Difference between revisions of "2012 CEMC Gauss (Grade 8) Problems/Problem 5"

(Created page with "==Problem== How many more coins does it take to make one dollar (100) using only nickels(5 coins) than it takes to make one dollar using only dimes(10 coins)?<!--Note: this is...")
 
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==Problem==
 
==Problem==
How many more coins does it take to make one dollar (100) using only nickels(5 coins) than it takes to make one dollar using only dimes(10 coins)?<!--Note: this is how the question was written-->
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If <math>\frac{8}{12} = \frac{\framebox {}}{3}</math>, then the value represented by <math>\framebox {}</math> is
  
<math>  \text{ (A) }\  15\qquad\text{ (B) }\ 10\qquad\text{ (C) }\ 75\qquad\text{ (D) }\ 64\qquad\text{ (E) }\ 54 </math>
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<math>  \text{ (A) }\  24\qquad\text{ (B) }\ 1\qquad\text{ (C) }\ 12\qquad\text{ (D) }\ 2\qquad\text{ (E) }\ 4 </math>
 
==Solution==
 
==Solution==
Since a dollar is <math>100</math> cents and a nickel is <math>5</math> cents, there are <math>\frac{100}{5} = 20</math> nickels in a dollar.
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<math>\frac{12}{3} = 4</math>, meaning we can divide the numerator and denominator by <math>4</math> to arrive at our answer.
  
Using this logic for dimes and the fact that a dime is <math>10</math> cents, there are <math>\frac{100}{10} = 10</math> dimes in a dollar.
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<math>\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}</math>
  
Thus, it takes <math>20 - 10 = \boxed{\textbf{(B) } 10}</math> more coins to make a dollar using only nickels than it takes to make one dollar using only dimes.
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Since the numerator is <math>2</math>, the answer is <math>\boxed {\textbf {(D) }2}</math>.
  
 
~anabel.disher
 
~anabel.disher

Revision as of 16:33, 20 April 2025

Problem

If $\frac{8}{12} = \frac{\framebox {}}{3}$, then the value represented by $\framebox {}$ is

$\text{ (A) }\  24\qquad\text{ (B) }\ 1\qquad\text{ (C) }\ 12\qquad\text{ (D) }\ 2\qquad\text{ (E) }\ 4$

Solution

$\frac{12}{3} = 4$, meaning we can divide the numerator and denominator by $4$ to arrive at our answer.

$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

Since the numerator is $2$, the answer is $\boxed {\textbf {(D) }2}$.

~anabel.disher