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

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==Problem 17==
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==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 <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?
 
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?
  
 
<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>
  
==Solution==
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==Solution 1==
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>
 
  
==Solution 2==
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Since Ella rides 5 times faster than Bella, the ratio of their speeds is 5:1. For Bella, we have D/R = t and for Ella, we have D/5r = t, however we know times for both girls must be the same, and so that means in D/5R = t, numerator becomes 5D(Ella travels 5 times the distance that Bella does). 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>.
  
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>.
 
  
==Solution 3==
 
  
We know that Ella's speed is 12.5 st/s (steps per second) and Bella's speed is 2.5 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 s the amount of steps she takes and d 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 2 (Use Answer Choices to our advantage)==
  
d = -12.5s + 10560
 
d = 2.5s
 
  
Subtracting both equations from each other gives us:
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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. For Bella it takes 4224 steps and for Ella it would take 1/5th those steps since Ella goes 5 times faster than Bella so it takes 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>. ~ Batmanstark
  
-15s + 10560 = 0
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== Video Solution ==
 
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https://www.youtube.com/watch?v=iXTtm7ePKYs
Solving that equation grants us:
 
 
 
s = 704
 
 
 
Since we are finding the number of steps they take until they meet each other, we can stop there. The solution is <math>\boxed{\textbf{(A) }704)</math>
 
  
 
==See Also==
 
==See Also==

Revision as of 22:59, 21 September 2021

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 D/R = t and for Ella, we have D/5r = t, however we know times for both girls must be the same, and so that means in D/5R = t, numerator becomes 5D(Ella travels 5 times the distance that Bella does). 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 $\boxed{\textbf{(A) }704}$.


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. For Bella it takes 4224 steps and for Ella it would take 1/5th those steps since Ella goes 5 times faster than Bella so it takes 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 $\boxed{\textbf{(A) }704}$. ~ Batmanstark

Video Solution

https://www.youtube.com/watch?v=iXTtm7ePKYs

See Also

2018 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AJHSME/AMC 8 Problems and Solutions

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