Difference between revisions of "2009 AMC 8 Problems"
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− | 1 | + | {{AMC8 Problems|year=2009}} |
+ | ==Problem 1== | ||
+ | Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? | ||
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14 </math> | <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 14 </math> | ||
− | 2 | + | ==Problem 2== |
+ | On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? | ||
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 112 </math> | <math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 112 </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 2|Solution]] | |
+ | ==Problem 3== | ||
+ | The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? | ||
− | + | ||
+ | <asy> | ||
import graph; /* this is a label */ | import graph; /* this is a label */ | ||
Label f; | Label f; | ||
Line 26: | Line 32: | ||
label("4", (-1, 20), black); | label("4", (-1, 20), black); | ||
dot((5,5), black+linewidth(5)); | dot((5,5), black+linewidth(5)); | ||
− | dot((10,10), black+linewidth(5)); | + | dot((10,10),black+linewidth(5)); |
− | dot((15, 15), black+linewidth(5)); | + | dot((15,15), black+linewidth(5)); |
dot((20,20), black+linewidth(5)); | dot((20,20), black+linewidth(5)); | ||
label("MINUTES", (11,-5), S); | label("MINUTES", (11,-5), S); | ||
− | label(rotate(90)*"MILES", (-5,11), W); | + | label(rotate(90)*"MILES", (-5,11), W);</asy> |
+ | |||
+ | <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 5.5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 6.5\qquad\textbf{(E)}\ 7 </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 3|Solution]] | |
− | 4 | + | ==Problem 4== |
− | + | The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure <b>cannot</b> be formed? | |
+ | <asy> | ||
defaultpen(linewidth(0.6)); | defaultpen(linewidth(0.6)); | ||
size(80); | size(80); | ||
Line 54: | Line 63: | ||
draw(shift(4s,0)*p); | draw(shift(4s,0)*p); | ||
draw(shift(4s,2r)*p); | draw(shift(4s,2r)*p); | ||
− | draw(shift(4s,4r)*p); | + | draw(shift(4s,4r)*p); |
+ | </asy> | ||
− | + | <asy> | |
size(350); | size(350); | ||
defaultpen(linewidth(0.6)); | defaultpen(linewidth(0.6)); | ||
Line 65: | Line 75: | ||
pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; | pair[] d={(17,0), (17,1), (17,2), (17,3), (17,4), (18,0), (18,1), (18,2), (18,3), (18,4), (19,0), (19,1), (19,2), (19,3), (19,4)}; | ||
pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; | pair[] e={(21,4), (22,1), (22,2), (22,3), (22,4), (23,0), (23,1), (23,2), (23,3), (23,4), (24,1), (24,2), (24,3), (24,4), (25,4)}; | ||
+ | |||
int i; | int i; | ||
for(int i=0; i<15; i=i+1) { | for(int i=0; i<15; i=i+1) { | ||
Line 72: | Line 83: | ||
draw(shift(d[i])*p); | draw(shift(d[i])*p); | ||
draw(shift(e[i])*p); | draw(shift(e[i])*p); | ||
− | }[/ | + | } |
+ | </asy> | ||
+ | <cmath> \textbf{(A)}\qquad\qquad\qquad\textbf{(B)}\quad\qquad\qquad\textbf{(C)}\:\qquad\qquad\qquad\textbf{(D)}\quad\qquad\qquad\textbf{(E)} </cmath> | ||
+ | |||
+ | [[2009 AMC 8 Problems/Problem 4|Solution]] | ||
− | 5 | + | ==Problem 5== |
+ | A sequence of numbers starts with <math>1</math>, <math>2</math>, and <math>3</math>. The fourth number of the sequence is the sum of the previous three numbers in the sequence: <math>1+2+3=6</math>. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence? | ||
<math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ 99 </math> | <math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ 99 </math> | ||
− | 6 | + | [[2009 AMC 8 Problems/Problem 5|Solution]] |
+ | ==Problem 6== | ||
+ | Steve's empty swimming pool will hold <math>24,000</math> gallons of water when full. It will be filled by <math>4</math> hoses, each of which supplies <math>2.5</math> gallons of water per minute. How many hours will it take to fill Steve's pool? | ||
<math> \textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 48 </math> | <math> \textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 48 </math> | ||
− | 7 | + | [[2009 AMC 8 Problems/Problem 6|Solution]] |
+ | |||
+ | ==Problem 7== | ||
+ | The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? | ||
− | + | <asy> | |
size(250); | size(250); | ||
defaultpen(linewidth(0.55)); | defaultpen(linewidth(0.55)); | ||
Line 104: | Line 125: | ||
draw((-1/10, i)--(13/30, i)); | draw((-1/10, i)--(13/30, i)); | ||
} | } | ||
− | label(" | + | label("$A$", A, SE); |
− | label(" | + | label("$B$", B, SE); |
− | label(" | + | label("$C$", C, SE); |
− | label(" | + | label("$D$", D, SE); |
− | label(" | + | label("$3$", (1/3,3), E); |
− | label(" | + | label("$3$", (1/3,9), E); |
− | label(" | + | label("$3$", (-3,0), S); |
label("Main", (-3,0), N); | label("Main", (-3,0), N); | ||
label(rotate(45)*"Aspen", A--C, SE); | label(rotate(45)*"Aspen", A--C, SE); | ||
− | label(rotate(63.43494882)*"Brown", A--D, NW); | + | label(rotate(63.43494882)*"Brown", A--D, NW);</asy> |
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4.5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 9 </math> | <math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4.5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 9 </math> | ||
− | 8 | + | [[2009 AMC 8 Problems/Problem 7|Solution]] |
+ | |||
+ | ==Problem 8== | ||
+ | The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area? | ||
+ | |||
<math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 99\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 110 </math> | <math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 99\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 110 </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 8|Solution]] | |
− | + | ==Problem 9== | |
+ | Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have? | ||
+ | |||
+ | <asy> | ||
defaultpen(linewidth(0.6)); | defaultpen(linewidth(0.6)); | ||
pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); | pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); | ||
draw(O--A--B--C--D--E--F--G--cycle); | draw(O--A--B--C--D--E--F--G--cycle); | ||
draw(O--D, dashed); | draw(O--D, dashed); | ||
− | draw(A--C, dashed); | + | draw(A--C, dashed);</asy> |
+ | |||
+ | <math> \textbf{(A)} \ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 29 </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 9|Solution]] | |
− | 10 | + | ==Problem 10== |
− | + | On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? | |
+ | <asy> | ||
unitsize(10); | unitsize(10); | ||
draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); | draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); | ||
Line 180: | Line 211: | ||
fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black); | fill((3,7)--(4,7)--(4,8)--(3,8)--cycle,black); | ||
fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black); | fill((5,7)--(6,7)--(6,8)--(5,8)--cycle,black); | ||
− | fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black); | + | fill((7,7)--(8,7)--(8,8)--(7,8)--cycle,black);</asy> |
+ | |||
+ | <math> \textbf{(A)}\ \frac{1}{16}\qquad\textbf{(B)}\ \frac{7}{16}\qquad\textbf{(C)}\ \frac{1}2\qquad\textbf{(D)}\ \frac{9}{16}\qquad\textbf{(E)}\ \frac{49}{64} </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 10|Solution]] | |
− | 11 | + | ==Problem 11== |
+ | The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of <math>1.43</math>. Some of the <math>30</math> sixth graders each bought a pencil, and they paid a total of <math>1.95</math>. How many more sixth graders than seventh graders bought a pencil? | ||
<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> | ||
− | 12 | + | [[2009 AMC 8 Problems/Problem 11|Solution]] |
+ | ==Problem 12== | ||
+ | The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? | ||
− | + | <asy> | |
unitsize(30); | unitsize(30); | ||
draw(unitcircle); | draw(unitcircle); | ||
Line 196: | Line 232: | ||
draw((0,0)--(cos(pi/6),sin(pi/6))); | draw((0,0)--(cos(pi/6),sin(pi/6))); | ||
draw((0,0)--(-cos(pi/6),sin(pi/6))); | draw((0,0)--(-cos(pi/6),sin(pi/6))); | ||
− | label(" | + | label("$1$",(0,.5)); |
− | label(" | + | label("$3$",((cos(pi/6))/2,(-sin(pi/6))/2)); |
− | label(" | + | label("$5$",(-(cos(pi/6))/2,(-sin(pi/6))/2));</asy> |
− | + | <asy> | |
unitsize(30); | unitsize(30); | ||
draw(unitcircle); | draw(unitcircle); | ||
Line 205: | Line 241: | ||
draw((0,0)--(cos(pi/6),sin(pi/6))); | draw((0,0)--(cos(pi/6),sin(pi/6))); | ||
draw((0,0)--(-cos(pi/6),sin(pi/6))); | draw((0,0)--(-cos(pi/6),sin(pi/6))); | ||
− | label(" | + | label("$2$",(0,.5)); |
− | label(" | + | label("$4$",((cos(pi/6))/2,(-sin(pi/6))/2)); |
− | label(" | + | label("$6$",(-(cos(pi/6))/2,(-sin(pi/6))/2));</asy> |
<math> \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{7}{9}\qquad\textbf{(E)}\ \frac{5}{6} </math> | <math> \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{7}{9}\qquad\textbf{(E)}\ \frac{5}{6} </math> | ||
− | 13 | + | [[2009 AMC 8 Problems/Problem 12|Solution]] |
+ | ==Problem 13== | ||
+ | A three-digit integer contains one of each of the digits <math>1</math>, <math>3</math>, and <math>5</math>. What is the probability that the integer is divisible by <math>5</math>? | ||
<math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6} </math> | <math> \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6} </math> | ||
− | 14 | + | [[2009 AMC 8 Problems/Problem 13|Solution]] |
+ | ==Problem 14== | ||
+ | Austin and Temple are <math>50</math> miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging <math>60</math> miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged <math>40</math> miles per hour on the return trip. What was the average speed for the round trip, in miles per hour? | ||
<math> \textbf{(A)}\ 46\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 54 </math> | <math> \textbf{(A)}\ 46\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 50\qquad\textbf{(D)}\ 52\qquad\textbf{(E)}\ 54 </math> | ||
− | 15 | + | [[2009 AMC 8 Problems/Problem 14|Solution]] |
+ | |||
+ | ==Problem 15== | ||
+ | A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, <math>1/4</math> cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make? | ||
<math> \textbf{(A)}\ 5\frac{1}8\qquad\textbf{(B)}\ 6\frac{1}4\qquad\textbf{(C)}\ 7\frac{1}2\qquad\textbf{(D)}\ 8\frac{3}4\qquad\textbf{(E)}\ 9\frac{7}8 </math> | <math> \textbf{(A)}\ 5\frac{1}8\qquad\textbf{(B)}\ 6\frac{1}4\qquad\textbf{(C)}\ 7\frac{1}2\qquad\textbf{(D)}\ 8\frac{3}4\qquad\textbf{(E)}\ 9\frac{7}8 </math> | ||
− | 16 | + | [[2009 AMC 8 Problems/Problem 15|Solution]] |
+ | |||
+ | ==Problem 16== | ||
+ | How many 3-digit positive integers have digits whose product equals <math>24</math>? | ||
<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ 24 </math> | <math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 21\qquad\textbf{(E)}\ 24 </math> | ||
− | + | [[2009 AMC 8 Problems/Problem 16|Solution]] | |
− | + | ==Problem 17== | |
+ | The positive integers <math>x</math> and <math>y</math> are the two smallest positive integers for which the product of <math>360</math> and <math>x</math> is a square and the product of <math>360</math> and <math>y</math> is a cube. What is the sum of <math>x</math> and <math>y</math>? | ||
− | [asy | + | <math> \textbf{(A)}\ 80\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 115\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 610 </math> |
+ | |||
+ | [[2009 AMC 8 Problems/Problem 17|Solution]] | ||
+ | |||
+ | ==Problem 18== | ||
+ | The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed? | ||
+ | |||
+ | <asy> | ||
unitsize(10); | unitsize(10); | ||
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle); | draw((0,0)--(7,0)--(7,7)--(0,7)--cycle); | ||
Line 251: | Line 305: | ||
fill((0,5)--(0,6)--(7,6)--(7,5)--cycle,black); | fill((0,5)--(0,6)--(7,6)--(7,5)--cycle,black); | ||
fill((0,3)--(0,4)--(7,4)--(7,3)--cycle,black); | fill((0,3)--(0,4)--(7,4)--(7,3)--cycle,black); | ||
− | fill((0,1)--(0,2)--(7,2)--(7,1)--cycle,black); | + | fill((0,1)--(0,2)--(7,2)--(7,1)--cycle,black);</asy> |
<math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 126 </math> | <math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 57\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 126 </math> | ||
− | 19 | + | [[2009 AMC 8 Problems/Problem 18|Solution]] |
+ | |||
+ | ==Problem 19== | ||
+ | Two angles of an isosceles triangle measure <math> 70^\circ </math> and <math> x^\circ </math>. What is the sum of the three possible values of <math>x</math>? | ||
<math> \textbf{(A)}\ 95\qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 180 </math> | <math> \textbf{(A)}\ 95\qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 165\qquad\textbf{(E)}\ 180 </math> | ||
− | 20 | + | [[2009 AMC 8 Problems/Problem 19|Solution]] |
− | + | ||
+ | ==Problem 20== | ||
+ | How many non-congruent triangles have vertices at three of the eight points in the array shown below? | ||
+ | <asy> | ||
dot((0,0)); | dot((0,0)); | ||
dot((0,.5)); | dot((0,.5)); | ||
Line 268: | Line 328: | ||
dot((1,.5)); | dot((1,.5)); | ||
dot((1.5,0)); | dot((1.5,0)); | ||
− | dot((1.5,.5)); | + | dot((1.5,.5));</asy> |
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math> | <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math> | ||
− | 21 | + | [[2009 AMC 8 Problems/Problem 20|Solution]] |
+ | |||
+ | ==Problem 21== | ||
+ | Andy and Bethany have a rectangular array of numbers greater than 0 with <math>40</math> rows and <math>75</math> columns. Andy adds the numbers in each row. The average of his <math>40</math> sums is <math>A</math>. Bethany adds the numbers in each column. The average of her <math>75</math> sums is <math>B</math>. What is the value of <math> \frac{A}{B} </math>? | ||
<math> \textbf{(A)}\ \frac{64}{225}\qquad\textbf{(B)}\ \frac{8}{15}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{225}{64} </math> | <math> \textbf{(A)}\ \frac{64}{225}\qquad\textbf{(B)}\ \frac{8}{15}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{225}{64} </math> | ||
− | 22 | + | [[2009 AMC 8 Problems/Problem 21|Solution]] |
+ | |||
+ | ==Problem 22== | ||
+ | How many whole numbers between 1 and 1000 do not contain the digit 1? | ||
<math> \textbf{(A)}\ 512\qquad\textbf{(B)}\ 648\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 800 </math> | <math> \textbf{(A)}\ 512\qquad\textbf{(B)}\ 648\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 800 </math> | ||
− | 23 | + | [[2009 AMC 8 Problems/Problem 22|Solution]] |
+ | |||
+ | ==Problem 23== | ||
+ | On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought <math>400</math> jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class? | ||
<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math> | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math> | ||
− | 24 | + | [[2009 AMC 8 Problems/Problem 23|Solution]] |
+ | |||
+ | ==Problem 24== | ||
+ | The letters <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> represent digits. If <math> \begin{tabular}{ccc}&A&B\\ +&C&A\\ \hline &D&A\end{tabular} </math> and <math> \begin{tabular}{ccc}&A&B\\ -&C&A\\ \hline &&A\end{tabular} </math>, what digit does <math>D</math> represent? | ||
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math> | <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9 </math> | ||
− | 25 | + | [[2009 AMC 8 Problems/Problem 24|Solution]] |
− | + | ||
+ | ==Problem 25== | ||
+ | A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is <math>1/2</math> foot from the top face. The second cut is <math>1/3</math> foot below the first cut, and the third cut is <math>1/17</math> foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? | ||
+ | <asy> | ||
+ | size(120); | ||
+ | |||
import three; | import three; | ||
real d=11/102; | real d=11/102; | ||
Line 308: | Line 385: | ||
label("1/2", (1,1,3/4), E); | label("1/2", (1,1,3/4), E); | ||
label("1/3", (1,1,1/3), E); | label("1/3", (1,1,1/3), E); | ||
− | label("1/17", (0,1,1/6-d/4), E); | + | label("1/17", (0,1,1/6-d/4), E);</asy> |
− | + | <asy> | |
+ | size(120); | ||
+ | |||
import three; | import three; | ||
real d=11/102; | real d=11/102; | ||
Line 333: | Line 412: | ||
label("A", (1+1/2, 1, 0), SE); | label("A", (1+1/2, 1, 0), SE); | ||
label("B", (2+1/2, 1, 0), SE); | label("B", (2+1/2, 1, 0), SE); | ||
− | label("C", (3+1/2, 1, 0), SE); | + | label("C", (3+1/2, 1, 0), SE);</asy> |
<math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math> | <math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math> | ||
+ | |||
+ | [[2009 AMC 8 Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2009|before=[[2008 AMC 8 Problems|2008 AMC 8]]|after=[[2010 AMC 8 Problems|2010 AMC 8]]}} | ||
+ | * [[AMC 8]] | ||
+ | * [[AMC 8 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 02:15, 7 September 2021
2009 AMC 8 (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?
Problem 2
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
Problem 3
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
Problem 4
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?
Problem 5
A sequence of numbers starts with , , and . The fourth number of the sequence is the sum of the previous three numbers in the sequence: . In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
Problem 6
Steve's empty swimming pool will hold gallons of water when full. It will be filled by hoses, each of which supplies gallons of water per minute. How many hours will it take to fill Steve's pool?
Problem 7
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
Problem 8
The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?
Problem 9
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
Problem 10
On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?
Problem 11
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of . Some of the sixth graders each bought a pencil, and they paid a total of . How many more sixth graders than seventh graders bought a pencil?
Problem 12
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
Problem 13
A three-digit integer contains one of each of the digits , , and . What is the probability that the integer is divisible by ?
Problem 14
Austin and Temple are miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
Problem 15
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water and 7 cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?
Problem 16
How many 3-digit positive integers have digits whose product equals ?
Problem 17
The positive integers and are the two smallest positive integers for which the product of and is a square and the product of and is a cube. What is the sum of and ?
Problem 18
The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed?
Problem 19
Two angles of an isosceles triangle measure and . What is the sum of the three possible values of ?
Problem 20
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
Problem 21
Andy and Bethany have a rectangular array of numbers greater than 0 with rows and columns. Andy adds the numbers in each row. The average of his sums is . Bethany adds the numbers in each column. The average of her sums is . What is the value of ?
Problem 22
How many whole numbers between 1 and 1000 do not contain the digit 1?
Problem 23
On the last day of school, Mrs. Awesome gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
Problem 24
The letters , , and represent digits. If and , what digit does represent?
Problem 25
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is foot from the top face. The second cut is foot below the first cut, and the third cut is foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
See Also
2009 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2008 AMC 8 |
Followed by 2010 AMC 8 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.