Manhattan GMAT Challenge Problem of the Week - 4 Mar 10

by , Mar 4, 2010

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8 ) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10

(B) 15

(C) 20

(D) 25

(E) 30

Solution

First, we set up two RT = D equations, one for the downhill run and one for the ride back up the mountain.

Downhill run: (x + 10)t = 300

Ride back up: (x 8)(t + 135) = 300

Technically, we just have to do some algebra & arithmetic from here on out. However, these equations are very difficult to solve in their current state. The tipoff for you is that the variable x does not represent, on its own, either the downhill or the uphill speed. Thus, the equations wind up being thorny (although still solvable).

However, we can reduce the complexity by creating a new variable, say r, that represents the speed on the ride back up. In other words, r = x 8. We can rewrite this equation as r + 8 = x, and thus the downhill speed, x + 10, can be re-expressed as r + 18. As youll see, this simplifies the algebra. In this sort of situation, when a variable such as x does not represent any real speed in the scenario, our instinct should be to replace x with another variable that does represent a real speed.

Downhill run: (r + 18)t = 300

Ride back up: r(t + 135) = 300

Now we can set the expressions on the left side equal to each other, since they both equal 300:

(r + 18)t = r(t + 135)

rt + 18t = rt + 135r

18t = 135r

2t = 15r

t = (15/2)r

Finally, we substitute back into either equation (well just pick the first). Since the numbers get large and we can see were going to get a quadratic, we might want to leave certain numbers factored as we go.

(r + 18)(15/2)r = 300

(15/2)[pmath]r^2[/pmath] + 135r = 300

(15/2)[pmath]r^2[/pmath] + 135r 300 = 0

Now divide by 15 and multiply throughout by 2.

[pmath]r^2[/pmath] + 18r 40 = 0

(r + 20)(r 2) = 0

Since r must be positive (it represents a speed), r must be 2. Thus, Lindseys downhill speed, in meters per second, is r + 18 = 20.

The correct answer is (C) 20.

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