In the figure above, car A and car B simultaneously begin traveling around a circular park with area of 4π square miles. Both cars start from the same point, the START location shown in the figure and travel with constant speed rates until they meet. Car A travels counter-clockwise and car B travels clockwise. Where along the track will they meet? (1) Car B travels 30 mph faster than car A. (2) Car A travels twice slower than car B.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) Knowing the area of the park, we can find its radius (πr² = 4π square miles, so r = 2 miles). Knowing the radius, we can find the length of the road (the circumference, it equals 2πr = 4π miles). To visualize this question, think about a circle as just a line segment joined at its two ends. Let’s first cut the circle at the starting point and make it into a straight line. Notice that cars A and B are at opposite ends of the line traveling toward each other. A [ ______________________ ] B

Let’s denote the speed rate of car A by x mph and the speed rate of car B by y mph. When they meet, the whole distance will be the sum of the paths they have travelled. Since they’ve started simultaneously, then they both travelled for the same time. Let’s denote it by t hours. This gives us the following relation: x × t + y × t = 4π

Statement (1) gives us the fact that y = x + 30. The relation transforms into xt + (x + 30)t = 4π. Let’s plug two values to see if we get different results. If x = 5, then t = π/10. So the cars would travel 5π/10 = π/2 and 35π/10 = 7π/2 miles respectively. If x = 10, then t = 4π/50. So the cars would travel 40π/50 = 4π/5 and 160π/50 = 16π/5 miles respectively.

As wee see these results define different points along the track. Therefore statement (1) by itself is NOT sufficient. Statement (2) gives us the fact that y = 2x. The relation transforms into xt + 2xt = 4π. So xt = 4π/3. Therefore we know that car A travells 4π/3 miles and car B travels 8π/3 miles. That gives us the definite point along the track, no matter what the values of x and t are. So statement (2) by itself is sufficient. Statement (2) by itself is sufficient to answer the question, while statement (1) by itself is not. The correct answer is B. ---------- I think the answer is laid out well, but conceptually I am struggling to wrap my mind around the fact that the first statement is insufficient.

Regardless of if the two speeds are 40 mph and 10 mph or 50 mph and 20 mph, won't the cars meet at the same place? I understand that if one car is faster, then it will be farther along the route than the other car, but in this case if one car is faster, then the other car becomes faster accordingly (i.e., speed of car A = speed of car B + 30).

Any input you could provide to help me understand the error in my thinking here would be really helpful. Thank you in advance for your time and help.

Any input you could provide to help me understand the error in my thinking here would be really helpful.

Quote:

If x = 5, then t = π/10. So the cars would travel 5π/10 = π/2 and 35π/10 = 7π/2 miles respectively. If x = 10, then t = 4π/50. So the cars would travel 40π/50 = 4π/5 and 160π/50 = 16π/5 miles respectively.

So in the first case the cars would travel 0.5π and 3.5π miles respectively. In the second case they would travel 0.8π and 3.2π miles respectively.

These two cases define different meeting points on the line AB and therefore different points on the circle.

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