The Sixth Annual West Windsor-Plainsboro Mathematics Tournament Saturday October 27th, 2018 Grade 6 Test RULES The test consists of 25 multiple choice problems and 5 short answer problems to be done in 40 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 (9 + 99 + 999)/9? A. 99 B. 100 C. 101 D. 123 E. 111 Solution: 123 (9 + 99 + 999)/9 = 1107/9 = 123. 2. Compute the following: 1 2 + 3 4 + 5 6 + 7 8 +... 144 + 145. A. 72 B. 72 C. 73 D. 145 E. 10585 Solution: 73 By grouping terms together the series becomes 1 + ( 2 + 3) + ( 4 + 5) +... + ( 144 + 145) = 1 + 1 + 1 +... + 1 = (145 + 1) 1/2 = 73. 3. Claude goes to the school store and buys a pencil and an eraser. If the total cost of his purchase is $5.50 and the pencil cost $5 more than the eraser, how much did the eraser cost? A. $0.00 B. $0.25 C. $5.00 D. $5.25 E. $5.50 Solution: $0.25 If x is the cost of an eraser, the cost of a pencil is x + $5.00. If the cost of an eraser and a pencil is $5.50 then x + x + $5.00 = $5.50 and 2x = $0.50 so x = $0.25. 4. David and his dad share their birthday on the same day! David just turned 17, and his dad is now exactly 3 times David s age. How old was his dad when David was born? A. 17 B. 18 C. 34 D. 35 E. 51 Solution: 34 If David s dad is 3 times as old as David, then he is 17 3 = 51 years old. David was born 17 years ago so hist dad s age then was 51 17 = 34 5. Bill is making a new fish tank. He takes a box with dimensions 6 cm by 4 cm by 4 cm, but places a sphere inside with a volume of 32 cubic centimeters, how many cubic centimeters of water is needed to fill the tank? A. 36 B. 45 C. 64 D. 72 E. 96 Solution: 64 The volume needed to fill the tank is the volume of the tank minus the volume of the sphere. This gives 6 4 4 32 = 96 32 = 64
6. In Math Club, there are three types of animals: Snakes, Cats, and Cows. Exactly 20% of Snakes attend every meeting, 30% of Cats attend every meeting, and 90% of Cows attend every meeting. Given that the number of each type of animal present is an natural number, what is the least amount of total members in Math Club? A. 13 B. 25 C. 30 D. 33 E. 140 Solution: 25 Converting the percentages to their most simplified factional equivalent results in 1 5, 3 10, and 9 10 This results in the denominator being the lest possible total number of members, as the fractions cannot be further reduced over the integers. 5 + 10 + 10 is 25 7. Bessie the Cow escapes from the farm. She heads 5 miles north and 12 miles east. Farmer Joe then goes in a straight line from his farm to Bessie s current location. How much more distance did Bessie travel than farmer Joe? A. 3 miles B. 4 miles C. 5 miles D. 6 miles E. 7 miles Solution: 4 Bessie the Cow travels a total of 5 + 12 = 17 miles while Farmer Joe travels a total of 12 2 + 5 2 = 13 so the difference is 17 13 = 4 8. Jimmy is baking cake, which has only flour, sugar, and baking soda. The recipe calls for three times as much flour as sugar, and half as much baking soda as sugar. If Jimmy uses 2/3 cups of baking soda, how many cups of ingredients did Jimmy use in total? A. 6 cups B. 7 cups C. 8 cups D. 9 cups E. 10 cups Solution: 6 cups. There is half as much baking soda as sugar, so the amount of sugar is 2 2/3 = 4/3. There is three times as much baking powder as sugar, so there is 3 4/3 = 4 cups of baking powder. So the total is 2/3 + 4/3 + 4 = 6 cups of ingredients. 9. In Mathville, for every dollar earned by a citizen, they must pay 15 cents to the government, but every $15 paid to the government results in a $1 refund. If Adam earns $1500 dollars, how much money does he have to pay to the government? A. $200.0 B. $202.50 C. $205.00 D. $207.50 E. $210.00 Solution: 210 Adam makes $1500 so he pays the government $1500 0.15/1 = $225. Adam then gets $225 1/15 = $15 in refunds. This means the total he pays the government is 225 15 = 210.
10. Andrew likes to build scale models in his free time. Recently, he decided to build a replica of the Empire state building, which is 1440 feet tall, at a scale of 1 inch to 40 feet. How tall is his scale model in inches? A. 3 inches B. 10 inches C. 36 inches D. 432 inches E. 57600 inches Solution: 36 inches. The height of the replica is 1440 feet 1 inch/40 feet= 36 inches. 11. Tommy joined cross-country in his school. In the first practice, he was able to run 8 miles in an hour. After a week of practicing, Tommy could run 10 miles in the same amount of time. If Tommy continued to increase the amount he could run by the same amount every week thereafter, how many miles would he be able to run after 4 more weeks? A. 10 miles B. 12 miles C. 14 miles D. 16 miles E. 18 miles Solution: 18 miles. Each week Tom will increase the distance by 2 miles, so after 4 weeks, he will be able to run 10 + 4 2 = 18 miles. 12. Jack and Jill are rounding numbers. They are given a number, 49.49, and decide to round it. Jack rounds it by first rounding to the hundredths place, then the tenths place, then the ones place, then the tens place, then the hundreds place. Jill rounds by rounding it to the nearest hundreds place. What is the difference between Jack s final number and Jill s final number? A. 0 B. 50 C. 100 D. 150 E. 200 Solution: 100 When Jack rounds, he will get, after each respective step, 49.50, then 49.5, then 50, then 50, and finally 100, while Jill will just round down to 0 so the difference is 100 0 = 100 13. The company SGSS is a massive company with 4 branches in each of the 9 biggest countries. In addition, each branch has 20 buildings each employing 20 workers. If SGSS pays all of their employees equally, and raised their wages by $0.20, then how much more money will they have to pay their employees? A. 2880 B. 3000 C. 3600 D. 4320 E. 6000 Solution: 2880 In 9 countries, there are 4 branches each, each with 20 buildings that have 20 workers that each get a $0.20 raise. This yields 9 4 20 20 $0.20 = 2880
14. Al is taking a test he did not study for. He can only correctly answer 10% of the problems, and will guess the answer for the rest. If the test is 120 multiple choice with four choices per answer, how many problems can he expect to get right? A. 12 B. 30 C. 36 D. 39 E. 42 Solution: 39 Al will correctly guess 120 0.1 = 12 questions. On the remaining 108 questions he will guess with a 1 4 chance, which gives an expected value of 108 1 4 = 27 questions. 12 + 27 = 39 questions. 15. Given a 5x5 grid, how many ways are there to color exactly 5 cells yellow, such that each row and each column contain exactly one yellow cell? A. 120 B. 240 C. 360 D. 720 E. 840 Solution: 120 There must be 1 square in each row. In the first row, there are 5 options to paint a square, then in the second row, there are 4 options, as it cannot be in the same column as the painted box in the first row. If this is continued through the rest of the rows, the total number of ways is 5! = 120 ways. 16. Fields on a farm are numbered 1 to 50. Bessie the cow goes around eating fields that are multiples of 2. Then she goes around and eats multiples of 3, then 4, then 5. How many fields are still full of grass? (Note: If Bessie ate the grass on a field, it is no longer full). A. 11 B. 12 C. 13 D. 14 E. 15 Solution: 14 Note that the multiples of 4 are included in the multiples of 2 meaning it is equivalent to finding the number of unique multiples of 2, 3, and 5 less than 50. Since all three are prime, we can count their multiples, subtract multiples that share two or more of the three as factors, and finally add multiples that have all three as factors. This yields 50 (25 + 16 + 10 8 3 5 + 1) = 50 36 = 14 fields. 17. You and a friend are playing a game with dice. If the dice lands on an even number, your friend gives you $3. If the dice lands on an odd number, you have to give your friend $2.37. You have time to play exactly 10 rounds. If you won exactly 6 rounds, how much money did you win? A. $7.26 B. $7.89 C. $8.52 D. $9.15 E. $9.88 Solution: 8.52 You win 6 round which give you 6 $3 = $18 and you lose 4 rounds which loose you 4 $2.37 = $9.48. This means overall you win $18 $9.48 = $8.52
18. Todd owns a restaurant with 20 tables. The tables can seat either 4 or 6 people. If 92 people can be seated at Todd s cafe, how many 4 people tables are there in Todd s restaurant? A. 14 B. 15 C. 16 D. 17 E. 18 Solution: 14 let: s = six-seat tables and f = four-seat tables s + f = 20 6s + 4f = 92 6(20-f) + 4f = 92 120-2f = 92-2f =-28 f = 14 19. What is the units digit of 2018 2018? A. 0 B. 2 C. 4 D. 6 E. 8 Solution: 4 Since the units digits of 2018 is 8 when the power it is taken to increases, the units digit of the value will cycle 8 to 4 to 2 to 6 and back to 8. Since 2018 is two more than a multiple of four, we take the second number in the cycle which is 4 20. A diagonal is defined as a line connecting two non-adjacent vertices of a polygon. How many diagonals does a polygon with 19 sides have? A. 152 B. 171 C. 304 D. 323 E. 342 Solution: 152 Each vertex is connected to 16 diagonals, as the lines connecting to adjacent vertices are not diagonals. This gives a total of 16 19 = 304, but each vertex is connects two vertices so each diagonal was counted twice, so the final answer is 304/2 = 152 21. Two sides of a triangle have lengths 13 and 15. If the altitude to the third side is 12, what is the length of the third side? A. 11 B. 12 C. 13 D. 14 E. 15 Solution: 30 The triangle can be divided by the altitude into two right triangles with hypotenuse 13 and 15 respectively, and one leg of 12 for both, which means that the legs on the base of the larger triangle are 5 and 9 respectively, so the total lenght is 14
22. How many digits does 11 11 have? A. 9 B. 10 C. 11 D. 12 E. 13 Solution: 12 Changing 11 into 10 1.1 and note that 1.1 11 < 10. This means that the total number of digits is equal to the total number of digits of 10 11 which has 12 digits. 23. I have a broken calculator that can only do two operations: divide by 3, or subtract 1. What is the minimum amount of operations it takes to get from 2018 to 0? A. 14 B. 16 C. 18 D. 20 E. 22 Solution: 16 Note how subtracting 1 three times and then dividing by 3 is equivalent to dividing by 3 and then subtracting 1. This means the most optimal way to reach 0 is to divide by 3 until it is no longer a multiple of three and subtract 1 until it is a multiple of 3 and repeat. This method takes 16 operations to reach 0 24. A cat is walking on the vertices of triangle ABC. The cat starts at vertex A and walks to an adjacent vertex at random every second. What is the probability that the cat is back at vertex A after 5 seconds? A. 1/16 B. 2/16 C. 3/16 D. 4/16 E. 5/16 5 Solution: If the cat moves clockwise, add one to its value, and if it moves counter-clockwise, subtract 16 one from its value. In order for the cat to return to vertex A, its final value must be a multiple of three. With 5 moves there is only 2 possible ways. 4 clockwise and 1 counter-clockwise, and vice versa. In each case, there are 5 ways to order the moves. The total number of possible outcomes is 2 5 as there are 2 possible moves every second. This yields the probability of 2 10 = 20 2 5 32 = 5 16 25. What is the tens digit of 5! + 6! + 7! +... + 999!? (The! is a factorial, and n! = n (n 1) (n 2) 2 1. For example, 4! = 4 3 2 1 = 24) A. 4 B. 5 C. 6 D. 7 E. 8 Solution: 8 Note that starting from 10!, the tens digit is 0, and will remain 0. This means the sum of 5! + 6! + 7! + 8! + 9! is equivalent. 5! = 120 has a tens digit of 2 and since (n + 1)! = (n + 1)n!, it follows that the tens digits of the 4 next terms are 2, 4, 2, and 8 respectively. Summing the numbers up results in 8
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 has to buy a birthday present for two of his friends, but he s forgotten which one wants which gift. He knows that one of them wants cat food, one wants an iphone 10, but he also remembers someone asking him for a hockey stick. If Adam randomly gives one of the three gifts to each of the two friends, what is the probability that both of them get the gift they want? (Note: Both friends do not want the same gift) 1 Solution: There is only 1 way in which the two friends both receive what they want. There are 3 gifts 6 Adam can give to his first friend, and 2 that he can give to his second friend. This means there are 6 total possible ways to hand out gifts making the probability 1 6 2. The numbers on the opposite faces of a die add up to 7. What is the sum of the numbers on the 4 faces touching the face with the number 1? Solution: 14 The 4 sides of the die adjacent to the side with the number 4 on it are 2 sets of opposite sides. This means their sum is 2 7 = 14 3. How many distinct ways can the letters B, E, S, S, I, E be rearranged such that it does not form the word Bessie? Solution: 179 6!/2!*2! = 720/4 = 180 total arrangements. Since the arrangement cannot be BESSIE, it becomes 179 arrangements. 4. Let x be the answer to this problem. If x is positive, what is x 2 6? Solution: 3 Since the solution of the questions is equal to both x and x 2 6, setting them equal results in x 2 6 = x which simplifies to x 2 x 6 which can be factored into (x 3)(x + 2) which gives the solutions x = 3, 2 and since x is positive, the answer is 3 5. A bee is riding on a train heading west. Once another train heading east on the same track is within 200 miles, the bee begins to fly to and from the two trains until they meet. The first train travels at 40 mi/h but stops for 3 minutes at a station before crossing paths with the other train, while the second train travels at 61 mi/h. If the bee travels 200 mi/h, how much distance does the bee travel? Solution: 400 When the first train stops, in that time, it could travel 40 3/60 = 2 miles. So the question is the equivalent of having the two trains start 202 miles apart, and not have any train stop. This means it will take the trains 202/(61 + 40) = 2 hours to meet. Since the bee is traveling at 200 mi/h, the bee will travel 400 miles in the 2 hour time.