When two people are traveling toward each other, we can add their rates together. So here D = the distance between them, R = their rates added together, and T = the time that each of them has traveled when they meet. I'll call A the rate of person A and B the rate of person B.
D = (A + B)T
If R is .4 from where A starts, then .4D = AT and .6D = BT. Let's find A in terms of B, as we want the distance from A's starting point.
.6D = 1.5AT = BT, or
1.5A = B, or
3A = 2B, or
A = (2/3)B.
So A travels at 2/3 the rate of B.
For the next step, I'll assume that "between the two points" means "between points P and Q".
The first time the two people meet, they have together traveled a distance of D. But the second time they meet, they have traveled another 2D - since they must go to points P and Q, then back again to some other point on the line. Each additional time adds another 2D, so for four meetings we have a total of 7D.
If they travel 7D, and A's rate is 2/3 that of D, A travels 2/5 of the distance and B travels 3/5 of the distance. (Their combined rate is 2 + 3, so A does 2/(2+3) and B does 3/(2+3).) So A travels 2/5 of 7D, or 2.8D. 2D is a roundtrip from P back to P, so the extra .8 leaves him .8D from point P upon the fourth meeting.
Really cool problem!
