Allen starts from X, goes to Y. At the same time Bob starts from Y and

IMO the answer must be C 126

as I tried to back solve this one C perfectly fit the statement above.

total distance 126km
in an hour both Allen and Bob covered 126km
that is 54+72=126

they meet for the first time Bob covered 18 more KM than Allen. 72-54=18.
so when they meet for the second time Bob covered 18*2= 36 more km

This is a good question. Definitely more towards the 700 range than the 600 range of the spectrum.

Rate * Time = Distance

Rate Allen =54
Rate Bob =72

They will be working together to cover the total distance from X to Y, so their combined rate, 54+72=126, will be used.

The key is figuring out how times Allen and Bob will collectively traverse the distance from X to Y.

It helps to conceptualize two people moving along this same pattern without factoring in their respective rates just yet.


X----------N----------Y

If the XY distance is drawn as above, the total distance can be represented as (X+N)+(N+Y). If both Allen and Bob meet at point N each time, during the first meeting one will have traveled X+N and the other will have traveled N+Y, for a total distance of (X+N) + (N+Y), or 1x the total distance.

As they pass the other, one will travel N+Y for the first time as the other travels X+N for the first time. Thus, they collectively make another combined trip of the full distance X to Y. They each turn around and cover the same ground they just traveled, for a third combined trip of the total distance. Thus, combined they total distance of 3x the distance from X to Y.

Set up a standard R*T=D equation with their combined rate and let the distance =3D to figure out the amount of time it takes for the guys to meet twice. Set everything in terms of D as we're trying to find out distance.

126* (3D/126)=3D. [T=D/R]

126*(D/42)=3D; Thus, the time it takes to meet twice can be represented as D/42.

Now use this representation of time and set the equations equal to each other with info given in the question.

72D/42=(54D/42)+36

18D/42=36

D=36(42)/18

D=2*42=84

Answer D

Well Actually OA = D

Here is my explanation
Let d be the distance between X and Y

A--> ---------------------- <---B
X---------------------------------Y

A traveled = 54
B traveled = 72

When they meet for 2nd time B = A + 36
B - A = 36 km

When they meet for the 2nd time then the total distance traveled by A & B combined would be = 3d --(i)

So if we consider total distance traveled by A & B = 54+72 = 126
126 =3d
d = 42

But the difference between B and A = 72 - 54 = 18
Given in the question is that the diff is 36 when they meet for the 2nd time.
In other words we can say double the one we calculated.

Thus Distance between X and Y = 42*2 = 84 kms

Hope you guys like my approach

1. At their first meeting, based on their respective speeds, the ratio of B's distance to A's distance is 4:3.
2. If there are 36 kms difference between their distances at the second meeting, there would have been
12 kms difference at the first, as the second meeting required twice the combined distance of the first.
3. 4x-3x=12➡x=12➡(7)(12)=84 kilometers.

How is it three times total distance (3d)?

Say distance between X and Y is D. When do they meet for the second time? When do they meet for the first time? When Allen is going to Y and Bob is going to X. They meet for the first time, somewhere in between X and Y. After that, Allen reaches Y and Bob reaches X. So both travel for distance D each.
Then, they meet second time when Allen is coming back from Y toward X and Bob is coming back from X toward Y. When they meet somewhere in the middle of X and Y again, they have together covered another D distance. So total they have covered distance 3D.

Originally:
X (Allen)________________________________________________________Y(Bob)
Distance covered together = 0

First Meet:
X ______________________(Allen)(Bob)__________________________________Y
Distance covered together = D

Reaching destination:
X (Bob)________________________________________________________Y(Allen)
Distance covered together = 2D

Second Meet:
X ______________________(Bob)(Allen)__________________________________Y
Distance covered together = 3D

Let 't' be the time taken by each of them.

Given, their rates are 72 and 54... and when they meet for the second time distance covered by B is 36 km more than by A.
Hence, 72t-54t = 36
or, t=2 hours

Also, total distance covered = 3*d
Distance covered by A is 54t and covered by B is 72t.
Hence, 72t + 54t = 126t = 126*2 = 3*d
d=84

\(D_b = D_a + 36\)

\(\frac{S_b}{t} = \frac{S_a}{t} + 36\)

\(\frac{72}{t} = \frac{54}{t} + 36\)

\(\frac{72}{t} - \frac{54}{t} = 36\)

\(\frac{(72t - 54t)}{t^{2}} = 36\)

\(18=36t\)

\(t=0.5\)

Total distance travelled by Bob and Allen = \(D_{ab} = \frac{S_{ab}}{t}\)
\(S_{ab}\) is the relative speed of Bob and Allen = 72 + 54 = 126 km/h
\(D_{ab}=\frac{126}{0.5}\)
\(D_{ab}=252\)

Bob and Allen together traversed the path XY three times until they met the second time.
So distance between X and Y = \(\frac{252}{3}=84km\)

I have a weird way of thinking about these problems that I've developed since Highschool:
The thought process basically goes like this:
Let B be the speed of Bob and A be the speed of Allen, let B be the distance between X and Y
B:A = 72: 54 = 4:3
Therefore the distance that Bob travels will always be 4/3 times the distance Allen travels:
By the time Allen meets Bob for the 2nd time, they have travelled a combined distance of 3D
therefore 3D/(7)*4 is the distance that Bob travelled and 3D/7*3 is the distance Allen travelled
The difference between their distances is: 3D/7 = 36, D = 84

We can let the distance between X and Y = d. In the total trip, we see that Allen covers a distance of d plus some more miles. Similarly, Bob also covers a distance of d, plus some more miles. Moreover, when they meet the second time, the sum of the surplus miles covered by Allen and Bob must also equal d; therefore the total distance that they cover is 3d.
We see that Bob travels 18 kmph faster than Allen, and, in order for Bob to travel 36 km more than Allen, they each must have traveled for 2 hours.

We also see that in 2 hours, Allen travels 54 x 2 = 108 km, and Bob travels 72 x 2 = 144 hm. We have argued that their total distance traveled must be three times the distance between X and Y. So we have:

108 + 144 = 3d

252 = 3d

84 = d

Answer: D

rate of B to A=72/54=4/3
if 7/7=combined distance traveled by A and B,
then 4/7-3/7=1/7=difference between distances traveled by A and B respectively
if d=distance between X and Y,
then 3d=combined distance of A and B at their second meeting
36 km=1/7*3d
d=84 km

Bob's rate - Allen's rate = 72-54 = 18 kph
Every hour Bob travels 18 more kilometers than Allen.
Implication:
For Bob to cover 36 more kilometers than Allen, Bob and Allen must each travel for 2 hours.

For Allen to meet Bob twice, Allen must travel from X to Y (meeting Bob along the way) and then partway back to X (meeting Bob again on the return trip).
In other words, Allen must travel the X-Y distance more than once but not quite twice.
In 2 hours Allen travels a total of 108 kilometers.
Thus:
Eliminate A and C, since each is greater than the 108k covered by Allen in 2 hours.
Eliminate E, since an X-Y distance of 42k implies that Allen travels the entire distance more than twice.

Similarly:
For Bob to meet Allen twice, Bob must travel from Y to X (meeting Allen along the way) and then partway back to Y (meeting Allen again on the return trip).
In other words, Bob must travel the Y-X distance more than once but not quite twice.
In 2 hours Bob travels a total of 144 kilometers.
Thus:
Eliminate B, since an X-Y distance of 72k implies that Bob travels the entire distance twice.