To solve this problem, let's assign variables to the unknowns.
Let's say the speed on the way out is "x" miles per hour (mph).
Since the speed on the return trip is 3 mph slower than the speed on the way out, the speed on the return trip can be represented as "x - 3" mph.
Now, we can use the formula: Speed = Distance / Time to find the time for each leg of the trip.
On the way to the island, the boat has to travel 60 miles at a speed of "x" mph.
So, the time taken for this leg of the trip is: Time = 60 miles / x mph, which can also be written as 60/x hours.
Similarly, on the way back, the boat has to travel 60 miles at a speed of "x - 3" mph.
So, the time taken for this leg of the trip is: Time = 60 miles / (x - 3) mph, which can also be written as 60/(x - 3) hours.
According to the given information, the total time taken for the trip is 9 hours.
Therefore, the equation can be set up as:
60/x + 60/(x - 3) = 9
To solve this equation, we need to find the values of "x" that satisfy it.
One way to solve it is by multiplying every term of the equation by the least common multiple (LCM) of the denominators of the fractions, which in this case is "x(x - 3)".
By doing so, the equation becomes:
60(x - 3) + 60x = 9x(x - 3)
Now, simplify and rearrange the equation:
60x - 180 + 60x = 9x^2 - 27x
Combine like terms:
120x - 180 = 9x^2 - 27x
Rearrange the equation to set it equal to zero:
9x^2 - 147x + 180 = 0
Now, we can either factor or use the quadratic formula to solve this equation.
By factoring, it becomes:
(3x - 4)(3x - 45) = 0
Setting each factor equal to zero:
3x - 4 = 0 or 3x - 45 = 0
Solving for "x" in each equation:
3x = 4 or 3x = 45
x = 4/3 or x = 45/3
x = 4/3 or x = 15
Since the speed of the boat cannot be negative, we can disregard the solution x = 4/3.
Therefore, the speed on the way out is x = 15 mph, and the speed on the return trip is (x - 3) = 12 mph.