A boat travels 105 km going with the current, then 60 km going against the current, in 12 hours. It then travels 60 km going with the current, then 132 km going against the current, in 15 hours. If the distance between two harbors is 120 km, how long does it take for the boat to have a round trip?

Let's define some variables to represent the problem:
- Let \( b \) be the speed of the boat in still water (km/h).
- Let \( c \) be the speed of the current (km/h).

### First Set of Equations
In the first part of the journey:
- The boat travels 105 km with the current and 60 km against the current in 12 hours.
- With the current, the effective speed of the boat is \( b + c \).
- Against the current, the effective speed of the boat is \( b - c \).

Using the formula for time \( \text{time} = \frac{\text{distance}}{\text{speed}} \), we can write:

\[
\frac{105}{b + c} + \frac{60}{b - c} = 12
\]

### Second Set of Equations
In the second part of the journey:
- The boat travels 60 km with the current and 132 km against the current in 15 hours.
- Again, using the formula for time, we have:

\[
\frac{60}{b + c} + \frac{132}{b - c} = 15
\]

### System of Equations

Now we have two equations:
1. \( \frac{105}{b + c} + \frac{60}{b - c} = 12 \)
2. \( \frac{60}{b + c} + \frac{132}{b - c} = 15 \)

We can solve this system for \( b \) and \( c \).

### Solving the System

I'll proceed to solve these two equations and determine \( b \) and \( c \), then calculate the round trip time between the two harbors, which are 120 km apart.The solution to the system of equations is:
- \( b = \frac{27}{2} = 13.5 \) km/h (the speed of the boat in still water),
- \( c = \frac{3}{2} = 1.5 \) km/h (the speed of the current).

### Calculating the Round Trip Time
The distance between the two harbors is 120 km, so for a round trip, the boat travels:
- 120 km with the current at a speed of \( 13.5 + 1.5 = 15 \) km/h, and
- 120 km against the current at a speed of \( 13.5 - 1.5 = 12 \) km/h.

Now, we can calculate the total time for the round trip.

\[
\text{Time} = \frac{120}{15} + \frac{120}{12}
\]It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you'd like, you can manually calculate the round trip time using the following steps:
1. Time for 120 km with the current:
   \[
   \frac{120}{15} = 8 \text{ hours}
   \]
2. Time for 120 km against the current:
   \[
   \frac{120}{12} = 10 \text{ hours}
   \]
   
The total round trip time is \( 8 + 10 = 18 \) hours.

Would you like more details or have any other questions? Here are 5 related questions to explore:

1. How does current speed affect travel time in different scenarios?
2. What would happen if the current were stronger or weaker?
3. How can we model similar problems for air travel considering wind speed?
4. What methods can be used to solve systems of non-linear equations?
5. How do different types of boats perform in still water vs. moving water?

**Tip**: When solving a system of equations, substitution or elimination methods are common, but sometimes graphing or numerical methods can also provide useful insights!

Discover how to solve a boat speed problem using systems of equations. This detailed explanation involves algebraic methods to calculate the boat's speed in still water and the current's speed. Suitable for Grades 10-12, learn to solve equations like 105/(b+c) + 60/(b-c) = 12.

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