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Difference between revisions of "1987 AJHSME Problems/Problem 24"

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==Solution==
 
==Solution==
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===Solution 1===
  
 
Let <math>c</math> be the number of questions correct, <math>w</math> be the number of questions wrong, and <math>b</math> be the number of questions left blank.  We are given that  
 
Let <math>c</math> be the number of questions correct, <math>w</math> be the number of questions wrong, and <math>b</math> be the number of questions left blank.  We are given that  
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Finally, we have <math>w=20-12-2=6</math>.  We want <math>c</math>, so the answer is <math>12</math>, or <math>\boxed{\text{D}}</math>.
 
Finally, we have <math>w=20-12-2=6</math>.  We want <math>c</math>, so the answer is <math>12</math>, or <math>\boxed{\text{D}}</math>.
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===Solution 2===
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If John answered 16 questions correctly, then he answered at most 4 questions incorrectly, giving him at least <math>16 \cdot 5 - 4 \cdot 2 = 72</math> points. Therefore, John did not answer 16 questions correctly. If he answered 12 questions correctly and 6 questions incorrectly (leaving 2 questions unanswered), then he scored <math>12 \cdot 5 - 6 \cdot 2 = 48</math> points. As all other options are less than 12, we conclude that 12 is the most questions John could have answered correctly, and the answer is <math>\boxed{\text{D}}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 01:42, 4 October 2015

Problem

A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answered correctly?

$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 16$

Solution

Solution 1

Let $c$ be the number of questions correct, $w$ be the number of questions wrong, and $b$ be the number of questions left blank. We are given that \begin{align} c+w+b &= 20 \\ 5c-2w &= 48 \end{align}

Adding equation $(2)$ to double equation $(1)$, we get \[7c+2b=88\]

Since we want to maximize the value of $c$, we try to find the largest multiple of $7$ less than $88$. This is $84=7\times 12$, so let $c=12$. Then we have \[7(12)+2b=88\Rightarrow b=2\]

Finally, we have $w=20-12-2=6$. We want $c$, so the answer is $12$, or $\boxed{\text{D}}$.

Solution 2

If John answered 16 questions correctly, then he answered at most 4 questions incorrectly, giving him at least $16 \cdot 5 - 4 \cdot 2 = 72$ points. Therefore, John did not answer 16 questions correctly. If he answered 12 questions correctly and 6 questions incorrectly (leaving 2 questions unanswered), then he scored $12 \cdot 5 - 6 \cdot 2 = 48$ points. As all other options are less than 12, we conclude that 12 is the most questions John could have answered correctly, and the answer is $\boxed{\text{D}}$.

See Also

1987 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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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