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Help solving problems involving rivers and boats -- Distance = Rate * Time

A reader wrote in, saying,

You don't have anything on motion or current problems, such as:

A motorboat can go 16 miles downstream on a river in 20 minutes. It takes 30 minutes for this boat to go back upstream the same 16 miles. Find the speed of the current.

I know it's probably really easy but I really don't understand these type of problems.

This is a "d=rt" problem -- that is, distance equals rate times time.

In some of these problems, you aren't told the distance -- this seems to make the problem especially tricky, but in fact, they are no harder. Just leave the distance as "d" and you'll see it cancels out, just before the answer is revealed.

In your problem d is 16. You can let r be the rate of the river, and b be the rate of the boat.

So, in d=rt fashion, you have two equations...

16=(b+r)(20/60)    -- (20 minutes is 20/60 hours, because we like to use miles and hours as our basic units, but you can suit yourself)

Then it becomes just a matter of solving the equations.

2/6b + 2/6r = 16
3/6b - 3/6r = 16
b + r = 48
b - r = 32

Subtracting the bottom equation from the top one gives us:

2r = 16

Adding the two equations gives us 2b=80, so b=40. Now let's check the answer.

With the current, the boat goes 40+8=48 miles an hour. So it goes 16 miles in a third of an hour. Good.
Against the current, the boat goes 48-8=32 miles an hour, so it goes 16 miles in half an hour. Also good.

Was this helpful? Can you see how d=rt does the trick for just about all problems of this type?

Internet references

Math League: ratio and proportion (Ratio, Comparing ratios, Proportion, Rate, Converting rates, Average rate of speed)

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