View source for 2011 AMC 8 Problems

==Problem 1==
Margie bought <math> 3 </math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?

<math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{(C) }\ \textdollar 2.50 \qquad \textbf{(D) }\ \textdollar 3.00 \qquad \textbf{(E) }\ \textdollar 3.50</math>


[[2011 AMC 8 Problems/Problem 1|Solution]]

==Problem 2==
Karl's rectangular vegetable garden is <math> 20 </math> feet by <math> 45 </math> feet, and Makenna's is <math> 25 </math> feet by <math> 40 </math> feet. Which of the following statements are true?

<math>\textbf{(A) }\text{Karl's garden is larger by 100 square feet.}</math>

<math>\textbf{(B) }\text{Karl's garden is larger by 25 square feet.}</math>
 
<math>\textbf{(C) }\text{The gardens are the same size.}</math> 

<math>\textbf{(D) }\text{Makenna's garden is larger by 25 square feet.}</math>

<math>\textbf{(E) }\text{Makenna's garden is larger by 100 square feet.}</math>



[[2011 AMC 8 Problems/Problem 2|Solution]]

==Problem 3==
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?<br />
<asy>
filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black);
filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black);
filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black);
draw((4,0)--(4,5));
draw((3,0)--(3,5));
draw((2,0)--(2,5));
draw((1,0)--(1,5));
draw((0,4)--(5,4));
draw((0,3)--(5,3));
draw((0,2)--(5,2));
draw((0,1)--(5,1));
</asy>

<math> \textbf{(A) }8:17 \qquad\textbf{(B) }25:49 \qquad\textbf{(C) }36:25 \qquad\textbf{(D) }32:17 \qquad\textbf{(E) }36:17</math>

[[2011 AMC 8 Problems/Problem 3|Solution]]

==Problem 4==
Here is a list of the numbers of fish that Tyler caught in nine outings last summer: <cmath>2,0,1,3,0,3,3,1,2.</cmath> Which statement about the mean, median, and mode is true?

<math>\textbf{(A) }\text{median} < \text{mean} < \text{mode} \qquad \textbf{(B) }\text{mean} < \text{mode} < \text{median} \\ \\ \textbf{(C) }\text{mean} < \text{median} < \text{mode} \qquad \textbf{(D) }\text{median} < \text{mode} < \text{mean} \\ \\ \textbf{(E) }\text{mode} < \text{median} < \text{mean}</math>


[[2011 AMC 8 Problems/Problem 4|Solution]]

==Problem 5==
What time was it <math>2011</math> minutes after midnight on January 1, 2011?

<math>\textbf{(A) }\text{January 1 at 9:31 PM}</math>

<math>\textbf{(B) }\text{January 1 at 11:51 PM}</math> 

<math>\textbf{(C) }\text{January 2 at 3:11 AM}</math> 

<math>\textbf{(D) }\text{January 2 at 9:31 AM}</math> 

<math>\textbf{(E) }\text{January 2 at 6:01 PM}</math>


[[2011 AMC 8 Problems/Problem 5|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 own a motorcycle?

<math> \textbf{(A) }20 \qquad\textbf{(B) }25 \qquad\textbf{(C) }45 \qquad\textbf{(D) }306 \qquad\textbf{(E) }351</math>


[[2011 AMC 8 Problems/Problem 6|Solution]]

==Problem 7==

Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?

<asy>
import graph; size(7.01cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.42,xmax=14.59,ymin=-10.08,ymax=5.26; 
pair A=(0,0), B=(4,0), C=(0,4), D=(4,4), F=(2,0), G=(3,0), H=(1,4), I=(2,4), J=(3,4), K=(0,-2), L=(4,-2), M=(0,-6), O=(0,-4), P=(4,-4), Q=(2,-2), R=(2,-6), T=(6,4), U=(10,0), V=(10,4), Z=(10,2), A_1=(8,4), B_1=(8,0), C_1=(6,-2), D_1=(10,-2), E_1=(6,-6), F_1=(10,-6), G_1=(6,-4), H_1=(10,-4), I_1=(8,-2), J_1=(8,-6), K_1=(8,-4); 
draw(C--H--(1,0)--A--cycle,linewidth(1.6)); draw(M--O--Q--R--cycle,linewidth(1.6)); draw(A_1--V--Z--cycle,linewidth(1.6)); draw(G_1--K_1--J_1--E_1--cycle,linewidth(1.6)); 
draw(C--D); draw(D--B); draw(B--A); draw(A--C); draw(H--(1,0)); draw(I--F); draw(J--G); draw(C--H,linewidth(1.6)); draw(H--(1,0),linewidth(1.6)); draw((1,0)--A,linewidth(1.6)); draw(A--C,linewidth(1.6)); draw(K--L); draw((4,-6)--L); draw((4,-6)--M); draw(M--K); draw(O--P); draw(Q--R); draw(O--Q); draw(M--O,linewidth(1.6)); draw(O--Q,linewidth(1.6)); draw(Q--R,linewidth(1.6)); draw(R--M,linewidth(1.6)); draw(T--V); draw(V--U); draw(U--(6,0)); draw((6,0)--T); draw((6,2)--Z); draw(A_1--B_1); draw(A_1--Z); draw(A_1--V,linewidth(1.6)); draw(V--Z,linewidth(1.6)); draw(Z--A_1,linewidth(1.6)); draw(C_1--D_1); draw(D_1--F_1); draw(F_1--E_1); draw(E_1--C_1); draw(G_1--H_1); draw(I_1--J_1); draw(G_1--K_1,linewidth(1.6)); draw(K_1--J_1,linewidth(1.6)); draw(J_1--E_1,linewidth(1.6)); draw(E_1--G_1,linewidth(1.6)); 
dot(A,linewidth(1pt)+ds); dot(B,linewidth(1pt)+ds); dot(C,linewidth(1pt)+ds); dot(D,linewidth(1pt)+ds); dot((1,0),linewidth(1pt)+ds); dot(F,linewidth(1pt)+ds); dot(G,linewidth(1pt)+ds); dot(H,linewidth(1pt)+ds); dot(I,linewidth(1pt)+ds); dot(J,linewidth(1pt)+ds); dot(K,linewidth(1pt)+ds); dot(L,linewidth(1pt)+ds); dot(M,linewidth(1pt)+ds); dot((4,-6),linewidth(1pt)+ds); dot(O,linewidth(1pt)+ds); dot(P,linewidth(1pt)+ds); dot(Q,linewidth(1pt)+ds); dot(R,linewidth(1pt)+ds); dot((6,0),linewidth(1pt)+ds); dot(T,linewidth(1pt)+ds); dot(U,linewidth(1pt)+ds); dot(V,linewidth(1pt)+ds); dot((6,2),linewidth(1pt)+ds); dot(Z,linewidth(1pt)+ds); dot(A_1,linewidth(1pt)+ds); dot(B_1,linewidth(1pt)+ds); dot(C_1,linewidth(1pt)+ds); dot(D_1,linewidth(1pt)+ds); dot(E_1,linewidth(1pt)+ds); dot(F_1,linewidth(1pt)+ds); dot(G_1,linewidth(1pt)+ds); dot(H_1,linewidth(1pt)+ds); dot(I_1,linewidth(1pt)+ds); dot(J_1,linewidth(1pt)+ds); dot(K_1,linewidth(1pt)+ds); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);</asy>

<math> \textbf{(A)}12\frac{1}{2}\qquad\textbf{(B)}20\qquad\textbf{(C)}25\qquad\textbf{(D)}33\frac{1}{3}\qquad\textbf{(E)}37\frac{1}{2} </math>

[[2011 AMC 8 Problems/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?

<math> \textbf{(A) }4 \qquad\textbf{(B) }5 \qquad\textbf{(C) }6 \qquad\textbf{(D) }7 \qquad\textbf{(E) }9 </math>


[[2011 AMC 8 Problems/Problem 8|Solution]]

==Problem 9==

Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?
<asy>
import graph; size(8.76cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.58,xmax=10.19,ymin=-4.43,ymax=9.63; 
draw((0,0)--(0,8)); draw((0,0)--(8,0)); draw((0,1)--(8,1)); draw((0,2)--(8,2)); draw((0,3)--(8,3)); draw((0,4)--(8,4)); draw((0,5)--(8,5)); draw((0,6)--(8,6)); draw((0,7)--(8,7)); draw((1,0)--(1,8)); draw((2,0)--(2,8)); draw((3,0)--(3,8)); draw((4,0)--(4,8)); draw((5,0)--(5,8)); draw((6,0)--(6,8)); draw((7,0)--(7,8)); label("$1$",(0.95,-0.24),SE*lsf); label("$2$",(1.92,-0.26),SE*lsf); label("$3$",(2.92,-0.31),SE*lsf); label("$4$",(3.93,-0.26),SE*lsf); label("$5$",(4.92,-0.27),SE*lsf); label("$6$",(5.95,-0.29),SE*lsf); label("$7$",(6.94,-0.27),SE*lsf); label("$5$",(-0.49,1.22),SE*lsf); label("$10$",(-0.59,2.23),SE*lsf); label("$15$",(-0.61,3.22),SE*lsf); label("$20$",(-0.61,4.23),SE*lsf); label("$25$",(-0.59,5.22),SE*lsf); label("$30$",(-0.59,6.2),SE*lsf); label("$35$",(-0.56,7.18),SE*lsf); draw((0,0)--(1,1),linewidth(1.6)); draw((1,1)--(2,3),linewidth(1.6)); draw((2,3)--(4,4),linewidth(1.6)); draw((4,4)--(7,7),linewidth(1.6)); label("HOURS",(3.41,-0.85),SE*lsf); label("M",(-1.39,5.32),SE*lsf); label("I",(-1.34,4.93),SE*lsf); label("L",(-1.36,4.51),SE*lsf); label("E",(-1.37,4.11),SE*lsf); label("S",(-1.39,3.7),SE*lsf); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);</asy>

<math> \textbf{(A)}2\qquad\textbf{(B)}2.5\qquad\textbf{(C)}4\qquad\textbf{(D)}4.5\qquad\textbf{(E)}5 </math>

[[2011 AMC 8 Problems/Problem 9|Solution]]

==Problem 10==

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

<math> \textbf{(A) } 3.0\qquad\textbf{(B) }3.25\qquad\textbf{(C) }3.3\qquad\textbf{(D) }3.5\qquad\textbf{(E) }3.75 </math>


[[2011 AMC 8 Problems/Problem 10|Solution]]

==Problem 11==

The graph shows the number of minutes studied by both Asha (black bar) and Sasha(grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
<asy>
size(300);
real i;
defaultpen(linewidth(0.8));
draw((0,140)--origin--(220,0));
for(i=1;i<13;i=i+1) {
draw((0,10*i)--(220,10*i));
}
label("$0$",origin,W);
label("$20$",(0,20),W);
label("$40$",(0,40),W);
label("$60$",(0,60),W);
label("$80$",(0,80),W);
label("$100$",(0,100),W);
label("$120$",(0,120),W);
path MonD=(20,0)--(20,60)--(30,60)--(30,0)--cycle,MonL=(30,0)--(30,70)--(40,70)--(40,0)--cycle,TuesD=(60,0)--(60,90)--(70,90)--(70,0)--cycle,TuesL=(70,0)--(70,80)--(80,80)--(80,0)--cycle,WedD=(100,0)--(100,100)--(110,100)--(110,0)--cycle,WedL=(110,0)--(110,120)--(120,120)--(120,0)--cycle,ThurD=(140,0)--(140,80)--(150,80)--(150,0)--cycle,ThurL=(150,0)--(150,110)--(160,110)--(160,0)--cycle,FriD=(180,0)--(180,70)--(190,70)--(190,0)--cycle,FriL=(190,0)--(190,50)--(200,50)--(200,0)--cycle;
fill(MonD,black);
fill(MonL,grey);
fill(TuesD,black);
fill(TuesL,grey);
fill(WedD,black);
fill(WedL,grey);
fill(ThurD,black);
fill(ThurL,grey);
fill(FriD,black);
fill(FriL,grey);
draw(MonD^^MonL^^TuesD^^TuesL^^WedD^^WedL^^ThurD^^ThurL^^FriD^^FriL);
label("M",(30,-5),S);
label("Tu",(70,-5),S);
label("W",(110,-5),S);
label("Th",(150,-5),S);
label("F",(190,-5),S);
label("M",(-25,85),W);
label("I",(-27,75),W);
label("N",(-25,65),W);
label("U",(-25,55),W);
label("T",(-25,45),W);
label("E",(-25,35),W);
label("S",(-26,25),W);</asy>

<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 </math>


[[2011 AMC 8 Problems/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?

<math> \textbf{(A) } \frac14 \qquad\textbf{(B) } \frac13 \qquad\textbf{(C) } \frac12 \qquad\textbf{(D) } \frac23 \qquad\textbf{(E) } \frac34 </math>


[[2011 AMC 8 Problems/Problem 12|Solution]]

==Problem 13==

Two congruent squares, <math>ABCD</math> and <math>PQRS</math>, have side length <math>15</math>. They overlap to form the <math>15</math> by <math>25</math> rectangle <math>AQRD</math> shown. What percent of the area of rectangle <math>AQRD</math> is shaded? 
<asy>
filldraw((0,0)--(25,0)--(25,15)--(0,15)--cycle,white,black);
label("D",(0,0),S);
label("R",(25,0),S);
label("Q",(25,15),N);
label("A",(0,15),N);
filldraw((10,0)--(15,0)--(15,15)--(10,15)--cycle,mediumgrey,black);
label("S",(10,0),S);
label("C",(15,0),S);
label("B",(15,15),N);
label("P",(10,15),N);</asy>

<math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 25 </math>

[[2011 AMC 8 Problems/Problem 13|Solution]]

==Problem 14==

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

<math> \textbf{(A) } \dfrac7{18} \qquad\textbf{(B) } \dfrac7{15} \qquad\textbf{(C) } \dfrac{22}{45} \qquad\textbf{(D) } \dfrac12 \qquad\textbf{(E) } \dfrac{23}{45} </math>


[[2011 AMC 8 Problems/Problem 14|Solution]]

==Problem 15==

How many digits are in the product <math>4^5 \cdot 5^{10}</math>?

<math> \textbf{(A) } 8 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12 </math>


[[2011 AMC 8 Problems/Problem 15|Solution]]

==Problem 16==

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

<math> \textbf{(A) } A = \dfrac9{16}B \qquad\textbf{(B) } A = \dfrac34B \qquad\textbf{(C) } A=B \qquad\textbf{(D) } A = \dfrac43B \\ \\ \textbf{(E) }A = \dfrac{16}9B </math>


[[2011 AMC 8 Problems/Problem 16|Solution]]

==Problem 17==

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> \textbf{(A) } 21\qquad\textbf{(B) }25\qquad\textbf{(C) }27\qquad\textbf{(D) }35\qquad\textbf{(E) }56 </math>


[[2011 AMC 8 Problems/Problem 17|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?

<math> \textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac5{12}\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac7{12}\qquad\textbf{(E) }\dfrac56 </math>


[[2011 AMC 8 Problems/Problem 18|Solution]]

==Problem 19==

How many rectangles are in this figure?

<asy>
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(0,0);
B=(20,0);
C=(20,20);
D=(0,20);
draw(A--B--C--D--cycle);
E=(-10,-5);
F=(13,-5);
G=(13,5);
H=(-10,5);
draw(E--F--G--H--cycle);
I=(10,-20);
J=(18,-20);
K=(18,13);
L=(10,13);
draw(I--J--K--L--cycle);</asy>

<math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 </math>


[[2011 AMC 8 Problems/Problem 19|Solution]]

==Problem 20==

Quadrilateral <math>ABCD</math> is a trapezoid, <math>AD = 15</math>, <math>AB = 50</math>, <math>BC = 20</math>, and the altitude is <math>12</math>. What is the area of the trapezoid?

<asy>
pair A,B,C,D;
A=(3,20);
B=(35,20);
C=(47,0);
D=(0,0);
draw(A--B--C--D--cycle);
dot((0,0));
dot((3,20));
dot((35,20));
dot((47,0));
label("A",A,N);
label("B",B,N);
label("C",C,S);
label("D",D,S);
draw((19,20)--(19,0));
dot((19,20));
dot((19,0));
draw((19,3)--(22,3)--(22,0));
label("12",(21,10),E);
label("50",(19,22),N);
label("15",(1,10),W);
label("20",(41,12),E);</asy>

<math> \textbf{(A) }600\qquad\textbf{(B) }650\qquad\textbf{(C) }700\qquad\textbf{(D) }750\qquad\textbf{(E) }800 </math>


[[2011 AMC 8 Problems/Problem 20|Solution]]

==Problem 21==

Students guess that Norb's age is <math>24, 28, 30, 32, 36, 38, 41, 44, 47</math>, and <math>49</math>. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?

<math> \textbf{(A) }29\qquad\textbf{(B) }31\qquad\textbf{(C) }37\qquad\textbf{(D) }43\qquad\textbf{(E) }48 </math>


[[2011 AMC 8 Problems/Problem 21|Solution]]

==Problem 22==

What is the '''tens''' digit of <math>7^{2011}</math>?

<math> \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7 </math>


[[2011 AMC 8 Problems/Problem 22|Solution]]

==Problem 23==

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?

<math> \textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }84\qquad\textbf{(E) }108 </math>


[[2011 AMC 8 Problems/Problem 23|Solution]]

==Problem 24==

In how many ways can 10001 be written as the sum of two primes?

<math> \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 </math>


[[2011 AMC 8 Problems/Problem 24|Solution]]

==Problem 25==
A circle with radius <math>1</math> 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,gray,black);
filldraw(Circle((0,0),1), mediumgray,black);
filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);</asy>

<math> \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{5}2 </math>


[[2011 AMC 8 Problems/Problem 25|Solution]]