2004 AMC 10A Problem 17

1. They run toward each other, so the distance between them shrinks by the sum of what they both run.
2. Before the first meeting. They start on opposite sides of the track, half a lap apart. So together they run half a lap.
3. Between the meetings. Right after meeting they are side by side, running apart. To meet again they must together cover the whole lap. So together they run one full lap.
4. The second stretch is twice as long in combined distance as the first. Their speeds never change, so it takes twice as long, and each girl runs twice as far as she did before.
5. Brenda ran $100$ meters in the first stretch, so she runs $200$ meters in the second.
6. In the second stretch Sally runs $150$ meters. Together they cover one lap:
$$\text{lap} = 200 + 150 = 350.$$
7. Check with the first stretch: half a lap is $175$, so Sally ran $75$ while Brenda ran $100$. In the second stretch the numbers are $150$ and $200$, the same ratio $3 : 4$. Good.

The answer is $\boxed{\textbf{(C)}\ 350}$.

When two people run toward each other, they meet when their distances add up to the gap between them. At steady speeds, each person's distance grows at the same rate as time, so comparing two time periods compares all the distances at once. You never need to know the actual speeds.

Call the lap length $L$ and use the fact that the ratio of their distances is the same in both stretches.

1. First stretch: Brenda runs $100$ and Sally runs $\tfrac{L}{2} - 100$.
2. Second stretch: Sally runs $150$ and Brenda runs $L - 150$.
3. Brenda's distance divided by Sally's distance must match:
$$\frac{100}{\tfrac{L}{2} - 100} = \frac{L - 150}{150}.$$
4. Cross-multiply: $15000 = \left(\tfrac{L}{2} - 100\right)(L - 150) = \tfrac{L^2}{2} - 175L + 15000$. So $\tfrac{L^2}{2} = 175L$, and $L = 350$.

Because they run toward each other, the distances they cover add up to the gap closed. From the start to the first meeting the gap is half the track, $\tfrac{L}{2}$. From the first meeting to the second, they must together cover a full lap, $L$.

The second stretch requires twice the combined distance of the first, so at constant speeds it takes twice as long, and each girl runs twice as far as she did in the first stretch. Brenda ran $100$ meters before the first meeting, so she runs $200$ meters between the meetings. Sally runs $150$ meters in that same stretch, and together they cover one lap:
$$L = 200 + 150 = 350.$$

The answer is $\boxed{\textbf{(C)}\ 350}$.