Problem 157 Solution

1. t=(w-700)*r
2. 3t=(2w-300)*r
   
   This step is the humdinger. If you draw the pathes of both ships you'll see that after the first meeting point the combined distance covered was the width of the river. At the second meeting the combined distance traveled was three times the width of the river. So, at the second meeting time, both boats traveled three times as far compared to the first meeting.
   
3. Equate 3t in equations (1) and (2): 3*(w-700)*r = (2w-300)*r
4. 3*(w-700) = (2w-300)
5. 3w-2100 = 2w-300
6. w=1800

w=width of river
s₁=speed of first boat
s₂=speed of second boat
t₁=time until boats meet the first time
t₂=time until boats meet the second time

I shall arbitrarily say that both boats are 700 yards from the shore of original of the first boat at time t₁.

The following equations can be inferred from the information given:

(1) w=t₁*(s₁+s₂)

(2) s₁*t₁=700

(3) w+300=s₁*t₂

(4) 2w-300=s₂*t₂

Adding equations (1) and (2) we get:

(5) 3w=t₂*(s₁+s₂)

Substituting s₁+s₂=w/t₁ from equation (1) we get:

3w=t₂*r/t₁

3w*t₁=r*t₂

(6) 3*t₁=t₂

Combining equations (3) and (6):

(7) w+300=3*s₁*t₁

Combining equations (2) and (7):

w+300=2100

w=1800

This problem was asked in the 'Ask Marilyn' column in the November 19, 2000, issue of Parade magazine.