# Thread: Need help with a word problem

1. ## Need help with a word problem

Once a month, a professor and his wife drive to Minneapolis, Minnesota to visit their relatives. The distance
between the Cedar Falls, Iowa gas station and their relatives’ house is 222 miles. The couple used to switch off
driving fairly often – they each drove half the time. When they did that, the trip took precisely 3 hours. Once
the professor learned how to grade papers while his wife was driving, he started to drive less during these trips.
His wife started driving for two-thirds of the time. When this happened, the trip took 2 hours 55 minutes.

I need help so I can set up an equation to find the professor's average driving speed.

2. Okay, we're going to have to use distance = rate x time. Let's do the following:

$\displaystyle r_1 = professor's rate$
$\displaystyle r_2 = his wife's rate$.

For the first part, we know the total distance is 222. We also know the total time is 180 minutes. We can write the following equations:

$\displaystyle d = 90r_{1}$
$\displaystyle (222-d) = 90 r_{2}$

Note that the wife has to drive the remaining distance, which can be represented by 222-d.Doing a little substitution (plugging in $\displaystyle 90 r_{1}$ for d in the second equation and simplifying, we get, eventually:

$\displaystyle 90 r_1 + 90 r_2 = 222$.

Now, we can set up similar equations in the second part, except the time has changed to 175 minutes. We know the professor drove one-third of the time, of 58 1/3 minutes, and his wife drove the rest of the time, or 116 2/3 minutes. This gives, eventually, the equation:

$\displaystyle 175 r_{1} + 350 r_{2} = 666$

(I multiplied everything - the times and distances - by 3, thus getting rid of the fractions (and ending up with the ominous 666 on the right.))

We have two equations, two unknowns. Solve it however you like - substitution, linear combinations, etc. You should be home free!

Remember that since we changed the times to minutes, your answers after solving the system will be in miles / minute. Multiply by 60 to convert to miles / hour (since there are 60 minutes in an hour).