2023 AMC 10A Problem 1

Question

Cities

A

and

B

are

45

miles apart. Alicia lives in

A

and Beth lives in

B.

Alicia bikes towards

B

at

18

miles per hour. Leaving at the same time, Beth bikes toward

A

at

12

miles per hour. How many miles from City

A

will they be when they meet?

20

24

25

26

27

They ride toward each other, so their speeds add; find the time until they meet

The meeting time is

\frac{45}{18 + 12}

hours; multiply Alice’s speed by that time

They ride toward each other, so their speeds add. That closes the

45

-mile gap at

18 + 12 = 30

mph, and they meet after

\frac{45}{30} = 1.5

hours. Alice starts at

A,

so by then she’s gone

18 \cdot 1.5 = 27

miles. Thus, E is the correct answer.

Problem 1 in other years