2018 AMC 8

Complete problem set with solutions and individual problem pages

Problem 17 Hard

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers 2 \tfrac{1}{2} feet with each step. How many steps will Bella take by the time she meets Ella?

  • A.

    704

  • B.

    845

  • C.

    1056

  • D.

    1760

  • E.

    3520

Answer:A

Solution 1 

Every 10 feet Bella goes, Ella goes 50 feet, which means a total of 60 feet. They need to travel that 60 feet 10560\div60=176 times to travel the entire 2 miles. Since Bella goes 10 feet 176 times, this means that she travels a total of 1760 feet. And since she walks 2.5 feet each step, 1760\div2.5=\boxed{\textbf{(A) }704}

 

Solution 2

We can turn 2 \tfrac{1}{2} into an improper fraction. It will then become \frac{5}{2}. Since Ella bikes 5 times faster, we multiply \frac{5}{2}\cdot 5=\frac{25}{2}. Then we add \frac{5}{2}+\frac{25}{2} to find the distance they walk and bike together for each step of Bella's: \frac{30}{2} = 15. This means that they travel 15 ft. of distance for each step that Bella takes. Divide 10,560 by 15 to find that Bella takes \boxed{\textbf{(A) }704} steps.

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