Betty and Ann are walking around a rectangular park with dimensions $600 \mathrm{~m}$ by $400 \mathrm{~m}$, as shown. They both begin at the top left corner of the park and walk at constant but different speeds. Betty walks in a clockwise direction and Ann walks in a counterclockwise direction. Points $P, Q, R, S, T$ divide the bottom edge of the park into six segments of equal length. When Betty and Ann meet for the first time, they are between $Q$ and $R$. Which

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-030.jpg?height=258&width=414&top_left_y=543&top_left_x=1300)
of the following could be the ratio of Betty's speed to Ann's speed?
(A) $5: 3$
(B) $9: 4$
(C) $11: 6$
(D) $12: 5$
(E) $17: 7$

## Part C: Each correct answer is worth 8.
Points $P, Q, R, S, T$ divide the bottom edge of the park into six segments of equal length, each of which has length $600 \div 6=100 \mathrm{~m}$.

If Betty and Ann had met for the first time at point $Q$, then Betty would have walked a total distance of $600+400+4 \times 100=1400 \mathrm{~m}$ and Ann would have walked a total distance of $400+2 \times 100=600 \mathrm{~m}$.

When they meet, the time that Betty has been walking is equal to the time that Ann has been walking and so the ratio of Betty's speed to Ann's speed is equal to the ratio of the distance that Betty has walked to the distance that Ann has walked.

That is, if they had met for the first time at point $Q$, the ratio of their speed's would be $1400: 600$ or $14: 6$ or $7: 3$.

Similarly, if Betty and Ann had met for the first time at point $R$, then Betty would have walked a total distance of $600+400+3 \times 100=1300 \mathrm{~m}$ and Ann would have walked a total distance of $400+3 \times 100=700 \mathrm{~m}$.

In this case, the ratio of their speed's would be $1300: 700$ or $13: 7$.

When Betty and Ann actually meet for the first time, they are between $Q$ and $R$.

Thus Betty has walked less distance than she would have had they met at $Q$ and more distance than she would have had they met at $R$.

That is, the ratio of Betty's speed to Ann's speed must be less than $7: 3$ and greater than $13: 7$.

We must determine which of the five given answers is a ratio that is less than $7: 3$ and greater than $13: 7$.

One way to do this is to convert each ratio into a mixed fraction.

That is, we must determine which of the five answers is less than $7: 3=\frac{7}{3}=2 \frac{1}{3}$ and greater than $13: 7=\frac{13}{7}=1 \frac{6}{7}$.

Converting the answers, we get $\frac{5}{3}=1 \frac{2}{3}, \frac{9}{4}=2 \frac{1}{4}, \frac{11}{6}=1 \frac{5}{6}, \frac{12}{5}=2 \frac{2}{5}$, and $\frac{17}{7}=2 \frac{3}{7}$.

Of the five given answers, the only fraction that is less than $2 \frac{1}{3}$ and greater than $1 \frac{6}{7}$ is $2 \frac{1}{4}$.

If Betty and Ann meet for the first time between $Q$ and $R$, then the ratio of Betty's speed to Ann's speed could be $9: 4$.

ANSWER: (B)

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