2025 AMC 12A Problems/Problem 1

The following problem is from both the 2025 AMC 10A #1 and 2025 AMC 12A #1, so both problems redirect to this page.

Problem

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at , traveling due north at an steady mile per hour. Betsy leaves on her bicycle from the same point at , traveling due east at a steady miles per hour. At what time will they be exactly the same distance from their common starting point?

Solution 1

We can see that at , Andy will be miles ahead. For every hour that they both travel, Betsy will gain miles on Andy. Therefore, it will take more hours for Betsy to catch up, and they will be at the same point at .

~vinceS

~minor LaTeX edits by zoyashaikh

Solution 2

You can look at this problem from both Andy's PoV and Betsy's PoV

Andys(A). Let be the number of hours after Andy starts. Then Andy has been traveling for hours, so he has gone miles, and Betsy has traveled miles since she started 1 hour later. Setting these equal, we get , which simplifies to , so and . Thus, Betsy catches up 3 hours after Andy starts. Since Andy started at 1:30, the catch-up time is . Answer: .

Betsy(B). hours after Betsy left, Andy has traveled miles, and Betsy has traveled miles. We are told these are equal, so . Solving, we get , so Andy and Betsy will be exactly the same distance from their common starting point two hours after Betsy leaves, or .

~kapiltheangel (2A) ~mithu542 (2B)

Solution 3 (bash)

We can use all the answer choices that we are given.

Let's use casework for each of the answers:

At 3:30, Andy will have gone miles. Betsy will have gone miles. At 3:45, we can just add the distance each of them goes in 1/4 hours. Therefore, Andy goes 18 miles, and Betsy goes 15 miles. At 4:00, we see from the same logic that Andy has gone 20 miles, and Betsy has gone 18 miles. At 4:15 we see that Andy has gone 22 miles, and Betsy has gone 21 miles. At E, 4:30, we see that both Andy and Betsy have gone 24 miles.

Now we see that is the correct answer.

~vgarg

Solution 4

We can see that Betsy travels 1 hour after Andy started. We have . Now we can find the total time Andy has taken once Betsy catches up:

So the answer is

OR

You can simply write it out. After one hour, where will Andy be located and where will Betsy be located based on their times?

~Boywithnuke(Goal: 10 followers)

~minor edits by ChickensEatGrass

~minor edits by RISHADA

Solution 5

The distance Andy travels can be represented by and Betsy with the equation The solution to this is so the answer is hours after therefore,the solution is .

~minor LaTeX edits by zoyashaikh

~minor edits by i_am_not_suk_at_math (saharshdevaraju 21:17, 6 November 2025 (EST)saharshdevaraju)

Solution 6 - Very Simple

Using the information that they move at a constant rate, we can create a small table based on their place (in miles) by hours. Andy moves at 8 miles an hour, as seen below, and Betsy moves at 12 miles per hour after one hour after Andy begins, as shown in the table. \begin{array}{c|c|c} \text{Time/Subject} & \text{Andy} & \text{Betsy} \\[6pt] \hline \text{1:30} & 0 & 0 \\[6pt] \text{2:30} & 8 & 0 \\[6pt] \text{3:30} & 16 & 12 \\[6pt] \text{4:30} & 24 & 24 \end{array}

Based on this very simple table, we can conclude that they will be the exact same difference from their mutual starting point at .

~i_am_not_suk_at_math (saharshdevaraju 21:17, 6 November 2025 (EST)saharshdevaraju)

~minor LaTeX edits by vinceS

~minor minor edits by A-V-N-I

Solution 7

OVERKILL using Calculus:

We define a function that represents the in their distances: Substituting the distance functions:

The time when their distances are equal is the time when .

We take the first derivative of with respect to time :

Since , the difference function is strictly increasing. This confirms that the point where is a \textbf{unique root}, meaning there is only one moment in time when .

We set the difference function to zero:

This calculus verification confirms that the unique solution occurs at hours.

The time is hours after :

-apex304

On

Video Solution (Fast and Easy)

https://youtu.be/fjpeOuUMOyc?si=ROuzlwgz7UByUx_9 ~ Pi Academy

Video Solution by Power Solve

https://www.youtube.com/watch?v=QBn439idcPo

Chinese Video Solution

https://www.bilibili.com/video/BV1852uBoE8K/

~metrixgo

Video Solution (Intuitive, Quick Explanation!)

https://youtu.be/c-UDo53KwFU

~ Education, the Study of Everything

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=dAeyV60Hu5c

Video Solution by Daily Dose of Math 🔥🔥🔥

https://youtu.be/LN5ofIcs1kY

~Thesmartgreekmathdude

Video Solution

https://youtu.be/gWSZeCKrOfU

~MK

Video solution

https://youtu.be/l1RY_C20Q2M

Easy Solution

https://www.youtube.com/watch?v=kHwBHvvvTbY

See Also

These problems are copyrighted © by the Mathematical Association of America.