2012AMC8

Sample questions from the AMC 8 Answers: 1) E 2) E 12) B 16) C 22) D American Mathematics Competitions amcinfo@maa.org • amc.maa.org 1-800-527-3690 American Mathematics Competitions is a program of The Mathematical Association of America and is sponsored by The Akamai Foundation Contributors Academy of Applied Sciences, American Mathematical Association of Two-Year Colleges, American Mathematical Society, American Statistical Association, Art of Problem Solving, Awesome Math, Casualty Actuarial Society, D.E. Shaw & Co., Delta Airlines, IDEA Math, Jane Street, Math for America, Mu Alpha Theta, National Council of Teachers of Mathematics, National Science Foundation, Pi Mu Epsilon, Robert Balles, Society of Industrial and Applied Math, W.H. Freeman and Company American Mathematics Competitions 8 Register now for the middle school level AMC 8 held in November of each year! Details inside. amc.maa.org 27th AMC 8 2011 5 11. The graph below 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? 120 100 80 60 40 20 0 Tu WM Th F M I N U T E S (A) 6 (B) 8 (C) 9 (D) 10 (E) 12 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? (A) 1 4 (B) 1 3 (C) 1 2 (D) 2 3 (E) 3 4 13. Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded? BA QP D S C R (A) 15 (B) 18 (C) 20 (D) 24 (E) 25 27th AMC 8 2011 6 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? (A) 7 18 (B) 7 15 (C) 22 45 (D) 1 2 (E) 23 45 15. How many digits are in the product 45 · 510 ? (A) 8 (B) 9 (C) 10 (D) 11 (E) 15 16. Let A be the area of a triangle with sides of length 25, 25, and 30. Let B be the area of a triangle with sides of length 25, 25, and 40. What is the relationship between A and B ? (A) A = 9 16 B (B) A = 3 4 B (C) A = B (D) A = 4 3 B (E) A = 16 9 B 17. Let w, x, y, and z be whole numbers. If 2w · 3x · 5y · 7z = 588, then what does 2w + 3x+ 5y + 7z equal? (A) 21 (B) 25 (C) 27 (D) 35 (E) 56 18. A fair six-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? (A) 1 6 (B) 5 12 (C) 1 2 (D) 7 12 (E) 5 6 27th AMC 8 2011 7 19. How many rectangles are in this figure? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12 20. Quadrilateral ABCD is a trapezoid, AD = 15, AB = 50, BC = 20, and the altitude is 12. What is the area of the trapezoid? BA D C 12 2015 50 (A) 600 (B) 650 (C) 700 (D) 750 (E) 800 21. Students guess that Norb’s age is 24, 28, 30, 32, 36, 38, 41, 44, 47, and 49. 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? (A) 29 (B) 31 (C) 37 (D) 43 (E) 48 22. What is the tens digit of 72011 ? (A) 0 (B) 1 (C) 3 (D) 4 (E) 7 27th AMC 8 2011 2 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 receive? (A) $1.50 (B) $2.00 (C) $2.50 (D) $3.00 (E) $3.50 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? (A) Karl’s garden is larger by 100 square feet. (B) Karl’s garden is larger by 25 square feet. (C) The gardens are the same size. (D) Makenna’s garden is larger by 25 square feet. (E) Makenna’s garden is larger by 100 square feet. 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? (A) 8 : 17 (B) 25 : 49 (C) 36 : 25 (D) 32 : 17 (E) 36 : 17 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, mode, and median of these numbers is true? (A) median < mean < mode (B) mean < mode < median (C) mean < median < mode (D) median < mode < mean (E) mode < median < mean 27th AMC 8 2011 2 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 receive? (A) $1.50 (B) $2.00 (C) $2.50 (D) $3.00 (E) $3.50 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? (A) Karl’s garden is larger by 100 square feet. (B) Karl’s garden is larger by 25 square feet. (C) The gardens are the same size. (D) Makenna’s garden is larger by 25 square feet. (E) Makenna’s garden is larger by 100 square feet. 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? (A) 8 : 17 (B) 25 : 49 (C) 36 : 25 (D) 32 : 17 (E) 36 : 17 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, mode, and median of these numbers is true? (A) median < mean < mode (B) mean < mode < median (C) mean < median < mode (D) median < mode < mean (E) mode < median < mean Administration of the AMC 8 Our contests are administered by a school or school-based organization and offer individual and school recognition for high scorers. The AMC 8 will be given in participating schools on the same date during the middle of November (please visit our website for specific dates). Administration on a later date is permitted in cases of academic conflicts or school closings The contest is only 40 minutes in length, making it easily administered within a class period. Students may not use calculators on the contest. To make the scoring official, we score all the contests in the AMC office. For more information about how to involve your school in the competitions, read the Frequently Asked Questions at amc.maa.org. Spanish, braille and large print editions are also available. Edyth May Sliffe Teaching Award The Sliffe Award is given each year to approximately fifty teachers from high-scoring AMC 8 schools. Winners of this award receive a recognition certificate, an award pin, a cash prize and a one-year membership to the Mathematical Association of America. A special reception during the NCTM spring meeting honors the award winners. Math Club Package For an additional $25 you can order an AMC 8 Math Club Package, which includes a Club Advisor’s Handbook (with a new set of problem worksheets taken from the AMC archives), plus a CD with all the contests given from 2001 through the most recent year, including the AMC 8, 10, and 12, the American Invitational Mathematics Examination (AIME), and the United States of America Mathematical Olympiad (USAMO). School and student awards for the AMC 8 Each participating school receives individual school results, school awards, and high scorers are listed on our website. An award plaque is given to all the perfect scorers, the top scoring student(s) in each state, and student awards are provided for other achievements. Please visit amc.maa.org for more details. Our Purpose The mission of the MAA Competitions is to increase interest in mathematics and to develop problem solving through a fun competition. Students gain the opportunity to learn and achieve through competition with students in their school and from around the world. Teachers and schools benefit from the chance to challenge students with interesting mathematical questions that are aligned with curriculum standards at all levels of difficulty. What are people saying about the AMC? “Our students look forward to competing in up to 20 different math contests each year, and the American Mathematics Competitions are the contests that the students strive to take every year. They generate great discussions and are a joy of the participating students.” –Mathematics Department Head, Michigan “We are thankful for the American Mathematics Competitions because it gives our students the opportunity to compete with other students who have the same passion for mathematics that they do.” –Mathematics Department Chair, Virginia What is the American Mathematics Competitions 8? Thank you for your interest in the middle school level AMC 8 contest! For over 60 years the AMC math contests have been the most respected school-based competitions in the nation. The AMC works with teachers, mathematicians, and professional organizations to provide high quality, challenging math problems aligned with curriculum standards. Many well- known colleges and universities request scores from our contests at the higher grade levels and use them for recruiting and admissions. The material covered on the AMC 8 is the middle school mathematics curriculum. Topics include probability, estimation, percentages, spatial visualization, everyday applications and reading and interpreting graphs. The AMC 8 contest is for students in the sixth through eighth grade, although accelerated fourth and fifth graders can also take part. AMC 8 eligibility extends to any student 14.5 years of age or younger, and not enrolled in grades 9 through 12. It is 25 questions in length, and is multiple choice with no penalty for guessing. The contest takes only 40 minutes. A student’s score is the number of problems correctly solved. The American Mathematics Competitions 8 is only the beginning. There are high school-level contests offered for grade 10 and below, as well as grade 12 and below. The AMC 10 and the AMC 12 contests are the first in the series that lead to the USA Team at the International Mathematical Olympiad. Encourage your students to reach for the stars!