# 2018 AMC 8 Problems/Problem 17

## Problem

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? $\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520$

## Solution 1

Since Ella rides $5$ times faster than Bella, the ratio of their speeds is $5:1$. For Bella, we have $\frac{d}{r}=t$ and for Ella, we have $\frac{d}{5r}=t$; however, we know the times for both girls must be the same, and so that means in $\frac{d}{5r}=t$, the numerator becomes $5d$ (Ella travels $5$ times the distance that Bella does). This means that Bella travels $\frac{1}{6}$ of the way, and $\frac{1}{6}$ of $10,560$ feet is $1,760$ feet. Bella also walks $2.5$ feet each step, and $1,760$ divided by $2.5$ is $\boxed{\textbf{(A) }704}$.

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## Solution 2 (Fast and Easy)

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, $1760\div2.5=\boxed{\textbf{(A) }704}$

~ alexdapog A-A

We know that Bella goes 2.5 feet per step and since Ella rides 5 times faster than Bella she must go 12.5 feet on her bike for every step of Bella's. For Bella, it takes 4,224 steps, and for Ella, it takes 1/5th those steps since Ella goes 5 times faster than Bella, taking her 844.8 steps. The number of steps where they meet therefore must be less than 844.8. The only answer choice less than it is $\boxed{\textbf{(A) }704}$.

## Solution 4

We can turn $2 \tfrac{1}{2}$ into a mixed number. It will then become 5/2. Since Ella bikes 5 times faster, we multiply 5/2 by 5 to get 25/2. Then we add 5/2 to it in order to find the distance they walk and bike together in total. After adding, you should get 30/2 which is equal to 15. This means that after 15 times, they will meet. So you have to divide 10,560 by 15. The answer should be $\boxed{\textbf{(A) }704}$.

## Video Solution (CREATIVE ANALYSIS!!!)

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## Video Solution

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