Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
ONLINE AMC 8 PREP WITH AOPS
Top scorers around the country use AoPS. Join training courses for beginners and advanced students.
VIEW CATALOG

Difference between revisions of "2011 AMC 8 Problems"

(→‎Problem 18)
(→‎Problem 17)
Line 144: Line 144:
 
Let <math>w</math>, <math>x</math>, <math>y</math>, and <math>z</math> be whole numbers. If <math>2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588</math>, then what does <math>2w + 3x + 5y + 7z</math> equal?
 
Let <math>w</math>, <math>x</math>, <math>y</math>, and <math>z</math> be whole numbers. If <math>2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588</math>, then what does <math>2w + 3x + 5y + 7z</math> equal?
  
−
<math> \text{(A) } 21\qquad\text{(B) }25\qquad\text{(C) }27\qquad\text{(D) }35\qquad\text{(E) }56 </math>
+
<math> \textbf{(A) } 21\qquad\textbf{(B) }25\qquad\textbf{(C) }27\qquad\textbf{(D) }35\qquad\textbf{(E) }56 </math>
  
  

Revision as of 18:09, 25 November 2011

Problem 1

Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie recieve?

$\text{(A)}\ \textdollar 1.50 \qquad \text{(B)}\ \textdollar 2.00 \qquad \text{(C)}\ \textdollar 2.50 \qquad \text{(D)}\ \textdollar 3.00 \qquad \text{(E)}\ \textdollar 3.50$


Solution

Problem 2

Karl's rectangular vegetable garden is $20$ feet by $45$ feet, and Makenna's is $25$ feet by $40$ feet. Whose garden is larger in area?

$\text{(A) Karl's garden is larger by 100 square feet.}$

$\text{(B) Karl's garden is larger by 25 square feet.}$

$\text{(C) The gardens are the same size.}$

$\text{(D) Makenna's garden is larger by 25 square feet.}$

$\text{(E) Makenna's garden is larger by 100 square feet.}$


Solution

Problem 3

Solution

Problem 4

Here is a list of the numbers of fish that Tyler caught in nine outings last summer: \[2,0,1,3,0,3,3,1,2.\] Which statement about the mean, median, and mode is true?

$\text{(A) median} < \text{mean} < \text{mode} \qquad \text{(B) mean} < \text{mode} < \text{median} \\ \\ \text{(C) mean} < \text{median} < \text{mode} \qquad \text{(D) median} < \text{mode} < \text{mean} \\ \\ \text{(E) mode} < \text{median} < \text{mean}$


Solution

Problem 5

What time was it $2011$ minutes after midnight on January 1, 2011?

$\text{(A) January 1 at 9:31PM}$

$\text{(B) January 1 at 11:51PM}$

$\text{(C) January 2 at 3:11AM}$

$\text{(D) January 2 at 9:31AM}$

$\text{(E) January 2 at 6:01PM}$


Solution

Problem 6

In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?

$\text{(A) 20} \qquad\text{(B) 25} \qquad\text{(C) 45} \qquad\text{(D) 306} \qquad\text{(E) 351}$


Solution

Problem 7

Solution

Problem 8

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?

$\text{(A) 4} \qquad\text{(B) 5} \qquad\text{(C) 6} \qquad\text{(D) 7} \qquad\text{(E) 9}$


Solution

Problem 9

Solution

Problem 10

The taxi fare in Gotham City is $2.40 for the first $\frac12$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?

$\text{(A) 3.0}\qquad\text{(B) 3.25}\qquad\text{(C) 3.3}\qquad\text{(D) 3.5}\qquad\text{(E) 3.75}$


Solution

Problem 11

Solution

Problem 12

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

$\text{(A) } \frac14 \qquad\text{(B) } \frac13 \qquad\text{(C) } \frac12 \qquad\text{(D) } \frac23 \qquad\text{(E) } \frac34$


Solution

Problem 13

Solution

Problem 14

There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

$\text{(A) } \dfrac7{18} \qquad\text{(B) } \dfrac7{15} \qquad\text{(C) } \dfrac{22}{45} \qquad\text{(D) } \dfrac12 \qquad\text{(E) } \dfrac{23}{45}$


Solution

Problem 15

How many digits are in the product $4^5 \cdot 5^{10}$?

$\text{(A) } 8 \qquad\text{(B) } 9 \qquad\text{(C) } 10 \qquad\text{(D) } 11 \qquad\text{(E) } 12$


Solution

Problem 16

Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$?

$\text{(A) } A = \dfrac9{16}B \qquad\text{(B) } A = \dfrac34B \qquad\text{(C) } A=B \qquad\text{(D) } A = \dfrac43B \\ \\ \text{(E) }A = \dfrac{16}9B$


Solution

Problem 17

Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?

$\textbf{(A) } 21\qquad\textbf{(B) }25\qquad\textbf{(C) }27\qquad\textbf{(D) }35\qquad\textbf{(E) }56$


Solution

Problem 18

A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?

$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac5{12}\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac7{12}\qquad\textbf{(E) }\dfrac56$


Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?

[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]

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


Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_8_Problems&oldid=43337"