A left P for Q at 10:00 am. At the same time B left Q for P.

hi here we can use , in Time and distance if both trains start at the same time
and travelled in opposite directions.
when you want to find time of both trains to meet,
we can multiply both the times and take square root of it.

Here one train reached at 54mins and another trains reached at 24 mins.

Multiply both times 54*24=1296.
square root of 1296 is 36mins.

from 1296 we dont need square root also because unit digit ends in 6 . only the number unit ends in 6 only have that square.

We can use and apply logic to save time in GMAT.

So option B is correct.

By Time you mean, The time they took after meeting eachother? Does this hold true always?

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.

the easiest way of doing this question :

meeting time T= sqrt of 24x36 =36
so, they will meet at 10.36 min form start.

Thanks, your explanation does clear things up. I did not understand the reasoning behind the equation written by Gracie.
I understood that:
1. A was travelling faster than B
2. point of intersection is closer to Q

yet, I did not see how to solve further this point.
Thanks one more time..would definitely remember this case. Kudos for you

Would be great if someone can explain how this method works

lets take D traveled by A ....and D1 traveled by B....Time traveled is " T "
D/T = S(a) =D = T*S(a)-----------1
D1/T = S(b) =D1 = T*S(b)----------2

now A need 24 min to travel what B has already covered ....(as they meet in a common point)
So D1/24 = S(a)...
substituting 2 here ...
T*S(b)/S(a) = 24 ------------3

similarly for B ...
D/54 = S(b)
substituting 1 here
T*S(a)/S(b) = 54 ------------4

if we multiply 3 and 4 we will get ..
T^2 = 24 * 54
T = 36

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