Given :
a paddle boat travels upstream = 12 miles
a paddle boat travels downstream = 20 miles
speed of the current = 2 miles per hour
Let the speed of the boat =x
Let the time taken = t
We know the distance for upstream D= (boat speed - current speed) * time
the distance for downstream D= (boat speed + current speed) * time
Plug in all the values and variables in the distance
For upstream 12 = (x-2)t, then [tex]t = \frac{12}{x-2}[/tex]
For downstream 20 = (x+2)t then [tex]t = \frac{20}{x+2}[/tex]
we got two equations for t, equate it and solve for x
[tex]\frac{12}{x-2} = \frac{20}{x+2}[/tex]
Cross multiply it
12(x+2) = 20(x-2)
12x + 24 = 20x - 40 (subtract 12x and add 40 on both sides)
64 = 8x
x= 8
the speed of the boat in still water = 8 miles per hour
We will take velocity of the boat Vb and velocity of the river Vr=2 mph
Distance D1=12 miles and distance D2=20 miles and t is the time.
First equation is (Vb-Vr) * t = D1 and the second is (Vb+Vr) * t = D2
When we replace data we have we get
(Vb-2) *t = 12 and (Vb+2) * t = 20
From the first equation t= 12/(Vb-2)
From the second equation t=20/(Vb+2)
If the left sides of equations are equal then the right sides are equal too,
and we get 12/(Vb-2)=20/(Vb+2) then we multiply crosswise and get
12*(Vb+2)=20*(Vb-2) => 12Vb+24=20Vb-40 => 20Vb-12Vb=40+24 =>
8Vb=64 => Vb=64/8=8mph Velocity ( or speed) of the boat is Vb= 8mph
Good luck!!!
