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

m (→‎Solution 3: LaTeX'ed the solution and uncentered the less important equations.)
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<math>\textbf{(A) }  0  \qquad        \textbf{(B) }  1  \qquad    \textbf{(C) }  2  \qquad  \textbf{(D) }  3  \qquad  \textbf{(E) }  4 </math>
 
<math>\textbf{(A) }  0  \qquad        \textbf{(B) }  1  \qquad    \textbf{(C) }  2  \qquad  \textbf{(D) }  3  \qquad  \textbf{(E) }  4 </math>
  
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==Solution 1==
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==Solution 1 (One Variable)==
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Let <math>x</math> be the number of 5-cent coins that Joe has. Therefore, he must have <math>(x+3)</math> 10-cent coins and <math>(23-(x+3)-x)</math> 25-cent coins. Since the total value of his collection is 320 cents, we can write
+
Let <math>x</math> be the number of <math>5</math>-cent coins that Joe has. Therefore, he must have <math>(x+3) \ 10</math>-cent coins and <math>(23-(x+3)-x) \ 25</math>-cent coins. Since the total value of his collection is <math>320</math> cents, we can write
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<math>5x + 10(x+3) + 25(23-(x+3)-x) = 320 \Rightarrow 5x + 10x + 30 + 500 - 50x = 320 \Rightarrow 35x = 210 \Rightarrow x = 6</math>  
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<cmath>\begin{align*}
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Joe has 6 5-cent coins, 9 10-cent coins, and 8 25-cent coins. Thus, our answer is  
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5x + 10(x+3) + 25(23-(x+3)-x) &= 320 \\
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<math>8-6 = \boxed{\textbf{(C) } 2}</math>
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5x + 10x + 30 + 500 - 50x &= 320 \\
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35x &= 210 \\
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x &= 6.
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\end{align*}</cmath>  
 +
Joe has six <math>5</math>-cent coins, nine <math>10</math>-cent coins, and eight <math>25</math>-cent coins. Thus, our answer is  
 +
<math>8-6 = \boxed{\textbf{(C) } 2}.</math>
  
 
~Nivek
 
~Nivek
  
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==Solution 2==
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==Solution 2 (Two Variables)==
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Let n be the number of 5 cent coins Joe has, d be the number of 10 cent coins, and q the number of 25 cent coins. We are solving for q - n.
 
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We know that the value of the coins add up to 320 cents.
 
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Thus, we have 5n + 10d + 25q = 320. Let this be (1).
 
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We know that there are 23 coins.
 
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Thus, we have n + d + q = 23. Let this be (2).
 
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We know that there are 3 more dimes than nickels, which also means that there are 3 less nickels than dimes.
 
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Thus, we have d - 3 = n.
 
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Plugging d-3 into the other two equations for n, (1) becomes 2d + q - 3 = 23 and (2) becomes 15d + 25q - 15 = 320. (1) then becomes 2d + q = 26, and (2) then becomes 15d + 25q = 335.
 
−
 
 
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Multiplying (1) by 25, we have 50d + 25q = 650 (or 25^2 + 25). Subtracting (2) from (1) gives us 35d = 315, which means d = 9.
 
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Plugging d into d - 3 = n, n = 6.
 
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Plugging d and q into the (2) we had at the beginning of this problem, q = 8.
 
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Thus, the answer is 8 - 6 = <math>\boxed{\textbf{(C) } 2}</math>.
 
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==Solution 3==
 
 
Let the number of <math>5</math>-cent coins be <math>x,</math> the number of <math>10</math>-cent coins be <math>x+3,</math> and the number of <math>25</math>-cent coins be <math>y.</math>
 
Let the number of <math>5</math>-cent coins be <math>x,</math> the number of <math>10</math>-cent coins be <math>x+3,</math> and the number of <math>25</math>-cent coins be <math>y.</math>
  
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Substitute <math>6</math> in for <math>x</math> into one of the equations to get <math>y=8.</math>
 
Substitute <math>6</math> in for <math>x</math> into one of the equations to get <math>y=8.</math>
  
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Finally, the answer is <math>8-6=\boxed{\textbf{(C) } 2}</math>.
+
Finally, the answer is <math>8-6=\boxed{\textbf{(C) } 2}.</math>
  
 
- mutinykids
 
- mutinykids
  
−
==Video Solution==
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==Solution 3 (Three Variables)==
 +
Let <math>n,d,</math> and <math>q</math> be the numbers of <math>5</math>-cent coins, <math>10</math>-cent coins, and <math>25</math>-cent coins in Joe's collection, respectively. We are given that
 +
<cmath>\begin{align*}
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n+d+q&=23, &(1) \\
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5n+10d+25q&=320, &(2) \\
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d&=n+3. &(3)
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\end{align*}</cmath>
 +
Substituting <math>(3)</math> into each of <math>(1)</math> and <math>(2)</math> and then simplifying, we have
 +
<cmath>\begin{align*}
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2n+q&=20, \hspace{17.5mm} &(1\star) \\
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3n+5q&=58. &(2\star)
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\end{align*}</cmath>
 +
Subtracting <math>(2\star)</math> from <math>5\cdot(1\star)</math> gives <math>7n=42,</math> from which <math>n=6.</math> Substituting this into either <math>(1\star)</math> or <math>(2\star)</math> produces <math>q=8.</math>
 +
 
 +
Finally, the answer is <math>q-n=\boxed{\textbf{(C) } 2}.</math>
 +
 
 +
~MRENTHUSIASM
 +
 
 +
==Video Solution (HOW TO THINK CREATIVELY!)==
 +
https://youtu.be/zbcnOfDJmQI
 +
 
 +
~Education, the Study of Everything
 +
 
 +
 
 +
 
 +
==Video Solutions==
 
https://youtu.be/ZiZVIMmo260
 
https://youtu.be/ZiZVIMmo260
 +
  
 
https://youtu.be/BLTrtkVOZGE
 
https://youtu.be/BLTrtkVOZGE
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~savannahsolver
 
~savannahsolver
  
−
== Video Solution ==
+
==Video Solution by OmegaLearn==
 
https://youtu.be/HISL2-N5NVg?t=1861
 
https://youtu.be/HISL2-N5NVg?t=1861
  
−
~ pi_is_3.14
+
~pi_is_3.14
  
 
== See Also ==
 
== See Also ==

Latest revision as of 14:14, 3 July 2023

Problem

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 1 (One Variable)

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

~Nivek

Solution 2 (Two Variables)

Let the number of $5$-cent coins be $x,$ the number of $10$-cent coins be $x+3,$ and the number of $25$-cent coins be $y.$

Set up the following two equations with the information given in the problem: \[5x+10(x+3)+25y=320 \Rightarrow 15x+25y+30=320 \Rightarrow 15x+25y=290\] \[x+x+3+y=23 \Rightarrow 2x+3+y=23 \Rightarrow 2x+y=20\]

From there, multiply the second equation by $25$ to get \[50x+25y=500.\]

Subtract the first equation from the multiplied second equation to get $35x=210,$ or $x=6.$

Substitute $6$ in for $x$ into one of the equations to get $y=8.$

Finally, the answer is $8-6=\boxed{\textbf{(C) } 2}.$

- mutinykids

Solution 3 (Three Variables)

Let $n,d,$ and $q$ be the numbers of $5$-cent coins, $10$-cent coins, and $25$-cent coins in Joe's collection, respectively. We are given that \begin{align*} n+d+q&=23, &(1) \\ 5n+10d+25q&=320, &(2) \\ d&=n+3. &(3) \end{align*} Substituting $(3)$ into each of $(1)$ and $(2)$ and then simplifying, we have \begin{align*} 2n+q&=20, \hspace{17.5mm} &(1\star) \\ 3n+5q&=58. &(2\star) \end{align*} Subtracting $(2\star)$ from $5\cdot(1\star)$ gives $7n=42,$ from which $n=6.$ Substituting this into either $(1\star)$ or $(2\star)$ produces $q=8.$

Finally, the answer is $q-n=\boxed{\textbf{(C) } 2}.$

~MRENTHUSIASM

Video Solution (HOW TO THINK CREATIVELY!)

https://youtu.be/zbcnOfDJmQI

~Education, the Study of Everything


Video Solutions

https://youtu.be/ZiZVIMmo260


https://youtu.be/BLTrtkVOZGE

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/HISL2-N5NVg?t=1861

~pi_is_3.14

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
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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