A boat can go 60 km downstream and 40 km upstream in 12 hours 30 minutes. It can go 84 km downstream and 63 km upstream in 18 hours 54 minutes. What is the speed (in km/h, to the nearest integer) of the boat in still water?

To solve this problem, let's use the concept of relative speed in still water, downstream, and upstream.

### Step 1: Define the variables
- Let \( b \) be the speed of the boat in still water (km/h).
- Let \( c \) be the speed of the current (km/h).

In downstream, the effective speed of the boat is \( b + c \) km/h (since the current aids the boat).
In upstream, the effective speed of the boat is \( b - c \) km/h (since the current opposes the boat).

### Step 2: Write the equations
Given two trips with different distances and times, we can create two equations based on the time taken.

#### First trip:
- Downstream: 60 km
- Upstream: 40 km
- Time: 12 hours 30 minutes = 12.5 hours

The time for downstream travel is \( \frac{60}{b + c} \) and for upstream is \( \frac{40}{b - c} \). The total time is 12.5 hours, so:

\[
\frac{60}{b + c} + \frac{40}{b - c} = 12.5
\]

#### Second trip:
- Downstream: 84 km
- Upstream: 63 km
- Time: 18 hours 54 minutes = 18.9 hours

The time for downstream travel is \( \frac{84}{b + c} \) and for upstream is \( \frac{63}{b - c} \). The total time is 18.9 hours, so:

\[
\frac{84}{b + c} + \frac{63}{b - c} = 18.9
\]

### Step 3: Solve the system of equations
We now have the system of two equations:

1. \[ \frac{60}{b + c} + \frac{40}{b - c} = 12.5 \]
2. \[ \frac{84}{b + c} + \frac{63}{b - c} = 18.9 \]

I will now solve this system to find \( b \) (the speed of the boat in still water) and \( c \) (the speed of the current).The solution to the system of equations gives:
- Speed of the boat in still water, \( b = 9.17 \) km/h (approximately 9 km/h when rounded to the nearest integer).
- Speed of the current, \( c = 4.17 \) km/h.

Thus, the speed of the boat in still water is approximately **9 km/h**.

Do you want more details on the solution, or do you have any questions?

Here are 5 related questions to consider:
1. How would the time change if the speed of the current increased?
2. What if the boat travels only downstream? How much time would it take?
3. How can we estimate the speed of the current using another method?
4. If the boat and current were traveling in opposite directions, how would this impact the effective speed?
5. How could this problem be extended to include acceleration in the water?

**Tip**: To solve such problems efficiently, always remember to break them down into smaller equations based on speed, distance, and time relationships.

Find out how to calculate the speed of a boat in still water using the concept of relative speed and linear equations. In this problem, a boat travels downstream and upstream at different distances and times, and we use algebraic equations to solve for the boat's speed. Suitable for students in Grades 9-10.

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