mvictor wrote:

I don't get it...

anyone can explain, maybe with graphs how to solve it?

Do you need an explanation for Gracie's solution?

If you need an explanation for Gracie's solution, since I wanted to know how her solution ticked, I quickly analyzed it from the ground up:

The first/long section is x, and the second/narrow section is y. The point about 2/3 through is where A and B met. A spent 24 minutes in section y, and B spent 54 minutes in section x.

P |-----------------------|------------| Q

1. Let's state the central idea Gracie's solution is built around: the distance A has to travel is the same distance B has to travel. Please keep this in the back of your mind for the next few steps.

2. For A, we know the time he spent in section y but not section x. As such, we can say that A spent \(\frac{time_x}{{time_x + 24}}\)% of his travel time covering section x, and \(\frac{24}{{time_x + 24}}\)% of his travel time covering section y.

3. For B, we know the time he spent in section x but not section y. As such, we can say that B spent \(\frac{54}{{time_y + 54}}\)% of his travel time covering section x, and \(\frac{time_y}{{time_y + 54}}\)% of his travel time covering section y.

4. Since A and B are traveling at a constant speed (this is an assumption- be extremely careful with assumptions), we know that both will share the same percent of their time in each section. As such, if we select a section, the amount of time each party spent in that section will be equal. Let's use section x going forward.

5. From (4), we have \(\frac{t}{{t + 24}} = \frac{54}{{t + 54}}\)

6. \(t * (t + 54) = 54 * (t + 24)\)

7. \(t^2 + 54t = 54t + 54*24\)

8. \(t^2 = 1296\)

9. \(t = 36\)

10. Since A spent t minutes traveling before meeting B, and t = 36, our answer must be 10:36, or B.

If anyone sees any errors in my reasoning, please let me know.