The Jones family took a 24-mile canoe ride down a river in two hours. After lunch the return trip back up the river took four hours. Find the rate of the canoe in still water.
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To find the rate of the canoe in still water, we need to consider the relative velocities. Let's assume the speed of the river is x miles per hour and the speed of the canoe in still water is y miles per hour.
During the downstream trip, the speed of the canoe relative to the river is the sum of the speed of the river and the speed of the canoe in still water. Therefore, the speed of the canoe during the downstream trip is (y + x) miles per hour.
During the upstream trip, the speed of the canoe relative to the river is the difference between the speed of the river and the speed of the canoe in still water. Therefore, the speed of the canoe during the upstream trip is (x - y) miles per hour.
Given that the downstream trip took 2 hours to cover 24 miles, we have the equation:
(y + x) * 2 = 24
Simplifying this equation, we get:
y + x = 12
Given that the upstream trip took 4 hours to cover 24 miles, we have the equation:
(x - y) * 4 = 24
Simplifying this equation, we get:
x - y = 6
Now, we can solve this system of equations to find the values of x and y.
Adding the two equations together, we get:
(y + x) + (x - y) = 12 + 6
Simplifying, we have:
2x = 18
Dividing both sides by 2, we find:
x = 9
Substituting the value of x back into one of the original equations, we find:
y + 9 = 12
Subtracting 9 from both sides, we get:
y = 3
Therefore, the rate of the canoe in still water is 3 miles per hour.
To find the rate of the canoe in still water, we need to determine its speed relative to the water. Let's denote the rate of the canoe in still water as "x" and the rate of the current of the river as "y".
During the downstream trip, the canoe's speed will be the sum of its own rate and the current's rate: x + y. Given that the family covered a distance of 24 miles in 2 hours, we can use the formula: distance = rate * time.
So, for the downstream trip, we have the equation: 24 = (x + y) * 2.
Now, for the upstream trip, the canoe's speed will be its own rate minus the current's rate: x - y. Since the family covered the same distance of 24 miles on the return trip, we can use a similar equation: 24 = (x - y) * 4.
Now, we have a system of two equations with two variables. We can solve this system to find the values of x and y.
Let's solve it:
From the first equation, we have: 24 = 2x + 2y (divide both sides by 2).
From the second equation, we have: 24 = 4x - 4y (divide both sides by 4).
Rearranging the first equation, we get: x = 12 - y.
Substituting this value of x into the second equation, we get: 24 = 4(12 - y) - 4y.
Expanding, we have: 24 = 48 - 4y - 4y.
Combining like terms, we get: 24 = 48 - 8y.
Move the term with y to the left side: 8y = 48 - 24.
Simplifying, we get: 8y = 24.
Divide both sides by 8: y = 3.
Now, substitute this value of y back into the equation x = 12 - y: x = 12 - 3 = 9.
Therefore, the rate of the canoe in still water is 9 mph.
c + r = 24/2
c - r = 24/4
add the equations (to eliminate r)