Driving to See Grandmother

Kim drove 84 miles to see her grandmother. Nine miles of the trip were inside the city limits. Her average speed outside the city was 12 mph faster than inside the city. The whole trip took 3 hours. How fast did Kim go while inside the city?


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We don’t know r, and we don’t know the individual t‘s, but we know that the total t is 3. So we fill in the boxes like this:

  d r t
in 9 r 9/r
out 84 – 9 = 75 r + 12 75/(r + 12)

And we get this: 9/r + 75/(r + 12) = 3
9(r + 12) + 75r = 3r(r + 12)
9r + 108 + 75r = 3r2 + 36r
3r2 + 36r – 9r – 75r – 108 = 0
3r2 – 48r – 108 = 0
3(r2 – 16r – 36) = 0
(r – 18)(r + 2) = 0
r = 18 or -2 (extraneous)
r = 18 in the city and r + 12 = 30 outside the city.

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