a motorboat traveling with the curent went 68 miles in 3hrs and 24mins. against the current, it took 4hrs and 15 mins to travel the same distance. find the rate of the motorboat in calm water and the rate of the current.

To find the rate of the motorboat in calm water and the rate of the current, we can use the concept of relative velocity.

Let's assume the rate of the motorboat in calm water is represented by 'b' (in miles per hour), and the rate of the current is represented by 'c' (in miles per hour).

When the motorboat is traveling with the current, the effective speed is equal to the sum of the rate of the motorboat and the rate of the current. Therefore, the speed with the current can be expressed as (b + c) miles per hour.

Similarly, when the motorboat is traveling against the current, the effective speed is the difference between the rate of the motorboat and the rate of the current. The speed against the current can be expressed as (b - c) miles per hour.

We are given that the distance traveled in both cases is 68 miles. Let's convert the time values to hours for consistency.

In the first scenario (with the current), the motorboat traveled the distance of 68 miles in 3 hours and 24 minutes, which is 3.4 hours.

Therefore, we have the equation: (b + c) × 3.4 = 68

In the second scenario (against the current), the motorboat took 4 hours and 15 minutes, which is 4.25 hours, to travel the same distance of 68 miles.

Thus, we have the equation: (b - c) × 4.25 = 68

Now, we have two equations:

1) (b + c) × 3.4 = 68
2) (b - c) × 4.25 = 68

To solve these equations, we can use a method called substitution or elimination. Let's use substitution.

From equation 1, we can rewrite it as:
b + c = 20 (dividing both sides by 3.4)

Now, we can substitute the value of (b + c) in equation 2:
(b - c) × 4.25 = 68
(b - 20) × 4.25 = 68 (substituting b + c with 20)

Expanding the equation, we get:
4.25b - 85 = 68

Next, let's isolate the variable 'b':
4.25b = 68 + 85
4.25b = 153

Finally, we can solve for 'b' by dividing both sides by 4.25:
b = 153 ÷ 4.25
b = 36

Now that we have the value of 'b', we can substitute it back into equation 1 to find the value of 'c':
36 + c = 20

Subtracting 36 from both sides, we get:
c = 20 - 36
c = -16

The rate of the motorboat in calm water is 36 mph, and the rate of the current is -16 mph.
Note: The negative sign for the current speed indicates that the current is flowing in the opposite direction to the motorboat.