Alright, so this seems to be one of those problems that gives a lot of trouble, but if you analyze it carefully it should be fine.
First of all, note that it makes sense that Elaine should have a faster rate, otherwise she can never reach Wayne.
Now, in the first five minutes, Wayne has covered a total distance of: 8*60*5 = 2400 ft.
(The reason is that 5 min = 5*60 = 300 sec. Each sec Wayne is covering a distance of 8 ft, so after 300 sec, it would be 300*8 = 2400 ft).
Now, after 5 min, Elaine notices that Wayne has forgot to take his lunch, and runs after him. The problem is that Wayne would continue to walk so we have to take that into account as well. So first let's summarize what we have after 5 min:
2400 ft unknown distance
covered
Elaine................................................................Wayne..............................final position (where they will meet).
(point A) (point B) (point C)
Now note that the time it takes Elaine to get from point A to point C (where she will meet Wayne), should be the same time it takes Wayne to go from point B to point C.
It will take Elaine 2400/12 = 200 s to get from point A to point B, and assuming the distance between B and C is equal to X (unknown), the time it takes her to get from point B to point C would be X/12. So the total time for her to get from point A to C would be 200 + (X/12) s
Now, the time it Wayne to move from point B to C would be X/8 (since according to the question, Wayne moves at the rate of 8 ft/s).
Therefore we can write:
200 + (X/12) = (X/15)
solve this and you will get X = 4800 ft.
So the total time that will take Elaine to reach Wayne would be: 200 S (time to get from A to B) + 4800/12 (time to get from point B to C) = 200 + 400 = 600 s or 600/60 = 10 min
Let's check!
Assuming 10 min is correct, after 10 min, Elaine would cover a distance of: 12*60*10 = 7200 ft
Also, Wayne has covered a distance of 2400 ft (initial position) + 4800 ft = 7200 ft
Since the final positions agree, we must have done it correctly!
Best,
Kaveh
