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Difference between revisions of "2018 AMC 8 Problems/Problem 17"

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==Problem 17==
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==Problem==
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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 <math>2</math> miles, which is <math>10,560</math> feet, and Bella covers <math>2 \tfrac{1}{2}</math> feet with each step. How many steps will Bella take by the time she meets Ella?
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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 <math>5</math> times as fast as Bella walks. The distance between their houses is <math>2</math> miles, which is <math>10,560</math> feet, and Bella covers <math>2 \tfrac{1}{2}</math> feet with each step. How many steps will Bella take by the time she meets Ella?
  
 
<math>\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520</math>
 
<math>\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520</math>
  
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==Solution 1==
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==Solution 1 (Fast and Easy)==
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Since Ella rides 5 times as fast as Bella, Ella rides at a rate of <math>\frac{25}{2}</math> or <math>12 \tfrac{1}{2}</math>. Together, they move <math>15</math> feet towards each other every unit. Dividing <math>10560</math> by <math>15</math> to find the number of steps Bella takes results in the answer of <math>\boxed{\textbf{(A) }704}</math>
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Every 10 feet Bella goes, Ella goes 50 feet, which means a total of 60 feet. They need to travel that 60 feet <math>10560\div60=176</math> 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, <math>1760\div2.5=\boxed{\textbf{(A) }704}</math>
  
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==Solution 2==
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==Solution 2 (Use Answer Choices to our advantage)==
  
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Since Ella rides 5 times faster than Bella, the ratio of their speeds is 5:1. This means that Bella travels 1/6 of the way, and 1/6 of 10560 feet is 1760 feet. Bella also walks 2.5 feet in a step, and 1760 divided by 2.5 is <math>\boxed{\textbf{(A) }704}</math>.
 
  
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==Solution 3==
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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 <math>\boxed{\textbf{(A) }704}</math>.
  
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We know that Ella's speed is <math>12.5</math> st/s (steps per second) and Bella's speed is <math>2.5</math> st/s. Let the speeds in which they are walking and riding at be the slope of two equations, one for Bella and one for Ella. We make <math>s</math> the amount of steps she takes and <math>d</math> distance from their houses. Since we are finding where they meet, we need to solve for the intersections of the two equations for Ella and Bella. The equations we get are:
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== Solution 3 (Fractions) ==
  
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<cmath>d = -12.5s + 10560</cmath>
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We can turn <math>2 \tfrac{1}{2}</math> into an improper fraction. It will then become <math>\frac{5}{2}</math>. Since Ella bikes 5 times faster, we multiply <math>\frac{5}{2}\cdot 5=\frac{25}{2}</math>. Then we add <math>\frac{5}{2}+\frac{25}{2}</math> to find the distance they walk and bike together for each step of Bella's: <math>\frac{30}{2} = 15</math>. 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  <math>\boxed{\textbf{(A) }704}</math> steps.
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<cmath>d = 2.5s</cmath>
 
  
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Subtracting both equations from each other gives us:
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==Video Solution (CREATIVE ANALYSIS!!!)==
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https://youtu.be/4xkFOM218Ro
  
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<cmath>-15s + 10560 = 0</cmath>
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~Education, the Study of Everything
  
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Solving that equation grants us:
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== Video Solution by OmegaLearn ==
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https://youtu.be/TkZvMa30Juo?t=1123
  
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<cmath>s = 704</cmath>
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~ pi_is_3.14
  
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Since we are finding the number of steps they take until they meet each other, we can stop there. The solution is
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==Video Solution==
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https://youtu.be/ycZ381n_1bQ
  
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<math>\boxed{\textbf{(A) }704}</math>
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~savannahsolver
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https://www.youtube.com/watch?v=UczCIsRzAeo  ~David
  
 
==See Also==
 
==See Also==

Latest revision as of 09:27, 10 April 2024

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

Solution 2 (Use Answer Choices to our advantage)

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 3 (Fractions)

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.

Video Solution (CREATIVE ANALYSIS!!!)

https://youtu.be/4xkFOM218Ro

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=1123

~ pi_is_3.14

Video Solution

https://youtu.be/ycZ381n_1bQ

~savannahsolver

https://www.youtube.com/watch?v=UczCIsRzAeo ~David

See Also

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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All AJHSME/AMC 8 Problems and Solutions

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