2023 AMC 10A Problem 1

1. Look at one hour first. Alicia rides $18$ miles toward Beth and Beth rides $12$ miles toward Alicia. So the gap between them shrinks by $18 + 12 = 30$ miles every hour.
2. The gap starts at $45$ miles. It takes
$$\frac{45}{30} = 1.5 \text{ hours}$$
for the gap to close, so they meet after $1.5$ hours.
3. The question asks for the distance from City $A$. Alicia starts at $A$, so we need how far she rode: $18 \times 1.5 = 27$ miles.
4. Check with Beth: she rides $12 \times 1.5 = 18$ miles, and $27 + 18 = 45$, the full distance.

The answer is $\boxed{\textbf{(E)}\ 27}$.

When two people move toward each other, you can pretend one of them stands still while the other moves at the two speeds added together. That gives the meeting time, and then distance = speed × time for each rider. The last step is to reread the question: it asks for the distance from $A$, which is Alicia's distance, not Beth's.

Both ride for the same amount of time, so the distances they cover are in the same ratio as their speeds, $18 : 12 = 3 : 2$. Split $45$ miles into $5$ equal parts of $9$ miles. Alicia gets $3$ parts, which is $27$ miles.