

We write a similar equation for Car B. It takes a certain amount of time to travel from 15 miles down to 10 miles. After that, its position decreases from 10 miles at a rate of 40 mph.

t_{B} delay = \frac{5}{20} = \frac{1}{4} hr

x_{B} (t) = 10 - 40 (t - \frac{1}{4}) = 20 - 40 t

Step 4: Solve for the meeting time

To find the exact time the two cars cross paths, we set their position equations equal to each other and solve for the time variable.

40 t - 3 = 20 - 40 t

80 t = 23

t = \frac{23}{80} hr

Step 5: Calculate the meeting distance

Finally, we substitute the meeting time back into Car A’s position equation to find the exact distance from town A where they meet.

x = 40 (\frac{23}{80}) - 3 = 11.5 - 3 = 8.5 miles

Correct answer: D. $8.5$
8.5 (Choice D). Since 8.5 is squarely between 5 and 10 miles, our initial assumption that they meet in the middle segment is correct, verifying the mathematical model.

Adapted Problems to Practice

1 / 10

What Formulas Do You Need for This Problem?

Distance, Speed, and Time:

d = v \times t

d is distance, v is constant speed or velocity, and t is time. Used to determine how long each car spends in a segment and how far a car travels within a specific time frame.

Midpoint Formula:

\frac{a + b}{2}

a and b are the coordinates of two points on a number line. When two objects travel toward each other at the exact same speed, they cover equal distances and will meet exactly at the mathematical midpoint of their starting positions.

Key Takeaway: Remember This Strategy for Next Time

A motion problem where speed limits change at specific locations or times.

Try: Break the journey into stages and find out where every object is at the exact moment a new stage begins.

Why: Synchronizing their positions to a single moment in time (like the 15-minute mark) turns a complex, multi-speed problem into a simple constant-speed problem.

Two objects moving towards each other at the exact same speed.

Try: Skip calculating the remaining time and simply average their current positions to find the meeting point.

Why: Equal speeds mean they will cover identical distances before crossing paths, making their meeting point exactly halfway between them.

Solving a piecewise math problem using continuous algebraic equations.

Try: Make an educated guess about which interval the solution lies in, construct the equations for that specific interval, and solve.

Why: If the calculated answer falls inside the assumed interval, the assumption is mathematically verified. If it falls outside, you simply know to test the adjacent interval.

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2025 AMC 8 Problem 19

In This Page:

2025 AMC8 Problem 19