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Difference between revisions of "2012 AMC 8 Problems/Problem 13"

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==Problem 13==
 
Jamar bought some pencils costing more than a penny each at the school bookstore and paid <math>  
 
Jamar bought some pencils costing more than a penny each at the school bookstore and paid <math>  
 
\textdollar 1.43 </math>. Sharona bought some of the same pencils and paid <math> \textdollar 1.87 </math>. How many more pencils did Sharona buy than Jamar?
 
\textdollar 1.43 </math>. Sharona bought some of the same pencils and paid <math> \textdollar 1.87 </math>. How many more pencils did Sharona buy than Jamar?
  
 
<math> \textbf{(A)}\hspace{.05in}2\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}4\qquad\textbf{(D)}\hspace{.05in}5\qquad\textbf{(E)}\hspace{.05in}6 </math>
 
<math> \textbf{(A)}\hspace{.05in}2\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}4\qquad\textbf{(D)}\hspace{.05in}5\qquad\textbf{(E)}\hspace{.05in}6 </math>
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==Solution 1==
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We assume that the price of the pencils remains constant. Convert <math> \textdollar 1.43 </math> and <math> \textdollar 1.87 </math> to cents.  Since the price of the pencils is more than one penny, we can find the price of one pencil (in cents) by taking the greatest common divisor of <math> 143 </math> and <math> 187 </math>, which is <math> 11 </math>. Therefore, Jamar bought <math> \frac{143}{11} \implies 13 </math> pencils and Sharona bought <math> \frac{187}{11} \implies 17 </math> pencils. Thus, Sharona bought <math> 17-13 = \boxed{\textbf{(C)}\ 4} </math>  more pencils than Jamar.
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==Solution 2==
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We find the difference between <math>1.43</math> and <math>1.87</math> is <math> 1.87-1.43 = 0.44 </math>, which is the extra cost of Sharona's pencils than Jamar's pencils. Because the difference between the amounts of the pencils they bought must be divided evenly by <math>0.44</math>, looking into the answers, <math>2</math> or <math>4</math> is possibly correct. It gives us the price of each pencil should be <math> 0.44/2=0.22 </math> or <math> 0.44/4=0.11 </math>, respectively. Then we find only <math>0.11</math> can be divided evenly by <math>1.43</math> and <math>1.87</math>. So the answer is <math> \boxed{\textbf{(C)}\ 4} </math>
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==Solution 3 (quick solution)==
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You can quickly tell, if you have memorized your times tables, that <math>187</math> and <math>143</math> are both multiples of <math>11</math>. Additionally, the difference is 44 cents. That means that there are <math> \boxed{\textbf{(C)}\ 4} </math> more pencils purchased by Sharona because <math>11*4 = 44</math>.
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==See Also==
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{{AMC8 box|year=2012|num-b=12|num-a=14}}
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{{MAA Notice}}

Latest revision as of 14:36, 28 December 2023

Problem 13

Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 1.43$. Sharona bought some of the same pencils and paid $\textdollar 1.87$. How many more pencils did Sharona buy than Jamar?

$\textbf{(A)}\hspace{.05in}2\qquad\textbf{(B)}\hspace{.05in}3\qquad\textbf{(C)}\hspace{.05in}4\qquad\textbf{(D)}\hspace{.05in}5\qquad\textbf{(E)}\hspace{.05in}6$

Solution 1

We assume that the price of the pencils remains constant. Convert $\textdollar 1.43$ and $\textdollar 1.87$ to cents. Since the price of the pencils is more than one penny, we can find the price of one pencil (in cents) by taking the greatest common divisor of $143$ and $187$, which is $11$. Therefore, Jamar bought $\frac{143}{11} \implies 13$ pencils and Sharona bought $\frac{187}{11} \implies 17$ pencils. Thus, Sharona bought $17-13 = \boxed{\textbf{(C)}\ 4}$ more pencils than Jamar.

Solution 2

We find the difference between $1.43$ and $1.87$ is $1.87-1.43 = 0.44$, which is the extra cost of Sharona's pencils than Jamar's pencils. Because the difference between the amounts of the pencils they bought must be divided evenly by $0.44$, looking into the answers, $2$ or $4$ is possibly correct. It gives us the price of each pencil should be $0.44/2=0.22$ or $0.44/4=0.11$, respectively. Then we find only $0.11$ can be divided evenly by $1.43$ and $1.87$. So the answer is $\boxed{\textbf{(C)}\ 4}$

Solution 3 (quick solution)

You can quickly tell, if you have memorized your times tables, that $187$ and $143$ are both multiples of $11$. Additionally, the difference is 44 cents. That means that there are $\boxed{\textbf{(C)}\ 4}$ more pencils purchased by Sharona because $11*4 = 44$.

See Also

2012 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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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