Difference between revisions of "2018 AMC 10A Problems/Problem 8"

(Solution)
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==Solution==
 
==Solution==
 
Let <math>x</math> be the number of 5-cent stamps that Joe has. Therefore, he must have <math>x+3</math> 10-cent stamps and <math>23-(x+3)-x</math> 25-cent stamps. Since the toal value of his collection is 320 cents, we can write
 
Let <math>x</math> be the number of 5-cent stamps that Joe has. Therefore, he must have <math>x+3</math> 10-cent stamps and <math>23-(x+3)-x</math> 25-cent stamps. Since the toal value of his collection is 320 cents, we can write
\begin{align*}
+
begin{align*}
 
5x+10(x+3)+25(23-(x+3)-x) & =320 \\
 
5x+10(x+3)+25(23-(x+3)-x) & =320 \\
 
5x+10(x+3)+25(20-2x) & =320 \\
 
5x+10(x+3)+25(20-2x) & =320 \\
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35x & =210 \\
 
35x & =210 \\
 
x=6 \
 
x=6 \
\end{align}*
+
end{align}*
 
Joe has 6 5-cent stamps, 9 10-cent stamps, and 8 25-cent stamps. Thus, our answer is  
 
Joe has 6 5-cent stamps, 9 10-cent stamps, and 8 25-cent stamps. Thus, our answer is  
 
<math>8-6=\boxed{2}</math>
 
<math>8-6=\boxed{2}</math>
  
~Nivek  
+
~Nivek
 +
 
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2018|ab=A|num-b=7|num-a=9}}
 
{{AMC10 box|year=2018|ab=A|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:06, 8 February 2018

Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?

$\textbf{(A) }   0   \qquad        \textbf{(B) }   1   \qquad    \textbf{(C) }   2   \qquad   \textbf{(D) }  3  \qquad  \textbf{(E) }   4$

Solution

Let $x$ be the number of 5-cent stamps that Joe has. Therefore, he must have $x+3$ 10-cent stamps and $23-(x+3)-x$ 25-cent stamps. Since the toal value of his collection is 320 cents, we can write begin{align*} 5x+10(x+3)+25(23-(x+3)-x) & =320 \\ 5x+10(x+3)+25(20-2x) & =320 \\ 5x+10x+30+500-50x & =320 \\ 35x & =210 \\ x=6 \ end{align}* Joe has 6 5-cent stamps, 9 10-cent stamps, and 8 25-cent stamps. Thus, our answer is $8-6=\boxed{2}$

~Nivek

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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 AMC 10 Problems and Solutions

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