Difference between revisions of "2013 AMC 12A Problems/Problem 2"

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== Problem 2 ==
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A softball team played ten games, scoring <math>1,2,3,4,5,6,7,8,9</math>, and <math>10</math> runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
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<math> \textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55 </math>
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==Solution==
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To score twice as many runs as their opponent, the softball team must have scored an even number.
 
To score twice as many runs as their opponent, the softball team must have scored an even number.
  
Therefore we can deduce that when they scored 1, 3, 5, 7, and 9 runs, they lost by one run, and
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Therefore we can deduce that when they scored an odd number of runs, they lost by one, and when they scored an even number of runs, they won by twice as much.
when they scored 2, 4, 6, 8, and 10 runs, they scored twice as many runs as their opponent.
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Therefore, the total runs by the opponent is <math>(2+4+6+8+10)+(1+2+3+4+5) = 45</math>, which is <math>C</math>
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==Video Solution==
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https://www.youtube.com/watch?v=2vf843cvVzo?t=91 (problem 2 starts at 1:31 in the video)
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~sugar_rush
  
Therefore, the total runs by the opponent is (2+4+6+8+10)+(1+2+3+4+5), which is 45, or C
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== See also ==
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{{AMC12 box|year=2013|ab=A|num-b=1|num-a=3}}
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{{MAA Notice}}

Latest revision as of 18:24, 23 August 2021

Problem 2

A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

$\textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55$

Solution

To score twice as many runs as their opponent, the softball team must have scored an even number.

Therefore we can deduce that when they scored an odd number of runs, they lost by one, and when they scored an even number of runs, they won by twice as much.

Therefore, the total runs by the opponent is $(2+4+6+8+10)+(1+2+3+4+5) = 45$, which is $C$

Video Solution

https://www.youtube.com/watch?v=2vf843cvVzo?t=91 (problem 2 starts at 1:31 in the video)

~sugar_rush

See also

2013 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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

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