Math 208, Section 7.1 Solutions: The Meaning of Division

Transcription

1 Math 208, Section 7.1 Solutions: The Meaning of Division 1. For each of the following story problems, write the corresponding numerical division problem, state which interpretation of division is involved (the how many groups? or the how many in each group?, exact division or with remainder), and solve the problem: a.) If 250 rolls are to be put in packages of 12, then how many packages of rolls can be made? ; how many groups (measurement division); with remainder; 20 full rolls with 10 left over. b.) If you have 500 stickers to give out equally to 23 children, then how many stickers will each child get? ; how many in each group (partitive division); with remainder; 21 stickers with 17 left over. c.) Given that 1 gallon is 8 pints, how many gallons of water is 45 pints of water? 45 8; how many groups (measurement division); exact division; gallons. d.) If your car used 12 gallons of gasoline to drive 350 miles, then how many miles per gallon did your car get? ; how many in each group (partitive); exact division; miles per gallon. e.) If you drove 170 miles at a constant speed and if it took you 3 hours, then how fast were you going? 170 3; how many in each group (partitive); exact division; miles per hour. f.) Given that 1 inch is 2.54 centimeters, how tall in inches is a man who is 186 centimeters tall? Given that 1 foot is 12 inches, how tall is the man in feet? is the answer to the first question, and then the answer to that problem 12; how many groups (measurement); exact division; inches or 6.1 feet (or 6 feet, 1.23 inches). g.) Sue needs to cut a piece of wood 0.4 of an inch thick, or just a little less thick. Sue s ruler shows sixteenths of an inch. How many sixteenths of an inch thick should Sue cut her piece of wood? 0.4 (1/16); how many groups (measurement division); with remainder; 6 sixteenths actually a little more to get 0.4 inches. 2. Write two story problems for 38 7, one for each of the two interpretations of division. Solve each problem and explain its answer. Susan has 38 brownies and she wants to give an equal amount to each of her 7 friends. How much of a brownie will each friend get? This is a how much in each group problem or a partitive division problem. Each friend will get 5 and 3/7 brownies. Page 1 of 5

2 Susan s recipe for macadamia nut cookies with white chocolate (yum!) calls for 7 cups of macadamia nuts. She has 38 cups of macadamia nuts on hand. How many recipes can she make? This is a how many groups or a measurement division problem. She can make 5 and 3/7 recipes, assuming that all other ingredients are available in any quantity. 3. Write and solve four different story problems for 7 3. a.) In the first story problem, the answer should be best expressed as 2, remainder 1. Explain why this is the best answer. 7 children want to go on a trip. Each car can hold one grown-up driver (there are enough drivers to go around) and 3 children. How many full cars will there be? How many children will go in the last (un-full) car? I think that 2, remainder 1 is a good way to express the answer here; because the 2 means that there are 2 full cars and the 1 child left over will have to go all by herself with a grown-up (booorrrring!). b.) In the second story problem, the answer should be best expressed as 2 1/3. Explain why this is the best answer. I have 7 cups of flour in my kitchen and I want to make a recipe that calls for 3 cups of flour. If there are plenty of all the other ingredients, how many recipes can I make if I use all the flour? Here, 2 and 1/3 seems the best answer. You want to use all the flour available, rather than making some full recipes and just having flour left over. It seems quite reasonable to express an answer as 1/3 of a recipe, since we do it all the time. c.) In the third story problem, the answer should be best expressed as Explain why this is the best answer. There are 3 children coming to my birthday party (I am new in town and don t have many friends) and I have $7.00 to spend on items for their party bags. What is the most I can spend on each bag, assuming all the bags have the same prizes? Here, I think $2.33 seems like the best answer, even though 2.33 is not exactly 7 3 but a little less. We don t normally talk in terms of thirds of a dollar but in dollars and cents so I wouldn t express the answer as 2 and 1/3 dollars. d.) In the fourth story problem, the answer should be best expressed as 3 (even though Explain why this is the best answer. 7 children want to go on a trip. Each car can hold one grown-up driver (there are enough drivers to go around) and 3 children. How many grown-up drivers will be needed? Here the answer is 3 obviously; you need 2 full cars (3 kids each) and one car with one child for a total of 3 cars. Each car needs one grown-up. 4. For each of the problems that follow, write the corresponding numerical division problem and solve the problem. Determine the best form (or forms) of the answer: a mixed number, a decimal, a whole number with remainder, or a whole number that is not equal to the solution of the numerical division problem. Briefly explain your answers. a.) For purposes of maintenance, a 58-mile stretch of road will be divided into 3 equal parts. How many miles of road are in each part? Page 2 of 5

3 19 and 1/3 miles. I think this answer could just as easily be expressed as a decimal, as long as you are willing to round off. However, 1/3 of a mile is a nice whole number of feet so it should be easy for the road crew to measure off. b.) For purposes of maintenance, a 58-mile stretch of road will be divided into sections of 15 miles. Each section of 15 miles will be the responsibility of a particular road crew. How many road crews are needed? You need 4 road crews. Obviously, is not four but does equal 3 with remainder 13. You can t just neglect the last 13 miles of road so you need a whole crew for that segment, even though they have a little less road to cover than the other crews. c.) You have 75 pencils to give out to a class of 23 children who insist on fair distribution. How many pencils will each child get? It makes no sense to break the pencils up since then some kids would get an eraser and no point and others would get points without erasers. So you d better just give the answer as 3 pencils per child. You will have 6 pencils left over but you can keep those in reserve for the kids who break their pencils or lose them or whatever. d.) You have 7 packs of chips to share equally among 4 hungry people. How many packs of chips will each person get? I think a fraction of a bag will make sense here since presumably the chips can be shared out equally (or almost) among the kids. So you can give the answer as 1 and ¾ bag per child. 6. a.) Is 0 5 defined or not? Write a story problem for 0 5, and use your story problem to discuss whether or not 0 5 is defined. 0 5 = 0, and it makes perfect sense. If I have a recipe that calls for 5 cups of flour and I have NO flour in my kitchen, then I can t make any recipes at all. Here, the first 0 (the dividend) is the number of cups of flour I have and the second 0 (the quotient) is the number of recipes I can make. b.) Is 5 0 defined or not? Write a story problem for 5 0, and use your story problem to discuss whether or not 5 0 is defined. If I have 5 cups of flour on hand (the dividend) and I have a recipe that doesn t require flour (the quotient), then how many recipes can I make? Notice that there is no way to answer this question! In these recipe problems, we usually assume that all the other ingredients are available in unlimited quantities. But if that is the case here, then I can make any number of recipes I want! Observe that this result is consistent with the fact that we usually dismiss division by zero as being undefined. 13. Explain clearly how to use the meanings of multiplication and division, as well as the following information, in order to determine how many grams 1 cup of water weighs: 1 quart = 4 cups; 1 liter = quarts; 1 liters weighs 1 kilogram; 1 kilogram = 1000 grams. How many grams does a cup of water weigh? We need to use the information given. Page 3 of 5

4 Let s start with the equation 1 liter = quarts. So a liter is just a little over a quart. This sort of computation is probably easier for children to follow if you convert it to whole numbers. Since 1 liter = quarts, we have that 1000 liters = 1056 quarts So 1 quart = liters = liters. (This is measurement division) 1056 Now there are 4 cups in a quart so 1 cup is ¼ of a quart, that is, ¼ liters = liters (I am rounding off in both of these computations, to the nearest thousandth). So 1 cup = liters. A liter weighs 1 kilogram, so 1 cup weighs kilograms. A kilogram is 1000 grams so kilograms = 237 grams. This is how much one cup of water weighs. 15. Susan has a 5-pound bag of flour and an old recipe of her grandmother s calling for one kilogram of flour. She reads on the bag of flour that it weighs 2.26 kilograms. She also reads on the bag of flour that one serving of flour is about ¼ cup and that there are about 78 servings in the bag of flour. a. Based on the information given, how many cups of flour should Susan use in her grandmother s recipe? Solve this problem, and explain how your use the meanings of multiplication and division in solving this problem. Susan s bag weighs 2.26 kilograms (note the comment at the end of the last problem, though) but her grandmother s recipe calls for only 1 kilogram. Let s first convert everything to fractions of the bag of flour: We have (fraction of a bag of flour to be used) (number of kilograms in one bag) = (number of kilograms to be used) Now we know the number of kilograms to be used: that s 1 kilogram (what the recipe calls for). We also know the number of kilograms in the bag: that s printed on the bag and it is So this is a measurement division problem ( how many groups? where a group is one bag of flour). The multiplication problem becomes? 2.26 = 1 1 so the fraction of the bag of flour that we want to use is = (I have rounded off to 2.26 the nearest thousandth. This is about 2/5 of the bag...) Now let s convert to servings. (Why the amounts are given in servings I do not know but maybe nutrition labels do this to make it sound like they have more in them.) There are 78 servings in a bag. We want to use bags of flour in the recipe. The equation (written as a multiplication) is (number of bags to be used) (number of servings in a bag) = (number of servings to be used) Here our group is one bag and the objects are servings. The equation is =? so the number of servings to be used is 34.5 servings. Once again I have rounded off. Page 4 of 5

5 Now how many cups is that?? Well, we know the number of servings and the number of servings in a cup: namely, 4. The equation is (number of cups to be used) (number of servings in a cup) = (number of servings to be used) In this equation the servings are the objects and the groups are the cups; there are four objects in a group. Numerically the equation is? 4 = So? = = This is just about 8 5 cups of flour. 4 8 note that at every stage it is important to keep track of what the groups are and what the objects are; otherwise you can get really confused about whether to multiply or divide. You can also let the units help you keep track of things: cups serving servings bag bags kilo kilos recipe = cups recipe If you kind of cancel the units you can see how it works out. This last kind of equation is called dimensional analysis in physics and it is very handy for keeping us doing the right thing at each stage multiplying or dividing. Page 5 of 5