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Difference between revisions of "2003 AMC 10A Problems/Problem 13"

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== Solution ==
 
== Solution ==
−
Let the numbers be <math>x</math>, <math>y</math>, and <math>z</math> in that order.  
+
=== Solution 1 ===
 +
Let the numbers be <math>x</math>, <math>y</math>, and <math>z</math> in that order. The given tells us that
  
−
<math>y=7z</math>
+
<cmath>\begin{eqnarray*}y&=&7z\\
 +
x&=&4(y+z)=4(7z+z)=4(8z)=32z\\
 +
x+y+z&=&32z+7z+z=40z=20\\
 +
z&=&\frac{20}{40}=\frac{1}{2}\\
 +
y&=&7z=7\cdot\frac{1}{2}=\frac{7}{2}\\
 +
x&=&32z=32\cdot\frac{1}{2}=16
 +
\end{eqnarray*}</cmath>
  
−
<math>x=4(y+z)=4(7z+z)=4(8z)=32z</math>
+
Therefore, the product of all three numbers is <math>xyz=16\cdot\frac{7}{2}\cdot\frac{1}{2}=28 \Rightarrow \boxed{\mathrm{(A)}\ 28}</math>.
  
−
<math>x+y+z=32z+7z+z=40z=20</math>
+
=== Solution 2 ===
 +
Alternatively, we can set up the system in [[equation]] form:
  
−
<math>z=\frac{20}{40}=\frac{1}{2}</math>
+
<cmath>\begin{eqnarray*}1x+1y+1z&=&20\\
 +
1x-4y-4z&=&0\\
 +
0x+1y-7z&=&0\\
 +
\end{eqnarray*}</cmath>
  
−
<math>y=7z=7\cdot\frac{1}{2}=\frac{7}{2}</math>
+
Or, in [[matrix]] form
 +
<math>
 +
\begin{bmatrix}
 +
1 & 1 & 1 \\
 +
1 & -4 & -4 \\
 +
0 & 1 & -7
 +
\end{bmatrix}
 +
\begin{bmatrix}
 +
x \\
 +
y \\
 +
z \\
 +
\end{bmatrix}
 +
=\begin{bmatrix}
 +
20 \\
 +
0 \\
 +
0 \\
 +
\end{bmatrix}
 +
</math>
  
−
<math>x=32z=32\cdot\frac{1}{2}=16</math>
+
To solve this matrix equation, we can rearrange it thus:
  
−
Therefore, the product of all three numbers is <math>xyz=16\cdot\frac{7}{2}\cdot\frac{1}{2}=28 \Rightarrow A</math>
+
<math>\begin{bmatrix}
 +
x \\
 +
y \\
 +
z \\
 +
\end{bmatrix}
 +
= \begin{bmatrix}
 +
1 & 1 & 1 \\
 +
1 & -4 & -4 \\
 +
0 & 1 & -7
 +
\end{bmatrix}
 +
^{-1}
 +
\begin{bmatrix}
 +
20 \\
 +
0 \\
 +
0 \\
 +
\end{bmatrix}
 +
</math>
  
−
Alternatively, we can set up the system in matrix form:
+
Solving this matrix equation by using [[inverse matrices]] and [[matrix multiplication]] yields
  
−
<math>x+y+z=20</math>
+
<math>\begin{bmatrix}
−
 
+
x \\
−
<math>x=4(y+z)=4y+4z</math>
+
y \\
−
 
+
z \\
−
<math>y=7z</math>
+
\end{bmatrix} =
 +
\begin{bmatrix}
 +
\frac{1}{2} \\
 +
\frac{7}{2} \\
 +
16 \\
 +
\end{bmatrix}
 +
</math>
  
−
is equivalent to
+
Which means that <math>x = \frac{1}{2}</math>, <math>y = \frac{7}{2}</math>, and <math>z = 16</math>. Therefore, <math>xyz = \frac{1}{2}\cdot\frac{7}{2}\cdot16 = 28 \Rightarrow \boxed{\mathrm{(A)}\ 28}</math>
  
−
<math>1x+1y+1z=20</math>
+
=== Solution 3 ===
 +
Let's denote the 3rd number as <math>x</math>, the 2nd as <math>7x</math>, and the 1st as <math>4(7x+x)</math> according to the information given in the problem. We know that all three numbers add up to <math>20</math>, so <math>4(8x)+7x+x = 20</math>. Solving the equation we get <math>x = \frac{1}{2}</math>. Then substitute this value of x to solve for the other two numbers. Lastly, we obtain <math>28</math> as the product of all three numbers.
  
−
<math>1x-4y-4z=0</math>
+
==Video Solution by WhyMath==
 +
https://youtu.be/3hY9YXqn5zg
  
−
<math>0x+1y-7z=0</math>
+
~savannahsolver
  
−
Or, in matrix form
+
==Video Solution==
−
<math>
+
https://www.youtube.com/watch?v=7V2k1WxooJ0  ~David
−
  \begin{bmatrix}
 
−
    1 & 1 & 1 \\
 
−
    1 & -4 & -4 \\
 
−
    0 & 1 & -7
 
−
  \end{bmatrix}
 
−
  \begin{bmatrix}
 
−
    x \\
 
−
    y \\
 
−
    z \\
 
−
  \end{bmatrix}
 
−
=
 
−
  \begin{bmatrix}
 
−
    20 \\
 
−
    0 \\
 
−
    0 \\
 
−
  \end{bmatrix}
 
−
</math>
 
−
To solve this matrix equation, we can rearrange it thus:
 
−
<math>
 
−
  \begin{bmatrix}
 
−
    x \\
 
−
    y \\
 
−
    z \\
 
−
  \end{bmatrix}
 
−
=
 
−
  \begin{bmatrix}
 
−
    1 & 1 & 1 \\
 
−
    1 & -4 & -4 \\
 
−
    0 & 1 & -7
 
−
  \end{bmatrix}
 
−
^{-1}
 
−
  \begin{bmatrix}
 
−
    20 \\
 
−
    0 \\
 
−
    0 \\
 
−
  \end{bmatrix}
 
−
</math>
 
−
Solving this matrix equation by using [[inverse matrices]] and [[matrix multiplication]] yields
 
−
<math>
 
−
  \begin{bmatrix}
 
−
    x \\
 
−
    y \\
 
−
    z \\
 
−
  \end{bmatrix}
 
−
=
 
−
\begin{bmatrix}
 
−
    \frac{1}{2} \\
 
−
    \frac{7}{2} \\
 
−
    16 \\
 
−
  \end{bmatrix}
 
−
</math>
 
−
Which means that x = \frac{1}{2}, y = \frac{7}{2}, and z = 16. Therefore, xyz = \frac{1}{2}\cdot\frac{7}{2}\cdot16 = 28
 
  
−
== See Also ==
+
== See also ==
−
*[[2003 AMC 10A Problems]]
+
{{AMC10 box|year=2003|num-b=12|num-a=14|ab=A}}
−
*[[2003 AMC 10A Problems/Problem 12|Previous Problem]]
 
−
*[[2003 AMC 10A Problems/Problem 14|Next Problem]]
 
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 14:45, 19 August 2023

Problem

The sum of three numbers is $20$. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three?

$\mathrm{(A) \ } 28\qquad \mathrm{(B) \ } 40\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 400\qquad \mathrm{(E) \ } 800$

Solution

Solution 1

Let the numbers be $x$, $y$, and $z$ in that order. The given tells us that

\begin{eqnarray*}y&=&7z\\ x&=&4(y+z)=4(7z+z)=4(8z)=32z\\ x+y+z&=&32z+7z+z=40z=20\\ z&=&\frac{20}{40}=\frac{1}{2}\\ y&=&7z=7\cdot\frac{1}{2}=\frac{7}{2}\\ x&=&32z=32\cdot\frac{1}{2}=16 \end{eqnarray*}

Therefore, the product of all three numbers is $xyz=16\cdot\frac{7}{2}\cdot\frac{1}{2}=28 \Rightarrow \boxed{\mathrm{(A)}\ 28}$.

Solution 2

Alternatively, we can set up the system in equation form:

\begin{eqnarray*}1x+1y+1z&=&20\\ 1x-4y-4z&=&0\\ 0x+1y-7z&=&0\\ \end{eqnarray*}

Or, in matrix form $\begin{bmatrix} 1 & 1 & 1 \\ 1 & -4 & -4 \\ 0 & 1 & -7 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} =\begin{bmatrix} 20 \\ 0 \\ 0 \\ \end{bmatrix}$

To solve this matrix equation, we can rearrange it thus:

$\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -4 & -4 \\ 0 & 1 & -7 \end{bmatrix} ^{-1} \begin{bmatrix} 20 \\ 0 \\ 0 \\ \end{bmatrix}$

Solving this matrix equation by using inverse matrices and matrix multiplication yields

$\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \\ \frac{7}{2} \\ 16 \\ \end{bmatrix}$

Which means that $x = \frac{1}{2}$, $y = \frac{7}{2}$, and $z = 16$. Therefore, $xyz = \frac{1}{2}\cdot\frac{7}{2}\cdot16 = 28 \Rightarrow \boxed{\mathrm{(A)}\ 28}$

Solution 3

Let's denote the 3rd number as $x$, the 2nd as $7x$, and the 1st as $4(7x+x)$ according to the information given in the problem. We know that all three numbers add up to $20$, so $4(8x)+7x+x = 20$. Solving the equation we get $x = \frac{1}{2}$. Then substitute this value of x to solve for the other two numbers. Lastly, we obtain $28$ as the product of all three numbers.

Video Solution by WhyMath

https://youtu.be/3hY9YXqn5zg

~savannahsolver

Video Solution

https://www.youtube.com/watch?v=7V2k1WxooJ0 ~David

See also

2003 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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All AMC 10 Problems and Solutions

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