2025 AMC 12A Problem 1

Question

Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at

1{:}30,

traveling due north at a steady

8

miles per hour. Betsy leaves on her bicycle from the same point at

2{:}30,

traveling due east at a steady

12

miles per hour. At what time will they be exactly the same distance from their common starting point?

3{:}30

3{:}45

4{:}00

4{:}15

4{:}30

Measure time from

1{:}30

; Betsy has been riding one hour less than Andy

If Andy has ridden

t

hours, set

8t = 12(t-1)

Let

t

be the number of hours since

1{:}30.

Andy has traveled

8t

miles north, and Betsy, who started an hour later, has traveled

12(t-1)

miles east.

Setting the distances equal,

8t = 12(t-1),

so

4t = 12

and

t = 3.

Three hours after

1{:}30

is

4{:}30.

Thus, the correct answer is E.

Problem 1 in other years