Another common math word problem that the Algebra student should know how to solve are the “boat-in-the-river” word problems. These are actually just a variation of the dreaded uniform motion word problems which I’ve discussed in two previous articles. With the boat in the river problems, we assume that the boat has a uniform speed in still water and that the speed of the water (or the speed of the river current) is constant. The Algebra student will be presented with a math word problem in which he must solve for the speed of the boat (in still water, implied), or the speed of the river current, or the time spent going upstream or the time going downstream or some combination of these variables.

With these problems, Saxon uses the following variables:

B = speed of the boat in still water

W = the speed of the water (or the rate of the river current)

Td = time spent going downstream

Tu = time spent going upstream

Dd = distance gone downstream

Du = distance gone upstream

As with other uniform motion word problems, an important algebraic equation to remember is:

Distance = Rate * Time

With the boat-in-the-river math problem, we modify this slightly when we are going downstream to get:

Dd = (B + W) * Td

That is, the speed (or rate) of the boat going downstream is the speed of the boat in still water plus the speed of the water (or river current). That is because when one is going downstream, the river works with you.

When a boat is going upstream, the river works against you so we have to change our distance equation to reflect this by subtracting the speed of the water from the speed of the boat to get our rate:

Du = (B – W) * Tu

The best way to understand how to solve these is to work through a couple examples.

Sample Boat-in-the-River Word Problem #1:

Hannah and Fred can go 45 miles downstream in their boat in the same time it takes them to go 15 miles upstream. If the speed of the current was 3 mph, what was the speed of their boat in still water and how long did it take for them to go 45 miles downstream?

With these kinds of word problems I recommend to the Algebra student that you always start out by writing down the two essential distance equations, before anything else:

Dd = (B + W) * Td

Du = (B – W) * Tu

Secondly we need to write down the information we know from dissecting the text of the word problem.

Dd = 45

Td = Tu = T

(Since we know that the two times are the same, let’s just use T for our time variable)

Du = 15

W = 3

Now we can plug in our known values to get our system of two equations:

45 = (B + 3) * T

15 = (B – 3) * T

Simplify and rearrange the equations so that they are easier to solve:

B*T + 3T = 45

B*T – 3T = 15

We see that we can easily add the two equations to get a value for B*T:

2B*T = 60

B*T = 30

We can now substitute B*T into the first equation (or the second, either would work) to solve for T:

30 + 3T = 45

3T = 15

T = 5

Which then tells us that B = 6.

The speed of the boat is 6 mph and the time spent going downstream (which is the same as the time spent going upstream) was 5 hours.

Sample Boat-in-the-River Word Problem #2:

The boat Murphy’s Law could go 91 miles downstream in 7 hours, but required 12 hours to go 84 miles upstream. What were the speed of the current and the speed of the boat in still water?

Our essential equations:

Dd = (B + W) * Td

Du = (B – W) * Tu

Known values from the word problem:

Dd = 91

Td = 7

Tu = 12

Du = 84

Plug in the values to get our equations:

91 = (B + W) * 7

84 = (B – W) * 12

Simplify and rearrange the equations:

7B + 7W = 91

12B – 12W = 84

We now must modify these two equations so that we can add them together to eliminate the W variable. We can do this by multiplying the top equation by 12 and the bottom equation by 7:

84B + 84W = 1092

84B – 84W = 588

Adding the two equations gives us:

168B = 1680

B = 10

Plug in the B to the original first equation to solve for W:

7(10) + 7W = 91

7W = 21

W = 3

The speed of the boat was 10 mph and the speed of the current (water) was 3 mph.

**Source**

John H. Saxon, Jr. Algebra 2. An Incremental Development. Third Edition.