2014 AMC 8

Complete problem set with solutions and individual problem pages

Problem 17 Hard

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first \frac{1}{2} mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last \frac{1}{2} mile in order to arrive just as school begins today?

  • A.

    4

  • B.

    6

  • C.

    8

  • D.

    10

  • E.

    12

Answer:B

Note that on a normal day, it takes him 1/3 hour to get to school. However, today it took \frac{1/2 \text{ mile}}{2 \text{ mph}}=1/4 hour to walk the first 1/2 mile. That means that he has 1/3 -1/4 = 1/12 hours left to get to school, and 1/2 mile left to go. Therefore, his speed must be \frac{1/2 \text{ mile}}{1/12 \text { hour}}=\boxed{6 \text{ mph}}, so \boxed{\text{(B) }6} is the answer.

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