Halloween (Oct. 31) Meeting (Multiple Topics) Topic There are a variety of math topics covered in the problems used for this meeting. Materials Needed Copies of the Halloween problem set (Problems and answers can be viewed here, but a more student-friendly version in larger font is available for download from www.mathcounts.org on the MCP Members Only page of the Club Program section.) Calculators Halloween candy for your students optional Meeting Plan Is it close to Halloween? If so, your students will enjoy working on these holiday-appropriate problems! Most of these problems were taken from previous Problems of the Week that are archived and available at www.mathcounts.org. If you are looking for more Halloween problems, there is another set of questions in the Club Resources from 2007 2008 that can be found on the MCP Members Only page in the Club Program section. Students can work individually or in groups on the problems. As soon as a group gets all of the correct answers (with help from you or the other groups, if necessary), you may have a special Halloween treat for its members. From the Oct. 29, 2007 Problem of the Week... 1. Bridgette wants to be a princess for Halloween. When she gets to the costume store, she realizes there are many options. There are five different crowns, eight different dresses and three different pairs of shoes. How many possible combinations are there for Bridgette s costume consisting of one crown, one dress and one pair of shoes? (For #2 and #3) Joseph and Dante went trick-or-treating and came back with A LOT of candy. However, when Joseph s little sister, Samantha, gets back, they find that not only does she have more candy, but she has better candy than they do. Samantha agrees to trade some of the candy, but they have to follow her trading rules. 3 Smarties packs = 1 fun-size candy bar 2 Tootsie Pops = 1 Skittles pack 15 candy corns = 1 Smarties pack 5 candy corns = 2 Bit-O-Honeys 3 Tootsie Pops = 1 fun-size candy bar 2. Based on Samantha s exchange rates, what item is the most valuable? 3. How many of the least-valuable item are required to get 1 of the most-valuable item? From the Oct. 27, 2003 Problem of the Week... 4. Before heading out to do some trick-or-treating, Molly had to choose which container to take with her to hold all of her candy. She had her plastic, spherical pumpkin with a radius of 5 inches, or a bag she found in the kitchen. The bag measured 5 inches by 9 inches by 11.5 inches and had the same shape as a rectangular prism when it was open. What is the volume, in cubic inches, of the container with the greatest volume? Express your answer as a decimal to the nearest tenth. 5. As Molly s mother was waiting for Molly to return from trick-or-treating, she started looking at the nutrition information on the bag from the candy she was distributing. For this bag of candy bars, a serving size is equivalent to four candy bars, there are eight servings per bag and 200 calories per serving. According to this information, how many calories are in one candy bar? How many calories are in all of the candy bars in an entire bag? 2008 2009 MATHCOUNTS Club Resource Guide 27 Club Resource Guide.pdf 27 8/18/08 11:24:15 AM
6. When Molly returned home from trick-or-treating, she spread all of her candy out on the kitchen table. She noticed that she had 14 Tootsie Pops, and she had at least one of exactly three different colors of Tootsie Pops (brown, red and purple). She had exactly twice as many purple as brown, and less than 25% of the 14 Tootsie Pops were brown. According to this information, what is the average value of the possible amounts of red Tootsie Pops she could have had? A few new ones... 7. Molly s mother started out the evening with three types of treats: candy bars, Tootsie Pops and Skittles packs in her Halloween candy bowl. The ratio candy bars:tootsie Pops:Skittles packs was 8:5:7, respectively. If she had exactly 40 Tootsie Pops, how many total treats were in the bowl? (A treat is one candy bar, one Tootsie Pop or one Skittles pack.) 8. Based on the information above, if the first trick-or-treater to Molly s house picked one treat from the bowl at random, what was the probability he was going to pick out a Skittles pack? Express your answer as a percent. 9. Molly was allowed to trick-or-treat from 6:30 p.m. until 8:45 p.m. If she averaged visiting 2 houses every 9 minutes during that time, how many houses did she visit? 10. Molly s mother had 6 different groups of kids with 3 or more kids in each group come to her door for candy. When looking at the numbers of kids per group, the average was 6, the unique mode was 4, the median was 4.5 and the range was 10. If the largest group to come to the door was as large as possible, how many kids were in the secondlargest group to visit Molly s house that night? Answers: 120 combinations; fun-size candy bar; 45 candy corns; 523.6 cubic inches (pumpkin); 50 calories, 1600 calories; 8 red Tootsie Pops; 160 treats; 35%; 30 houses; 5 kids **Complete solutions to the Problems of the Week are available in the Problem of the Week Archive section of www.mathcounts.org.** Possible Next Steps Ask the students to come up with some Halloween-based questions that can be shared with the elementary schools in your area or with your math classes. You can give students free reign or provide them with a list of math topics and ask them to create a Halloween-related problem for each one. They can work in small groups or together as a big team. If your students come up with some creative ideas for Halloween problems, we would love to see them. Perhaps one of your students problems can be used in a Problem of the Week or future Club Resource Guide. Please send them to info@mathcounts.org with the subject line MATHCOUNTS Club Program. Be sure to let us know who the author is so that we can give proper credit if the problem is used in MATHCOUNTS materials! 28 2008 2009 MATHCOUNTS Club Resource Guide Club Resource Guide.pdf 28 8/18/08 11:24:15 AM
Halloween Meeting Problem Set 1. Bridgette wants to be a princess for Halloween. When she gets to the costume store, she realizes there are many options. There are five different crowns, eight different dresses and three different pairs of shoes. How many possible combinations are there for Bridgette s costume consisting of one crown, one dress and one pair of shoes? (For #2 and #3) Joseph and Dante went trick-or-treating and came back with A LOT of candy. However, when Joseph s little sister, Samantha, gets back, they find that not only does she have more candy, but she has better candy than they do. Samantha agrees to trade some of the candy, but they have to follow her trading rules. 3 Smarties packs = 1 fun-size candy bar 2 Tootsie Pops = 1 Skittles pack 15 candy corns = 1 Smarties pack 5 candy corns = 2 Bit-O-Honeys 3 Tootsie Pops = 1 fun-size candy bar 2. Based on Samantha s exchange rates, what item is the most valuable? 3. How many of the least-valuable item are required to get 1 of the most-valuable item? 4. Before heading out to do some trick-or-treating, Molly had to choose which container to take with her to hold all of her candy. She had her plastic, spherical pumpkin with a radius of 5 inches, or a bag she found in the kitchen. The bag measured 5 inches by 9 inches by 11.5 inches and had the same shape as a rectangular prism when it was open. What is the volume, in cubic inches, of the container with the greatest volume? Express your answer as a decimal to the nearest tenth. 5. As Molly s mother was waiting for Molly to return from trick-or-treating, she started looking at the nutrition information on the bag from the candy she was distributing. For this bag of candy bars, a serving size is equivalent to four candy bars, there are eight servings per bag and 200 calories per serving. According to this information, how many calories are in one candy bar? How many calories are in all of the candy bars in an entire bag? Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
6. When Molly returned home from trick-or-treating, she spread all of her candy out on the kitchen table. She noticed that she had 14 Tootsie Pops, and she had at least one of exactly three different colors of Tootsie Pops (brown, red and purple). She had exactly twice as many purple as brown, and less than 25% of the 14 Tootsie Pops were brown. According to this information, what is the average value of the possible amounts of red Tootsie Pops she could have had? 7. Molly s mother started out the evening with three types of treats: candy bars, Tootsie Pops and Skittles packs in her Halloween candy bowl. The ratio candy bars:tootsie Pops:Skittles packs was 8:5:7, respectively. If she had exactly 40 Tootsie Pops, how many total treats were in the bowl? (A treat is one candy bar, one Tootsie Pop or one Skittles pack.) 8. Based on the information above, if the first trick-or-treater to Molly s house picked one treat from the bowl at random, what was the probability he was going to pick out a Skittles pack? Express your answer as a percent. 9. Molly was allowed to trick-or-treat from 6:30 p.m. until 8:45 p.m. If she averaged visiting 2 houses every 9 minutes during that time, how many houses did she visit? 10. Molly s mother had 6 different groups of kids with 3 or more kids in each group come to her door for candy. When looking at the numbers of kids per group, the average was 6, the unique mode was 4, the median was 4.5 and the range was 10. If the largest group to come to the door was as large as possible, how many kids were in the second-largest group to visit Molly s house that night? **Answers to these problems are on page 28 of the 2008-2009 Club Resource Guide.** Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
Halloween Solutions (2008-2009 MCP Club Resource Guide) Problem 1. Any of the five crowns can be paired with any of the eight different dresses. This means there are 5 8 = 40 combinations of crowns and dresses. Any of these combinations can be paired with any of the pairs of shoes. This means there are 40 3 = 120 combinations of crowns, dresses and shoes. Problems 2 and 3. Let s see if we can get all of these items equal to each other. Notice that the fun-size candy bar is the same as 3 Smarties and also the same as 3 Tootsie Pops. Therefore, 1 candy bar = 3 Smarties = 3 Tootsie Pops. (We can also deduce that 1 Smarties = 1 Tootsie Pop.) From the third equation, we seen that 15 candy corns equal 1 Smarties pack, so 45 candy corns equal 3 Smarties packs, and we have: 1 candy bar = 3 Smarties = 3 Tootsie Pops = 45 candy corns. Multiplying the fifth equation by 9, we see 45 candy corns equal 18 Bit-O-Honeys, so we have: 1 candy bar = 3 Smarties = 3 Tootsie Pops = 45 candy corns = 18 Bit-O-Honeys. The only thing we re missing are the Skittles. We know 2 Tootsie Pops equal 1 Skittles, so 3 Tootsie Pops equal 1.5 Skittles. Therefore, our final comparison using all of the items is: 1 candy bar = 3 Smarties = 3 Tootsie Pops = 45 candy corns = 18 Bit-O-Honeys = 1.5 Skittles. The item with the greatest value is then the fun-size candy bar. The item with the least value is the candy corn, and it takes 45 candy corns to equal 1 fun-size candy bar. Problem 4. Since the pumpkin is a sphere with radius 5 inches, the volume is (4/3)(pi)(5 3 ) = 523.6 cubic inches. The volume of the bag is (5)(9)(11.5) = 517.5 cubic inches, so it seems that Molly should go with the pumpkin! Problem 5. If there are 200 calories per serving and a serving is four candy bars, then one candy bar has 200 4 = 50 calories. The bag, however, has eight servings at 200 calories each, which is 8 200 = 1600 calories! Problem 6. We know that she has at least one of each color. Since less than 25% of the Tootsie Pops are brown, she must have 1, 2 or 3 brown ones. (Four brown ones would be over 28% of the Pops.) If she has one brown, she has 2(1) = 2 purples and therefore 14 (1 + 2) = 11 red. If she has two brown, she has 2(2) = 4 purples and therefore 14 (2 + 4) = 8 red. If she has three brown, she has 2(3) = 6 purples and therefore 14 (3 + 6) = 5 red. The average value of 11, 8 and 5 is (11 + 8 + 5) 3 = 8 red Tootsie Pops. Problem 7. Since the ratio candy bars:tootsie Pops:Skittles packs was 8:5:7 and there were 40 Tootsie Pops, we can see that the numbers in the original ratio must each be multiplied by 8 to get to the actual numbers since 40 5 =8. (Since Tootsie Pops is the second item listed in the ratio, it correlates to the 5 in the original ratio.) Now we can see that the actual numbers for these items was 64:40:56, which is a total of 64 + 40 + 56 = 160 treats. Problem 8. Since there are 160 treats and 56 of them are Skittles packs, the probability that the first treat selected will be a Skittles pack is 56/160 = 35%. Problem 9. From 6:30 p.m. until 8:45 p.m. is 2 hours, 15 minutes. This is 120 + 15 = 135 minutes, which is 135 9 = 15 nine-minute periods. If she visited two houses every nine minutes, then she visited 15 2 = 30 houses. Problem 10. Molly s mom had six different groups of kids come to the door. Since the average size per group was 6 kids, she must have had a total of 6 6 = 36 kids. Since the unique mode was 4 kids, there must have been at least two groups of 4 kids. Since the median was 4.5, then Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set
the third-largest and fourth-largest groups together had 9 kids. So they either had 4 and 5 kids or 3 and 6 kids. However, since we know there were at least two groups with 4 kids, this second scenario is not possible and we must have: (, 4, 4, 5,, ). The range of the group was 10. So we have either (3, 4, 4, 5,, 13) or (4, 4, 4, 5,, 14). Since the largest group to come to the door was as large as possible, then we are going to look at the second scenario: (4, 4, 4, 5,, 14). Remember that we determined there were a total of 36 kids, which means the missing number is 36 4 4 4 5 14 = 5, which would give us (4, 4, 4, 5, 5, 14). Because all of our criteria are still being met, this is our solution and the second-largest group has 5 kids. Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set