Question 290676
There are two cities, A and B, on the opposite sides of a big lake. 
Two motorboats leave at exactly the same time: one leaves A going directly to B,
 while the other leaves B going directly to A. They each travel at constant speed(s).

The boats meet for the first time 500 meters from A.
 After each boat reaches its destination it immediately turns around and travels
 directly back to its starting point. 
Their second meeting takes place 300 meters from B.
How wide is the lake between A and B?
 What is the ratio between the speeds of the two boats?
:
A >--------------------500*---------------< B
Let d = dist from A to B towns
Meet 1st time
A boat travels 500m, 
B boat travels (d-500)
:
:
B >--------------------*300---------------< A
Meet the 2nd time
A boat travels (d-500) + 300 = d-200
B boat travels 500 + (d-300) = d+200
:
Ratio of distance traveled by the two boats remain the same, therefore:
{{{500/(d-500)}}} = {{{(d-200)/(d+200)}}}
Cross multiply
500(d+200) = (d-500)*(d-200)
500d + 100000 = d^2 - 700d + 100000
0 = d^2 - 700d - 500d + 100000 - 100000
d^2 - 1200d  = 0
d(d - 1200) = 0
d = 0
d = 1200 meters from town A to town B
:
Find the ratio of the boats' speed using {{{500/(d-500)}}}
Replace d with 1500
{{{500/(1200-500)}}} = {{{500/700}}} = 5:7, ratio of the speeds
:
:
Check solution by finding the ratio of {{{(d-200)/(d+200)}}}
Replace d with 1200
{{{(1200-200)/(1200+200)}}} = {{{1000/1400}}} = {{{10/14}}} = 5:7 ratio