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Difference between revisions of "2004 AMC 8 Problems"

(→‎Problem 20)
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Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
 
Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
  
−
<math> \textbf{(A)}2\qquad\textbf{(B)}4\qquad\textbf{(C)}6\qquad\textbf{(D)}8\qquad\textbf{(E)}10 </math>
+
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10 </math>
  
 
[[2004 AMC 8 Problems/Problem 4|Solution]]
 
[[2004 AMC 8 Problems/Problem 4|Solution]]
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The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
 
The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
  
−
<math> \textbf{(A)}4\qquad\textbf{(B)}7\qquad\textbf{(C)}8\qquad\textbf{(D)}15\qquad\textbf{(E)}16 </math>
+
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 </math>
  
 
[[2004 AMC 8 Problems/Problem 5|Solution]]
 
[[2004 AMC 8 Problems/Problem 5|Solution]]
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After Sally takes <math>20</math> shots, she has made <math>55\%</math> of her shots. After she takes <math>5</math> more shots, she raises her percentage to <math>56\%</math>. How many of the last <math>5</math> shots did she make?
 
After Sally takes <math>20</math> shots, she has made <math>55\%</math> of her shots. After she takes <math>5</math> more shots, she raises her percentage to <math>56\%</math>. How many of the last <math>5</math> shots did she make?
  
−
<math> \textbf{(A)}1\qquad\textbf{(B)}2\qquad\textbf{(C)}3\qquad\textbf{(D)}4\qquad\textbf{(E)}5 </math>
+
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>
  
 
[[2004 AMC 8 Problems/Problem 6|Solution]]
 
[[2004 AMC 8 Problems/Problem 6|Solution]]
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Find the number of two-digit positive integers whose digits total <math>7</math>.
 
Find the number of two-digit positive integers whose digits total <math>7</math>.
  
−
<math> \mathrm{(A)\ 6 }\qquad\mathrm{(B)\ 7 }\qquad\mathrm{(C)\ 8 }\qquad\mathrm{(D)\ 9 }\qquad\mathrm{(E)\ 10 } </math>
+
<math> \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10 </math>
  
 
[[2004 AMC 8 Problems/Problem 8|Solution]]
 
[[2004 AMC 8 Problems/Problem 8|Solution]]
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numbers is <math>48</math>. What is the average of the last three numbers?
 
numbers is <math>48</math>. What is the average of the last three numbers?
  
−
<math> \mathrm{(A)\ 55 }\qquad\mathrm{(B)\ 56 }\qquad\mathrm{(C)\ 57 }\qquad\mathrm{(D)\ 58 }\qquad\mathrm{(E)\ 59 } </math>
+
<math> \textbf{(A)}\ 55\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 57\qquad\textbf{(D)}\ 58\qquad\textbf{(E)}\ 59</math>
  
 
[[2004 AMC 8 Problems/Problem 9|Solution]]
 
[[2004 AMC 8 Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
 +
Handy Aaron helped a neighbor <math>1 \frac14</math> hours on Monday, <math>50</math> minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid <math>\textdollar 3</math> per hour. How much did he earn for the week?
 +
 +
<math>\textbf{(A)}\ \textdollar 8 \qquad \textbf{(B)}\ \textdollar 9 \qquad \textbf{(C)}\ \textdollar 10 \qquad \textbf{(D)}\ \textdollar 12 \qquad \textbf{(E)}\ \textdollar 15</math>
 +
 +
[[2004 AMC 8 Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
  
−
The numbers <math>-2</math>, <math>4</math>, <math>6</math>, <math>9</math> and <math>12</math> are rearranged according to these rules:
+
The numbers <math>-2, 4, 6, 9</math> and <math>12</math> are rearranged according to these rules:
 
          
 
          
 
         1. The largest isn’t first, but it is in one of the first three places.  
 
         1. The largest isn’t first, but it is in one of the first three places.  
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What is the average of the first and last numbers?
 
What is the average of the first and last numbers?
  
−
<math> \mathrm{(A)\ 3.5 }\qquad\mathrm{(B)\ 5 }\qquad\mathrm{(C)\ 6.5 }\qquad\mathrm{(D)\ 7.5 }\qquad\mathrm{(E)\ 8 } </math>
+
<math>\textbf{(A)}\ 3.5 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6.5 \qquad \textbf{(D)}\ 7.5 \qquad \textbf{(E)}\ 8</math>
 +
 
 +
[[2004 AMC 8 Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
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more but leaves the phone on, how many more hours will the battery last?
 
more but leaves the phone on, how many more hours will the battery last?
  
−
<math> \mathrm{(A)\ 7 }\qquad\mathrm{(B)\ 8 }\qquad\mathrm{(C)\ 11 }\qquad\mathrm{(D)\ 14 }\qquad\mathrm{(E)\ 15 } </math>
+
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15</math>
 +
 
 +
[[2004 AMC 8 Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
 +
[[2004 AMC 8 Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
 +
[[2004 AMC 8 Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
 +
[[2004 AMC 8 Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
 +
[[2004 AMC 8 Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
 +
[[2004 AMC 8 Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
 +
 +
[[2004 AMC 8 Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
 +
 +
[[2004 AMC 8 Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
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From: http://faculty.wiu.edu/JR-Olsen/wiu/contests/AMC/pastAMC/AMC-2010-Disc/AMC%208/Contests%20and%20Solutions/2004AMC8.pdf
 
From: http://faculty.wiu.edu/JR-Olsen/wiu/contests/AMC/pastAMC/AMC-2010-Disc/AMC%208/Contests%20and%20Solutions/2004AMC8.pdf
 +
 +
[[2004 AMC 8 Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
 +
 +
[[2004 AMC 8 Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
 +
 +
[[2004 AMC 8 Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
 +
[[2004 AMC 8 Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
 +
[[2004 AMC 8 Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
 +
 +
[[2004 AMC 8 Problems/Problem 25|Solution]]

Revision as of 04:56, 24 December 2012

Problem 1

Ona map, a $12$-centimeter length represents $72$ kilometers. How many kilometers does a $17$-centimeter length represent?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 102\qquad\textbf{(C)}\ 204\qquad\textbf{(D)}\ 864\qquad\textbf{(E)}\ 1224$

Solution

Problem 2

How many different four-digit numbers can be formed be rearranging the four digits in $2004$?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 81$

Solution

Problem 3

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for $18$ people. If they shared, how many meals should they have ordered to have just enough food for the $12$ of them?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 18$

Solution

Problem 4

The following information is needed to solve problems 4, 5 and 6.

Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament.

Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$

Solution

Problem 5

The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16$

Solution

Problem 6

After Sally takes $20$ shots, she has made $55\%$ of her shots. After she takes $5$ more shots, she raises her percentage to $56\%$. How many of the last $5$ shots did she make?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Problem 7

An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?

$\textbf{(A)}\ 134\qquad\textbf{(B)}\ 155\qquad\textbf{(C)}\ 176\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 243$

Solution

Problem 8

Find the number of two-digit positive integers whose digits total $7$.

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10$

Solution

Problem 9

The average of the five numbers in a list is $54$. The average of the first two numbers is $48$. What is the average of the last three numbers?

$\textbf{(A)}\ 55\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 57\qquad\textbf{(D)}\ 58\qquad\textbf{(E)}\ 59$

Solution

Problem 10

Handy Aaron helped a neighbor $1 \frac14$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\textdollar 3$ per hour. How much did he earn for the week?

$\textbf{(A)}\ \textdollar 8 \qquad \textbf{(B)}\ \textdollar 9 \qquad \textbf{(C)}\ \textdollar 10 \qquad \textbf{(D)}\ \textdollar 12 \qquad \textbf{(E)}\ \textdollar 15$

Solution

Problem 11

The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules: 

        1. The largest isn’t first, but it is in one of the first three places. 
        2. The smallest isn’t last, but it is in one of the last three places. 
        3. The median isn’t first or last.

What is the average of the first and last numbers?

$\textbf{(A)}\ 3.5 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6.5 \qquad \textbf{(D)}\ 7.5 \qquad \textbf{(E)}\ 8$

Solution

Problem 12

Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes. If she doesn’t talk any more but leaves the phone on, how many more hours will the battery last?

$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

"Two thirds pf the people in a room are seated in three fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?"

From: http://faculty.wiu.edu/JR-Olsen/wiu/contests/AMC/pastAMC/AMC-2010-Disc/AMC%208/Contests%20and%20Solutions/2004AMC8.pdf

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

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