Difference between revisions of "2012 AMC 10A Problems/Problem 8"
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− | == Problem | + | {{duplicate|[[2012 AMC 12A Problems|2012 AMC 12A #6]] and [[2012 AMC 10A Problems|2012 AMC 10A #8]]}} |
+ | |||
+ | == Problem == | ||
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? | The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? | ||
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math> | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | Let the three numbers be equal to <math>a</math>, <math>b</math>, and <math>c</math>. We can now write three equations: | ||
+ | |||
+ | <math>a+b=12</math> | ||
+ | |||
+ | <math>b+c=17</math> | ||
+ | |||
+ | <math>a+c=19</math> | ||
+ | |||
+ | Adding these equations together, we get that | ||
+ | |||
+ | <math>2(a+b+c)=48</math> and | ||
+ | |||
+ | <math>a+b+c=24</math> | ||
+ | |||
+ | Substituting the original equations into this one, we find | ||
+ | |||
+ | <math>c+12=24</math> | ||
+ | |||
+ | <math>a+17=24</math> | ||
+ | |||
+ | <math>b+19=24</math> | ||
+ | |||
+ | Therefore, our numbers are 12, 7, and 5. The middle number is <math>\boxed{\textbf{(D)}\ 7}</math> | ||
+ | |||
+ | ==Solution 2 (Faster)== | ||
+ | |||
+ | Let the three numbers be <math>a</math>, <math>b</math> and <math>c</math> and <math>a<b<c</math>. We get the three equations: | ||
+ | |||
+ | <math>a+b=12</math> | ||
+ | |||
+ | <math>a+c=17</math> | ||
+ | |||
+ | <math>b+c=19</math> | ||
+ | |||
+ | To isolate <math>b</math>, We add the first and last equations and then subtract the second one. | ||
+ | |||
+ | <math>(a+b)+(b+c)-(a+c) = 12+19-17 \Rightarrow 2b=14 \Rightarrow b = 7</math> | ||
+ | |||
+ | Because <math>b</math> is the middle number, the middle number is <math>\boxed{\textbf{(D)}\ 7}</math> | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/sU5NAg6bhxI | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | == See Also == | ||
+ | |||
+ | {{AMC10 box|year=2012|ab=A|num-b=7|num-a=9}} | ||
+ | {{AMC12 box|year=2012|ab=A|num-b=5|num-a=7}} | ||
+ | |||
+ | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 12:55, 1 July 2023
- The following problem is from both the 2012 AMC 12A #6 and 2012 AMC 10A #8, so both problems redirect to this page.
Contents
Problem
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
Solution 1
Let the three numbers be equal to , , and . We can now write three equations:
Adding these equations together, we get that
and
Substituting the original equations into this one, we find
Therefore, our numbers are 12, 7, and 5. The middle number is
Solution 2 (Faster)
Let the three numbers be , and and . We get the three equations:
To isolate , We add the first and last equations and then subtract the second one.
Because is the middle number, the middle number is
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
See Also
2012 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 |
2012 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.