It takes a motorboat 1 1/3 hours to go 20 miles downstream and 2 2/9 hours to return.Find the rate of the current andthe rate of the boat in still water.
let rate of boat in still water be x mph
let the rate of the current be y mph
(4/3)(x+y) = 20
times 3
4x + 4y = 60
divide by 4
x + y = 15
(20/9)(x-y) = 20
times 9
20x - 20y = 180
divide by 20
x - y = 9
add them
2x = 24
x = 12
in x+y = 15 , if x = 12, then y = 3
speed of boat in still water: 12 mph
speed of current : 3 mph
Verify my answer by testing it for the given conditions.
To find the rate of the current and the rate of the boat in still water, we can set up a system of equations based on the given information.
Let's assume that the rate of the boat in still water is represented by "b" (in miles per hour) and the rate of the current is represented by "c" (also in miles per hour).
When the boat is going downstream (with the current), the effective rate of the boat is increased by the rate of the current, so the boat's speed is (b + c) miles per hour. It takes 1 1/3 hours (or 4/3 hours) to travel 20 miles downstream, so we can set up the equation:
(1) (b + c) * (4/3) = 20
Similarly, when the boat is going upstream (against the current), the effective rate of the boat is decreased by the rate of the current, so the boat's speed is (b - c) miles per hour. It takes 2 2/9 hours (or 20/9 hours) to travel back the same 20 miles upstream, so we have another equation:
(2) (b - c) * (20/9) = 20
Now, we can solve this system of equations to find the values of "b" and "c".
First, let's simplify equation (1) by multiplying both sides by 3/4:
(b + c) = 20 * (3/4) = 15
Similarly, let's simplify equation (2) by multiplying both sides by 9/20:
(b - c) = 20 * (9/20) = 9
Now, we have a system of equations:
(b + c) = 15 ---(3)
(b - c) = 9 ---(4)
To find the values of "b" and "c", we can add equations (3) and (4):
(b + c) + (b - c) = 15 + 9
2b = 24
b = 12
Substituting the value of "b" back into equation (3), we can solve for "c":
12 + c = 15
c = 15 - 12
c = 3
So, the rate of the boat in still water is 12 miles per hour, and the rate of the current is 3 miles per hour.