Distance Formula Word Problems
by Chuck
(Corning)
If a person can travel 20 miles upstream in 10 hours and the same distance downstream in 2.5 hours, find the rate of the current.
Karin from Algebra Class Says:
If a person can travel 20 miles upstream in 10 hours and the same distance downstream in 2.5 hours, find the rate of the current.
This problem is a little tricky.
First we want to use the distance formula to find his overall rate for both directions.
Upstream:
d = rt
20 = r10
20/10 = 10r/10
2 mi/h = r
His total rate, including the current is 2 mi/h.
Downstream:
d = rt
20 = r2.5
20/2.5 = 2.5r/2.5
8 mi/h = r
His total rate going downstream is 8 mi/hr.
Now we need to think about what is happening in this problem.
We have the swimmer who is swimming at x mph. We don't really know his swimming speed. Then if he's traveling downstream, we can add the speed of the current. If he's traveling upstream, then we subtract the speed of the current. (The speed of the current is y since we don't know that information.)
So,
Downstream:
Swimming speed (x) + current (y) = 8 mph
or
x+y = 8
Upstream:
Swimming speed (x) - current (y) = 2 mph
or
x - y = 2
Now we have a system of equations and we can solve to find the current (y).
Since I am solving for y, I am going to solve my first equation for x and substitute into the second equation.
x + y = 8
x + y - y = 8 - y
x = 8-y
Now substitute this into the second equation:
x - y = 2
(8 - y) - y = 2
8 - 2y = 2
8 - 8 - 2y = 2 - 8
-2y = -6
-2y/-2 = -6/-2
y = 3
Since y equals the current of the water, the current is 3 mph.
There are two steps. First you must use the distance the formula to find his overall rate in each direction. Then you must write a system of equations to find the rate of the current.
I hope this helps,
Karin