PSAT Math Multiple-Choice Question 607: Answer and Explanation

Question: 607

Towns A and B are 200 miles apart. Caitlin starts driving from Town A to Town B at 3 P.M. at a rate of 30 miles per hour. Hannah starts driving from Town B to Town A at 4 P.M. on the same day at a rate of 40 miles per hour. At what time will they meet (to the nearest minute)?

• A. 3:42 P.M.
• B. 5:29 P.M.
• C. 6:26 P.M.
• D. 7:32 P.M.

Correct Answer: C

Explanation:

(C) The women will meet when their positions are equal, so we need to come up with equations to model each of their positions. First, notice that Caitlin leaves a full hour before Hannah. In that first hour, she'll travel 30 miles since she's traveling at 30 mph. Therefore, the women start out 170 miles apart at 4:00 P.M.

Let's say that Caitlin starts at position 0, while Hannah starts at position 170. Caitlin is ­moving toward 170, so she's moving in the positive direction at 30 mph. Keeping in mind that distance = rate × time, Caitlin's position, s, can then be described as:

s = 0 + 30t = 30t

On the other hand, Hannah is traveling from position 170 toward position 0, so she's traveling in the negative direction. Therefore, her position can be described as:

s = 170 - 40t

In order to solve for t, we must set the women's positions equal to one another:

30t = 170 - 40t

Adding 40t to both sides results in:

70t = 170

Dividing by 70 tells us:

t = 2.429

Because our rates were in miles per hour, this time is in hours. Therefore, it takes the women two full hours and a fraction of a third hour, so they meet sometime between 6 and 7. This is enough information to narrow down the solution to choice (C).

To find the exact time, we can figure out how many minutes 0.429 hours is by multiplying 0.429 hours by 60 minutes/hour. 0.429(60) = 25.74 minutes. Therefore, Caitlin and Hannah meet 2 hours and 26 minutes after the time Hannah started traveling, 4:00 P.M. So the women arrive at the same place at 6:26 P.M.