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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 directly proportional to the amount (or volume) of silver, we must have a constant quotient.
 
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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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