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Difference between revisions of "1997 AJHSME Problems/Problem 22"

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==Problem==
 
==Problem==
  
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A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth 200. How much is a three-inch cube of silver worth?
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A two-inch cube <math>(2\times 2\times 2)</math> of silver weighs 3 pounds and is worth 200 dollars. How much is a three-inch cube of silver worth?
  
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<math>\text{(A)}\ 300\text{ dollars} \qquad \text{(B)}\ 375\text{ dollars} \qquad \text{(C)}\ 450\text{ dollars} \qquad \text{(D)}\ 560\text{ dollars} \qquad \text{(E)}\ 675\text{ dollars}</math>
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<math>\textbf{(A) }\text{300 dollars} \qquad \textbf{(B) }\text{375 dollars} \qquad \textbf{(C) }\text{450 dollars} \qquad \textbf{(D) }\text{560 dollars}\qquad \textbf{(E) }\text{675 dollars}</math>
  
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==Solution 1==
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==Solution==
  
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The 2x2x2 cube of silver can be divided into <math>8</math> equal cubes that are 1x1x1. Each smaller cube is worth <math>\frac{200}{8} = 25</math> dollars.
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The two-inch cube has a volume of <math>8</math> cubic inches, and the three-inch cube has a volume of <math>27</math> cubic inches. Thus, the three-inch cube has a weight that is <math>\frac{27}{8}</math> times that of the two-inch cube. Then its value is <math>\frac{27}{8} \cdot 200 = \boxed{\textbf{(E) }\text{675 dollars}}</math>.
  
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To create a 3x3x3 cube of silver, you need <math>27</math> of those 1x1x1 cubes. The cost of those <math>27</math> cubes is <math>27 \cdot 25 = 675</math> dollars, which is answer <math>\boxed{E}</math>
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~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
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==Solution 2==
 
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Since price is proportional to the amount (or volume) of silver, and volume is proportional to the cube of the side, the price ought to be proportional to the cube of the side.
 
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Setting up a proportion:
 
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<math>\frac{200}{2^3} = \frac{x}{3^3}</math>
 
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<math>x = 200 \cdot \frac{3^3}{2^3} = 675</math>, which is answer <math>\boxed{E}</math>
 
  
 
== See also ==
 
== See also ==

Latest revision as of 13:39, 4 April 2024

Problem

A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth 200 dollars. How much is a three-inch cube of silver worth?

$\textbf{(A) }\text{300 dollars} \qquad \textbf{(B) }\text{375 dollars} \qquad \textbf{(C) }\text{450 dollars} \qquad \textbf{(D) }\text{560 dollars}\qquad \textbf{(E) }\text{675 dollars}$

Solution

The two-inch cube has a volume of $8$ cubic inches, and the three-inch cube has a volume of $27$ cubic inches. Thus, the three-inch cube has a weight that is $\frac{27}{8}$ times that of the two-inch cube. Then its value is $\frac{27}{8} \cdot 200 = \boxed{\textbf{(E) }\text{675 dollars}}$.

~ cxsmi

See also

1997 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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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