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Difference between revisions of "2001 AMC 10 Problems/Problem 22"

(→‎Solution)
(→‎Solution)
Line 23: Line 23:
 
label("$w$",(2.5,2.5));</asy>
 
label("$w$",(2.5,2.5));</asy>
  
−
==Solution==
+
==Solutions==
 +
 
 +
===Solution 1===
  
 
We know that <math> y+z=2v </math>, so we could find one variable rather than two.
 
We know that <math> y+z=2v </math>, so we could find one variable rather than two.
Line 76: Line 78:
 
Thus <math> 66-18-25=66-43=v=23 </math>.  
 
Thus <math> 66-18-25=66-43=v=23 </math>.  
 
   
 
   
 +
Since we needed <math> 2v </math> and we know <math> v=23 </math>, <math> 23 \times 2 = \boxed{\textbf{(D)}\ 46} </math>.
 +
 +
===Solution 2===
 +
 +
<math> v+24+w=43+v </math>
 +
 +
<math> 24+w=43 </math>
 +
 +
<math> w=19 </math>
 +
 +
<asy>
 +
unitsize(10mm);
 +
defaultpen(linewidth(1pt));
 +
for(int i=0; i<=3; ++i)
 +
{
 +
draw((0,i)--(3,i));
 +
draw((i,0)--(i,3));
 +
}
 +
label("$25$",(0.5,0.5));
 +
label("$z$",(1.5,0.5));
 +
label("$21$",(2.5,0.5));
 +
label("$18$",(0.5,1.5));
 +
label("$x$",(1.5,1.5));
 +
label("$y$",(2.5,1.5));
 +
label("$v$",(0.5,2.5));
 +
label("$24$",(1.5,2.5));
 +
label("$19$",(2.5,2.5));</asy>
 +
 +
<math> 44+x=24+x+z </math>
 +
<math> z=20 </math>
 +
 +
<asy>
 +
unitsize(10mm);
 +
defaultpen(linewidth(1pt));
 +
for(int i=0; i<=3; ++i)
 +
{
 +
draw((0,i)--(3,i));
 +
draw((i,0)--(i,3));
 +
}
 +
label("$25$",(0.5,0.5));
 +
label("$20$",(1.5,0.5));
 +
label("$21$",(2.5,0.5));
 +
label("$18$",(0.5,1.5));
 +
label("$x$",(1.5,1.5));
 +
label("$y$",(2.5,1.5));
 +
label("$v$",(0.5,2.5));
 +
label("$24$",(1.5,2.5));
 +
label("$19$",(2.5,2.5));</asy>
 +
 +
The magic sum is determined by the bottom row. <math> 25+20+21=66 </math>.
 +
 +
Solving for <math> y </math>:
  
−
Since we needed <math> 2v </math> and we know <math> v=23 </math>, <math> 23 \times 2 = \boxed{\textbf{(D)}\ 46} </math>.
+
<math> y=66-19-21=66-40=26 </math>.
 +
 
 +
To find our answer, we need to find <math> y+z </math>. <math> y+z=20+26 = \boxed{\textbf{(D)}\ 46} </math>.

Revision as of 19:54, 16 March 2011

Problem

In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v$, $w$, $x$, $y$, and $z$. Find $y + z$.

$\textbf{(A)}\ 43 \qquad \textbf{(B)}\ 44 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 47$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$w$",(2.5,2.5));[/asy]

Solutions

Solution 1

We know that $y+z=2v$, so we could find one variable rather than two.

$v+24+w=43+v$

$24+w=43$

$w=19$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]

$44+x=24+x+z$ $z=20$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$20$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]

The sum per row is $25+21+20=66$.

Thus $66-18-25=66-43=v=23$.

Since we needed $2v$ and we know $v=23$, $23 \times 2 = \boxed{\textbf{(D)}\ 46}$.

Solution 2

$v+24+w=43+v$

$24+w=43$

$w=19$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]

$44+x=24+x+z$ $z=20$

[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$20$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]

The magic sum is determined by the bottom row. $25+20+21=66$.

Solving for $y$:

$y=66-19-21=66-40=26$.

To find our answer, we need to find $y+z$. $y+z=20+26 = \boxed{\textbf{(D)}\ 46}$.

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