a boat travels 60 miles to an island and 60 miles back again. The return trip is 3 mph slower than the speed on the way out. The total time of the trip took 9 hours. Find the speed going to the island and the speed of the return trip.

let the rate took when travelling to the island be r. So,the rate for return trip is r-3.

time taken to the island=60/r
time taken back from island=60/r-3
60/r+60/r-3 =9=total time
find L.C.D ,
so after solving,u get r=15mph
speed to the island=15mph
speed from island =15-3=12mph
u can check ur answer.

To solve this problem, we can use the formula:

Time = Distance / Speed

Let's assign variables to the speed going to the island and the speed of the return trip.

Let's say the speed going to the island is x mph.

Since the return trip is 3 mph slower, the speed of the return trip would be (x - 3) mph.

According to the problem, the boat travels 60 miles to the island and 60 miles back, so the total distance covered is 120 miles (60 + 60).

Now, let's calculate the time it takes to go to the island and back.

The time taken to go to the island is:

Time going = Distance / Speed = 60 / x hours

The time taken to return is:

Time back = Distance / Speed = 60 / (x - 3) hours

According to the problem, the total time for the trip is 9 hours.

So we can write the equation:

Time going + Time back = 9

Substituting the respective expressions for time:

60 / x + 60 / (x - 3) = 9

Now, let's solve this equation to find the value of x:

Multiplying the equation through by x(x - 3) to clear the denominators:

60(x - 3) + 60x = 9x(x - 3)

Simplifying the equation:

60x - 180 + 60x = 9x² - 27x

120x - 180 = 9x² - 27x

Rearranging the equation:

9x² - 147x + 180 = 0

This is a quadratic equation. We can now solve it by factoring or by using the quadratic formula. Factoring might be a bit complex in this case, so I will use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, a = 9, b = -147, and c = 180. Plugging these values into the quadratic formula, we get:

x = (-(-147) ± √((-147)² - 4*9*180)) / (2*9)

Simplifying further:

x = (147 ± √(21609 - 6480)) / 18

x = (147 ± √15129) / 18

x = (147 ± 123) / 18

Now, let's calculate both possible values of x:

For x = (147 + 123) / 18 = 270 / 18 = 15

For x = (147 - 123) / 18 = 24 / 18 = 4/3

Since the speed of the boat cannot be negative or less than the speed of walking, we can conclude that the speed going to the island is 15 mph.

Therefore, the speed of the return trip would be 15 - 3 = 12 mph.