Rate Problems

A boat can travel 16 miles up a river in 2 hours.  The same boat can travel 36 miles downstream in 3 hours.  What is the speed of the boat in still water?  What is the speed of the current?

What are we trying to find in this problem?
We want to find two things-- the speed of the boat in still water and the speed of the current.  Each of these things will be represented by a different variable:

B = speed of the boat in still water
C = speed of the current

Since we have two variables, we will need to find a system of two equations to solve.

How do we find the two equations we need?
Rate problems are based on the relationship Distance = (Rate)(Time).

To organize our work, we'll make a chart of the distance, rate and time that the boat travels going both upstream and downstream.  The chart will give us the information about distance, rate and time that we need to write our two equations.

Here's what the chart looks like before we put any of our information in it:

 Distance Rate Time upstream downstream

Let's look at the words of the problem.

A boat can travel 16 miles up a river in 2 hours.  We'll put 16 in our chart for the distance upstream, and we'll put 2 in the chart for the time upstream.

The same boat can travel 36 miles downstream in 3 hours.  We'll put 36 in our chart for the distance downstream, and we'll put 3 in the chart for the time downstream.

Our chart now looks like this:

 Distance Rate Time upstream 16 2 downstream 36 3

Now let's think about the rate the boat travels.  We know that if the boat were on a still lake, its motor would propel it at a rate of B miles per hour.  But the boat is not on a still lake; it's moving upstream and downstream on a river.  If the boat is traveling upstream, the current (which is C miles per hour) will be pushing against the boat, and the boat's speed will decrease by C miles per hour.  The resulting speed of the boat (traveling upstream) is B-C miles per hour.  On the other hand, if the boat is traveling downstream, the current will be pushing the boat faster, and the boat's speed will increase by C miles per hour.  The resulting speed of the boat (traveling downstream) is B+C miles per hour.  We'll put this information in our chart:

 Distance Rate Time upstream 16 B-C 2 downstream 36 B+C 3

Each row in the chart will give us an equation.

Going upstream, Distance = (Rate)(Time), so 16 = (B-C)(2)
Going downstream, Distance = (Rate)(Time), so 36 = (B+C)(3)

Now we'll solve the system of equations:

16 = 2(B-C)
36 = 3(B+C)

If we divide both sides of the first equation by 2, it will become 8 = B-C.
If we divide both sides of the second equation by 3, it will become 12 = B+C.

We'll add these equations together to find our solution:

8  =  B-C
12 = B+C
20 = 2B
10 = B

The speed of the boat in still water is 10 miles per hour.

To find the speed of the current, we can substitute 10 for the B in any of our equations.  We'll choose the easiest equation to work with:

12 = B+C
12 = 10+C
2 = C

The speed of the current is 2 miles per hour.