Difference between revisions of "2018 AMC 8 Problems/Problem 1"

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==Problem 1==
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==Problem==
An amusement park has a collection of scale models, with ratio <math>1 : 20</math>, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?
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An amusement park has a collection of scale models, with a ratio <math> 1: 20</math>, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?
  
 
<math>\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20</math>
 
<math>\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20</math>
  
==Solution==
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==Solution 1==
  
You can set up a ratio: <math>\frac{1}{20}=\frac{x}{289}</math>. Cross multiplying, you get <math>20x=289</math>. You divide by 20 to get <math>x=14.45</math>. The closest integer is <math>14</math> or <math>\textbf{(A) }14</math>
 
  
==See Also==
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You can see that since the ratios of real building's heights to the model building's height is <math>1:20</math>. We also know that the U.S Capitol is <math>289</math> feet in real life, so to find the height of the model, we divide by 20. That gives us <math>14.45</math> which rounds to 14. Therefore, to the nearest whole number, the duplicate is <math>\boxed{\textbf{(A)}14\text{ feet}}</math>.  ~avamarora.
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==Solution 2==
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We can compute <math>\frac{289}{20}</math> and round our answer to get <math>\boxed{\textbf{(A)}14}</math>.
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It is basically Solution 1 without the ratio calculation. However, Solution 1 is referring further to the problem.
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==Solution 3==
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We know that <math> 20 \cdot 14 = 280 ,</math> and that <math> 20 \cdot 15 = 300 .</math> These are the multiples of <math>20</math> around <math>289 ,</math> and the closest one of those is <math>280.</math> Therefore, the answer is <math> \dfrac {280} {20} = \boxed{\textbf{(A) }14} .</math>
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==See also==
 
{{AMC8 box|year=2018|before=First Problem|num-a=2}}
 
{{AMC8 box|year=2018|before=First Problem|num-a=2}}
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:47, 18 January 2021

Problem

An amusement park has a collection of scale models, with a ratio $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?

$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Solution 1

You can see that since the ratios of real building's heights to the model building's height is $1:20$. We also know that the U.S Capitol is $289$ feet in real life, so to find the height of the model, we divide by 20. That gives us $14.45$ which rounds to 14. Therefore, to the nearest whole number, the duplicate is $\boxed{\textbf{(A)}14\text{ feet}}$. ~avamarora.

Solution 2

We can compute $\frac{289}{20}$ and round our answer to get $\boxed{\textbf{(A)}14}$. It is basically Solution 1 without the ratio calculation. However, Solution 1 is referring further to the problem.

Solution 3

We know that $20 \cdot 14 = 280 ,$ and that $20 \cdot 15 = 300 .$ These are the multiples of $20$ around $289 ,$ and the closest one of those is $280.$ Therefore, the answer is $\dfrac {280} {20} = \boxed{\textbf{(A) }14} .$

See also

2018 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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