2017 AMC 12A Problem 13

Question

Driving at a constant speed, Sharon usually takes

180

minutes to drive from her house to her mother’s house. One day Sharon begins the drive at her usual speed, but after driving

\dfrac{1}{3}

of the way, she hits a bad snowstorm and reduces her speed by

20

miles per hour. This time the trip takes her a total of

276

minutes. How many miles is the drive from Sharon’s house to her mother’s house?

132

135

138

141

144

Her usual drive is

3

hours, so the distance equals

3

times her usual speed

The final two-thirds takes

276-\dfrac13\cdot180=216

minutes at the reduced speed

Let the distance be

d

miles and the usual speed

r

mph. Since the usual trip is

3

hours,

d=3r.

The first

\dfrac{1}{3}

of the drive takes

\dfrac{1}{3}\cdot180=60

minutes at speed

r,

so the remaining

\dfrac{2}{3}

takes

276-60=216

minutes

=\dfrac{18}{5}

hours at speed

r-20.

That final portion covers

\dfrac{2}{3}d=2r

miles, so

2r=(r-20)\cdot\dfrac{18}{5}.

Solving gives

10r=18r-360,

so

r=45

and

d=3\cdot45=135.

Thus, the correct answer is B.

Problem 13 in other years