The Sixth Annual West Windsor-Plainsboro Mathematics Tournament Saturday October 27th, 2018 Grade 4 Test RULES The test consists of 15 multiple choice problems and 5 short answer problems to be done in 30 minutes. All multiple choice answers should be marked on the Scantron mark sheet. No marks on this paper will be graded. All short answers you write should be in exact, simplified form, unless otherwise stated in the question, following the rules below: All fractions must be simplified. All fractional answers must be expressed in the form a b where a and b share no common factors other than 1. For example, if the answer is 9 5, then 1 4 18 5 (mixed fraction) or 10 are NOT ACCEPTABLE. Any question that involves the number π in the answer must be written using the symbol π. In particular, decimal approximations of π are NOT ACCEPTABLE. All radicals must be simplified. Radicals in the denominator of a final answer, a perfect n th power under a n th root, or fractions/decimals under a radical are NOT ACCEPTABLE. You can ask your proctor a question, but if they think answering your question will give you an unfair advantage on the test over other contestants, they don t have to respond. After the testing has completed, please go to your selected mini-event rotation (directions will be given for this after the main contest is over). DO NOT FLIP THIS PAGE TO BEGIN THE TEST UNTIL INSTRUCTED TO DO SO BY YOUR PROCTOR(S).
Multiple-Choice Questions Answers to this section of the test should go on the scantron multiple-choice form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORED. 1. What is the value of 15 10 + 15 / 10? A. 37.5 B. 50 C. 77.5 D. 151.5 E. 175.5 Solution: 151.5 (15 10) + (15/10) = 150 + 1.5 = 151.5 2. This year, 2018, is the 6th annual WWPMT. In what year did the 1st WWPMT occur? A. 2011 B. 2012 C. 2013 D. 2014 E. 2015 Solution: 2013 2018 6 + 1 = 12 3. Max and Chris are playing tennis. If Max is able to hit 50 tennis balls in one minute, and Chris can hit 45 tennis balls in one minute, how many more tennis balls can Max hit than Chris in 3 minutes? A. 10 balls B. 15 balls C. 20 balls D. 25 balls E. 30 balls Solution: 15 (50 45) 3 = 5 3 = 15 4. Daphne loves to eat pocky! She is gifted a bag with 100 pieces of pocky and eats 6 pieces a day. After 12 days, how many pieces of pocky will be left over? A. 12 pieces B. 28 pieces C. 36 pieces D. 72 pieces E. 172 pieces Solution: 28 100 12 6 = 100 72 = 28 5. You find yourself out of money after using all your 20 dollars on 42 Twist bars and some Himhey bars. If each Twist bar costs 30 cents and each Himhey bar costs 20 cents, how many Himhey bars did you buy? A. 31 B. 32 C. 33 D. 35 E. 37 Solution: 37 0.3 42 + 0.2h = 20 0.2h = 20 12.6 h = 37
6. Sam is 7 years older than her sister Eva. 12 years ago, Eva was half as old as her sister. How old is Eva now? A. 10 years old B. 12 years old C. 14 years old D. 19 years old E. 26 years old Solution: 19 B = 7 + E (B 12) 1 2 = (E 12) B 2 6 = E 12 E 2 + 7 2 6 = E 12 E 2 = 9.5 E = 19 7. Mason has 84 graphing calculators, and Eric has 1 3 the amount of graphing calculators as Mason. Matthew has 5 times the number of graphing calculators that Eric has. How many graphing calculators does Matthew have? A. 84 calculators B. 90 calculators C. 132 calculators D. 140 calculators E. 160 calculators Solution: 140 calculators Since Mason has 84 calculators, Eric has 84/3 = 28 calculators. Matthew has 5 times as many as Eric, so he has 28*5 = 140 calculators. 8. John s teacher assigned him a book to read for homework. He needs to read 113 pages of the book. If he has already read pages 41 to 132, how many more pages does he need to read? A. 19 pages B. 20 pages C. 21 pages D. 22 pages E. 23 pages Solution: 21 pages He has already read 132 41 + 1 = 92 pages, so he needs to read 21 more pages 9. Pratyoy has 56 normal pizza slices and 36 large pizza slices. If Pratyoy eats half of his normal pizza slices and a third of his large pizza slices, how many pizza slices does Pratyoy have left in total? A. 4 B. 12 C. 28 D. 40 E. 52 Solution: 52 Pratyoy eats half of his normal pizza slices, and half of 56 is 28. He eats a third of his large pizza slices, and one-third of 36 is 12. 28 + 12 = 40. Pratyoy ate 40 pizza slices. 56 + 36 = 92 There were 92 total pizza slices. 92-40 = 52. There are 52 pizza slices left in total.
10. Mary has 88 marbles numbered 1 to 88. If she picks one out, what is the chance that the number on it is less than 45 and even? 5 1 3 1 1 A. 44 B. 8 C. 22 D. 4 E. 2 Solution: 1/4 44 marbles are less than 45 and 22 of those are even so it s 22/88 or 1/4 11. A colony of ants line up in a square formation. When 21 more ants join the colony, they reconfigure the square into a new square with a side length of 3 more ants than the original square. How many ants were in the colony originally? A. 4 ants B. 9 ants C. 16 ants D. 25 ants E. 36 ants Solution: 4 The problem can be written as a 2 + 21 = (a + 3) 2. After expanding: a 2 + 21 = a 2 + 6a + 9 6a + 9 = 21 6a = 12 a = 2 Since there were a 2 ants in the original colony, there were 4 ants in the colony. 12. Ronak has 100 tiles labeled with the numbers 1-100. How many tiles does he have that are multiples of 2, 7, or both? A. 44 B. 53 C. 54 D. 57 E. 64 Solution: 57 If there are a tiles that are multiples of 2, b tiles that are multiples of 7, and c tiles that are multiples of both, therefore multiples of 14, then the total number of tiles that fit the condition are a + b c, since the multiples of both of them were counted twice, once in a and once in b, so it needs to be subtracted once from the final sum. We find that a is 50 since 2*50 = 100, b is 14 since the highest multiple of 7 under 100 is 7*14=98, and c is 7 since the highest multiple of 14 is 14*7=98.The final answer is then 50 + 14 7 = 57 tiles 13. Adam is multiplying 7s together on his calculator, but then realizes that he forgot to let go of enter, which resulted in 7 7 7... If Adam counts the 7s and finds out that there were exactly 83 of them, what is the units digit of his final answer? A. 1 B. 3 C. 5 D. 7 E. 9 Solution: 3 If you start multiplying 7 s together and look only at the last digit, you notice a pattern of: 7 3 1 7 3 1... This pattern cycles every three terms, so 7 83 has the same last digit as 7 2, which ends in 3. Therefore, 7 83 ends in the digit 3
14. How many ways are there to color a 3 by 3 grid using only black and white? A. 81 ways B. 128 ways C. 512 ways D. 729 ways E. 1024 ways Solution: 512 There are two ways to color each of 9 squares, so the total number of ways to color the 3x3 square is 2 9 = 512 15. Let S = 1 + 2 + 3... + 999. What is the sum of the digits of S? A. 12 B. 15 C. 27 D. 39 E. 45 Solution: 27 If you write out S as : S = 1 + 2 + 3 +... + 998 + 999 AND S = 999 + 998 + 997 +... + 2 + 1 2S= 1000 + 1000 + 1000 +... +1000 + 1000 with 999 thousands Therefore, S = 999*1000/2 = 999*500 = 499500 The sum of the digits of S is 4+9+9+5 = 27
Short Answer Questions Answers to this section of the test should go on the short answer questions answer form given to you. NOTHING WRITTEN ON THIS TEST PAPER WILL BE SCORED. 1. Adam is finally graduating from elementary school! As he is the shortest member of his graduating class, he has been chosen to organize everything! Adam needs to buy 2 hats for each of the 20 students, but he misreads the instructions and buys one hat for every two students. How many more hats does Adam need to buy to correct his mistake? Solution: 30 hats Adam bought 1 hat for every two students, indicating that he bought, 20/2 = 10 hats. He needs to buy two hats for each student, which is 20*2 = 40 hats, so to make up the difference, he needs to buy another 40-10 = 30 hats 2. Ansh likes to build random shapes out of toothpicks. He builds a square with a side length of 1 toothpick, and another square with a side length of 3 toothpicks. If the combined area of the two squares is 250 square centimeters, what is the length of one toothpick in centimeters? Solution: 5 The areas of the two squares, if s is the length of one toothpick, is s 2 + (3s) 2 = 10s 2 = 250. s 2 = 25 s = 5 3. David decides to write a book which unsurprisingly becomes the best seller on the WWP Times! Each copy of the book costs $2 to print and sells for $42. If David earns all of the profits from every book and wants to earn $10, 000 from selling this book, how many books does he need to sell? Solution: 250 books Each book sells for a net profit of 42-2 = $40. 10000 40 = 250 books Since he needs to earn $10,000, he needs to sell exactly
4. Daniel is on the bus and has reached halfway home when he realizes that he left his phone at school. He immediately jumps off the bus and gets on his bike. Daniel bikes at a rate of 20 mph and reaches school only to realize that his phone was in his pocket. Disappointed, he immediately bikes all the way back to his house at half his previous speed. If he bikes for 45 minutes over the entire trip, how many miles away from the school is Daniel s house? Solution: 6 While traveling to the school, Daniel covers half the distance at a speed of 20mph, meaning that it takes 0.5d 20 d hours. On the way back, he covers the full distance at 10mph which gives a time of 10 hours. Adding them together gives d 40 + d 10 = d 8 = 3 4 hours. Therefore, d = 8*3/4 = 6 miles. 5. Richard got some money and decided to go out into town and spend it. He starts feeling hungry and spends 1 2 of it on ramen noodles. After that, he goes to an arcade, where he spends 2 3 of his remaining money. He played at the arcade for a couple hours before spending another 1 5 of his money on transportation. Finally, he returns home and donates the remaining $80 to charity. How much money did Richard receive at the beginning? Solution: 600 After the Ramen Noodles, he has 1/2 his money left. After the arcade, he has 1 3 1 2 = 1 6 of his money left. After transportation, he has 4 5 1 6 = 2 2 15 of his money left. That 15 of his money is the last $80, meaning that his starting money was 15 2 80 = $600