Difference between revisions of "1999 AMC 8 Problems/Problem 11"

(Problem)
Line 1: Line 1:
 
==Problem==
 
==Problem==
  
Each of the five numbers 1,4,7,10, and 13  
+
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is
is placed in one of the five squares
 
so that the sum of the three numbers
 
in the horizontal row equals the sum of the three numbers
 
in the vertical column. The largest possible value for the
 
horizontal or vertical sum is
 
(A) 20 (B) 21 (C) 22 (D) 24 (E) 30
 
  
==solution==
+
<asy>
 +
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
 +
draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);
 +
</asy>
  
(D) 24: The largest sum occurs when 13 is placed in the center. This sum is 13 + 10 + 1 = 13 + 7 + 4 = 24.
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<math>\text{(A)}\ 20 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30</math>
Note: Two other common sums, 18 and 21, are possible.
 
  
OR
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==Solution==
 +
===Solution 1===
 +
The largest sum occurs when <math>13</math> is placed in the center. This sum is <math>13 + 10 + 1 = 13 + 7 + 4 = \boxed{\text{(D)}\ 24}</math>. Note: Two other common sums, <math>18</math> and <math>21</math>, are also possible.
 +
 
 +
===Solution 2===
  
 
Since the horizontal sum equals the vertical sum, twice this sum will be the sum
 
Since the horizontal sum equals the vertical sum, twice this sum will be the sum
 
of the five numbers plus the number in the center. When the center number is
 
of the five numbers plus the number in the center. When the center number is
13, the sum is the largest, [10 + 4 + 1 + 7 + 2(13)]=2 = 48=2 = 24. The other
+
<math>13</math>, the sum is the largest, <cmath>[10 + 4 + 1 + 7 + 2(13)]=2S\\ 48=2S\\ S=\boxed{\text{(D)}\ 24}</cmath>
 +
The other
 
four numbers are divided into two pairs with equal sums.
 
four numbers are divided into two pairs with equal sums.
  

Revision as of 12:42, 23 December 2012

Problem

Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is

[asy] draw((0,0)--(3,0)--(3,1)--(0,1)--cycle); draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle); [/asy]

$\text{(A)}\ 20 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 22 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30$

Solution

Solution 1

The largest sum occurs when $13$ is placed in the center. This sum is $13 + 10 + 1 = 13 + 7 + 4 = \boxed{\text{(D)}\ 24}$. Note: Two other common sums, $18$ and $21$, are also possible.

Solution 2

Since the horizontal sum equals the vertical sum, twice this sum will be the sum of the five numbers plus the number in the center. When the center number is $13$, the sum is the largest, \[[10 + 4 + 1 + 7 + 2(13)]=2S\\ 48=2S\\ S=\boxed{\text{(D)}\ 24}\] The other four numbers are divided into two pairs with equal sums.

see also

1999 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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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