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Faster (Posted on 2004-11-29) Difficulty: 2 of 5
You traveled three hours at a certain speed. If you had taken 1 minute less to do each mile you would have gone 30 miles farther than you actually did.

How far did you go?

See The Solution Submitted by Charlie    
Rating: 3.4545 (11 votes)

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Solution Answer + Solution | Comment 1 of 10
The answer is 60 miles. Here is the solution:

There might be a more intuitive way to think about this, but it’s too early for that in my book =) Instead of thinking of the rates as x mph and y mph, I thought of them as 1mile/x and 1mile/y where y is one minute less than x. So I decided to convert all of the time in the problem to minutes for simplicity.

R1 = 1 mile / t minutes
R2 = 1 mile / (t-1) minutes

180min*R1 = 180/t miles = D
180min*R2 = 180/(t-1) miles = D+30

180/D = t
180/(D+30) = t-1
180/(D+30) = 180/D – 1 = (180-D)/D

180D = (180-D)(D+30) = 180D + 5400 – D^2 – 30D
0 = D^2 + 30D – 5400

D = [-b +/- sqrt(b^2 – 4ac)]/(2a)
D = [-30 +/- sqrt(900 + 21600)]/2
D = (-30 +/- 150)/2 = 60

So you traveled for 3 hours at a rate of 1mile/3min = 20 mph, covering a distance of 60 miles.

If you had taken one minute less to do each mile, you would have traveled for 3 hours at a rate of 1mile/2min = 30 mph, covering a distance of 90 miles, which is indeed 30 miles more than 60.


  Posted by nikki on 2004-11-29 14:03:58
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