0.. DIVIDING FRACTIONS Excerpt from: Mathematics for Elementary Teachers, First Edition, by Sybilla Beckmann. Copyright c 00, by Addison-Wesley 0. Dividing Fractions In this section, we will discuss the two interpretations of division for fractions, and we will see why the standard invert and multiply procedure for dividing fractions gives answers to fraction division problems that agree with what we expect from the meaning of division. The Two Interpretations of Division for Fractions Let s review the meaning of division for whole numbers, and see how to interpret division for fractions. The how many groups? interpretation With the how many groups? interpretation of division, means the number of groups we can make when we divide objects into groups with objects in each group. In other words, tells us how many groups of we can make from. Similarly, with the how many groups? interpretation of division, 5 means the number of groups we can make when we divide 5 of an object into groups with of an object in each group. In other words, 5 tells us how many groups of we can make from 5. For example, suppose you are making popcorn balls and each popcorn ball requires of a cup of popcorn. If you have = 5 of a cup of popcorn, then how many popcorn balls can you make? In this case you want to divide 5 of a cup of popcorn into groups (balls) with of a cup of popcorn in each group. According to the how many groups? interpretation of division, you can make popcorn balls. 5
The how many in one (each) group? interpretation With the how many in each group? interpretation of division, means the number of objects in each group when we distribute objects equally among groups. In other words, is the number of objects in one group if we use objects to evenly fill groups. When we work with fractions, it often helps to think of how many in each group? division story problems as asking how many are in one whole group?, and it helps to think of filling groups or part of a group. So in the context of fractions, we will usually refer to the how many in each group? interpretation as how many in one group?. With the how many in one group? interpretation of division, is the number of objects in one group when we distribute of an object equally among of a group. A clearer way to say this is: is the number of objects (or fraction of an object) in one whole group when of an object fills of a group. For example, suppose you pour of a pint of blueberries into a container and this fills of the container. How many pints of blueberries will it take to fill the whole container? In this case, of a pint of blueberries fills (i.e., is distributed equally among) of a group (a container). So according to the how many in one group? interpretation of division, the number of pints of blueberries in one whole group (one full container) is One way to better understand fraction division story problems is to think about replacing the fractions in the problem with whole numbers. For example, if you have pints of blueberries and they fill containers, then how many pints of blueberries are in each container? We solve this problem by dividing, according to the how many in each group? interpretation. Therefore if we replace the pints with of a pint, and the containers with of a container, we solve the problem in the same way as before: now becomes. Here is another way to think about the problem. Because of the container is filled, and because this amount is of a pint, therefore of the
0.. DIVIDING FRACTIONS number of pints in a full container is Therefore of a pint. In other words: number of pints in full container = number of pints in full container = Dividing by Versus Dividing in In mathematics, language is used much more precisely and carefully than in everyday conversation. This is one source of difficulty in learning mathematics. For example, consider the two phrases: dividing by, dividing in. You may feel that these two phrases mean the same thing, however, mathematically, they do not. To divide a number, say 5, by means to calculate 5. Remember that we read A B as A divided by B. We would divide 5 by if we wanted to know how many half cups of flour are in 5 cups of flour, for example. (Notice that there are 0 half-cups of flour in 5 cups of flour, not.) On the other hand, to divide a number in half means to find half of that number. So to divide 5 in half means to find of 5. One half of 5 means 5. So dividing in is the same as dividing by. The Invert and Multiply Procedure for Fraction Division Although division with fractions can be difficult to interpret, the procedure for dividing fractions is quite easy. To divide fractions, such as and 6 5 we can use the familiar invert and multiply method in which we invert the divisor and multiply by it: = = = 9 8
and 6 5 = 6 5 = 6 5 = 6 5 = 0 = 5 reciprocal Another way to describe this invert and multiply method for dividing fractions is in terms of the reciprocal of the divisor. The reciprocal of a fraction C is the fraction D. In order to divide fractions, we should multiply D C by the reciprocal of the divisor. So in general, A B C D = A B D C = A D B C Explaining Why Invert and Multiply is Valid by Relating Division to Multiplication The procedure for dividing fractions is easy enough to carry out, but why is it a valid method? Before we answer this question in general, consider a special case. Recall that every whole number is equal to a fraction (for example, 6 = 6 ). Therefore we can apply the invert and multiply procedure to whole numbers as well as to fractions. According to this procedure, = = = = Notice that this result, that =, agrees with our findings earlier in this chapter: that we can describe fractions in terms of division, namely that A = A B. B In general, why is the invert and multiply procedure a valid way to divide fractions? One way to explain this is to relate fraction division to fraction multiplication. Recall that every division problem is equivalent to a multiplication problem (actually two multiplication problems): A B =? is equivalent to (or B? = A). So? B = A =?
0.. DIVIDING FRACTIONS 5 is equivalent to? =. () Now remember that we want to explain why the invert and multiply rule for fraction division is valid. This rule says that ought to be equal to Let s check that this fraction works in the place of the? in Equation. In other words, let s check that if we multiply times, then we really do get : = ( ) ( ) = = ( ) ( ) = Therefore the answer we get from the invert and multiply procedure really is the answer to the original division problem. Notice that the line of reasoning above applies in the same way when other fractions replace the fractions and used above. It will still be valuable to explore fraction division further, interpreting fraction division directly rather than through multiplication. Class Activity 0A: Explaining Invert and Multiply by Relating Division to Multiplication Using the How Many Groups? Interpretation to Explain Why Invert And Multiply Is Valid Above, we explained why the invert and multiply procedure for dividing fractions is valid by considering fraction division in terms of fraction multiplication. Now we will explain why the invert and multiply procedure is valid by working with the how many groups? interpretation of division. Consider the division problem The following is a story problem for this division problem: How many cups of water are in of a cup of water?
6 Or, said another way: How many times will we need to pour cup of water into a container that holds cup of water in order to fill the container? From the diagram in Figure we can say right away that the answer to this problem is one and a little more because one half cup clearly fits in two thirds of a cup, but then a little more is still needed to fill the two thirds of a cup. But what is this little more? Remember the original question: we want to know how many cups of water are in of a cup of water. So the answer should be of the form so and so many cups of water. This means that we need to express this little more as a fraction of cup of water. How can we do that? By subdividing both the and the into common parts, namely by using common denominators. / cup / cup / cup = /6 cup Figure : How Many / Cups of Water Are in / Cup? / cup = /6 cup When we give and the common denominator of 6, then, as on the right of Figure, the cup of water is made out of parts ( sixths of a cup of water), and the cup of water is made out of parts ( sixths of a cup of water), so the little more we were discussing above is just one of those parts. Since cup is parts, and the little more is part, the little more is of the cup of water. This explains why = : there s a whole cup plus another of the cup in of a cup of water. To recap: we are considering the fraction division problem in terms of the story problem how many cups of water are in of a cup of water? If we give and the common denominator of 6, then we can rephrase the problem as how many of a cup are in of a cup? But in terms of 6 6 Figure??, this is equivalent to the problem how many s are in? which is the problem, whose answer is =. Notice that is exactly the same answer we get from the invert and multiply procedure for fraction division: = = =
0.. DIVIDING FRACTIONS 7 So the invert and multiply procedure gives the same answer to that we arrive at by using the how many groups? interpretation of division. The same line of reasoning will work for any fraction division problem A B C D Thinking logically, as above, and interpreting A C as how many C cups B D D of water are in A cups of water?, we can conclude that B A B C D = A D B D B C B D = (A D) (B C) = A D B C The final expression, A D, is the answer provided by the invert and multiply B C procedure for dividing fractions. Therefore we know that the invert and multiply procedure gives answers to division problems that agree with what we expect from the meaning of division. Class Activity 0B: How Many Groups? Fraction Division Problems Using the How Many in One Group? Interpretation to Explain Why Invert And Multiply Is Valid Above, we saw how to use the how many groups? interpretation of division to explain why the invert and multiply procedure for fraction division is valid. We can also use the how many in one group? interpretation for the same purpose. This interpretation, although perhaps more difficult to understand, has the advantage of showing us directly why we can multiply by the reciprocal of the divisor in order to divide fractions. Consider the following how many in one group? story problem for 5 : You used of can of paint to paint of a wall. How many cans 5 of paint will it take to paint the whole wall? This is a how many in one group? problem because we can think of the paint as filling of the wall. We can also see that this is a division problem 5 by writing the corresponding number sentence: 5 (amount to paint the whole wall) =
8 Therefore amount to paint the whole wall = 5 We will now see why it makes sense to solve this problem by multiplying by the reciprocal of, namely by 5. Let s focus on the wall to be painted, 5 as shown in Figure. Think of dividing the wall into 5 equal sections, of the / can of paint is divided equally among parts the amount of paint for the full wall is 5 times the amount in one part Figure : The Amount of Paint Needed for the Whole Wall is 5 of the Can Used to Cover of the Wall 5 which you painted with the can of paint. If you used a can of paint to paint sections, then each of the sections required or cans of paint. To determine how much paint you will need for the whole wall, multiply the amount you need for one section by 5. So you can determine the amount of paint you need for the whole wall by multiplying the can of paint by and then multiplying that result by 5, as summarized in Table. But to multiply a number by and then multiply it by 5 is the same as multiplying the number by 5. Therefore we can determine the number of cans of paint you need for the whole wall by multiplying by 5: 5 = 5 6 This is exactly the invert and multiply procedure for dividing. It 5 shows that you will need 5 of a can of paint for the whole wall. 6
0.. DIVIDING FRACTIONS 9 use can paint for 5 of the wall or or use 6 can paint for 5 of the wall use 5 6 5 5 can paint for whole wall use in one step: can paint for of the wall 5 use 5 6 5 5 can paint for whole wall Table : Determining How Much Paint to Use for a Whole Wall if Paint Covers of the Wall 5 Can of
0 The argument above works when other fractions replace and, thereby 5 explaining why A B C D = A B D C In other words, to divide fractions, multiply the dividend by the reciprocal of the divisor. Class Activity 0C: How Many in One Group? Fraction Division Problems Class Activity 0D: Are These Division Problems? Exercises for Section 0. on Dividing Fractions. Write a how many groups? story problem for 5. Use the story 7 problem and a diagram to help you solve the problem.. Write a how many in one group? story problem for. Use the situation of the story problem to help you explain why the answer is =.. Annie wants to solve the division problem story problem: by using the following I need cup of chocolate chips to make a batch of cookies. How many batches of cookies can I make with of a cup of chocolate chips? Annie draws a diagram like the one in Figure. Explain why it would be easy for Annie to misinterpret her diagram as showing that =. How should Annie interpret her diagram so as to conclude that =?. Which of the following are solved by the division problem? For those that are, which interpretation of division is used? For those that are not, determine how to solve the problem, if it can be solved. (a) of a bag of jelly worms make worms are in one bag? a cup. How many cups of jelly
0.. DIVIDING FRACTIONS / cup makes one batch / cup left Figure : How Batches of Cookies Can We Make With of a Cup of Chocolate Chips if Batch Requires Cup of Chocolate Chips? (b) of a bag of jelly worms make a cup. How many bags of jelly worms does it take to make one cup? (c) You have of a bag of jelly worms and a recipe that calls for of a cup of jelly worms. How many batches of your recipe can you make? (d) You have of a cup of jelly worms and a recipe that calls for of a cup of jelly worms. How many batches of your recipe can you make? (e) If of a pound of candy costs of a dollar, then how many pounds of candy should you be able to buy for dollar? (f) If you have of a pound of candy and you divide the candy in, then how much candy will you have in each portion? (g) If of a pound of candy costs $, then how many dollars should you expect to pay for of a pound of candy? 5. Frank, John, and David earned $ together. They want to divide it equally, except that David should only get a half share, since he did half as much work as either Frank or John did (and Frank and John worked equal amounts). Write a division problem to find out how much Frank should get. Which interpretation of division does this story problem use? 6. Bill leaves a tip of $.50 for a meal. If the tip is 5% of the cost of the meal, then how much did the meal cost? Write a division problem to solve this. Which interpretation of division does this story problem use?
7. Compare the arithmetic needed to solve the following problems. (a) What fraction of a cup measure is filled when we pour in cup of water? (b) What is one quarter of cup? (c) How much more is cup than cup? (d) If cup of water fills of a plastic container, then how much water will the full container hold? 8. Use the meanings of multiplication and division to solve the following problems. (a) Suppose you drive 500 miles every half year in your car. At the end of years, how many miles will you have driven? (b) Mo used 8 ounces of liquid laundry detergent in 6 weeks. If Mo continues to use laundry detergent at this rate, how much will he use in a year? (c) Suppose you have a ounce bottle of weed killer concentrate. The directions say to mix two and a half ounces of weed killer concentrate with enough water to make a gallon. How many gallons of weed killer will you be able to make from this bottle? 9. The line segment below is of a unit long. Show a line segment that is 5 of a unit long. Explain how this problem is related to fraction division. unit Answers To Exercises For Section 0. on Dividing Fractions. A simple how many groups? story problem for 5 is how many 5 7 7 of a cup of water are in cup of water? Figure shows cup of water and shows 5 of a cup of water shaded. The shaded portion is divided 7 into 5 equal parts and the full cup is 7 of those parts. So the full cup is 7 of the shaded part. Thus there are 7 of 5 of a cup of water in 5 5 7 cup of water, so 5 = 7. 7 5
0.. DIVIDING FRACTIONS cup each piece is 5 of the shaded of a cup 5 7 portion Figure : Showing Why 5 = 7 by Considering How Many 5 7 5 7 Water are in Cup of Water of a Cup of. A how many in one group? story problem for is if ton of dirt fills a truck full, then how many tons of dirt will be needed to fill the truck completely full?. We can see that this is a how many in one group? type of problem because the ton of dirt fills of a group (the truck) and we want to know the amount of dirt in whole group. Figure 5 shows a truck bed divided into equal parts with of those parts filled with dirt. Since the parts are filled with ton of dirt, each of the parts must contain of a ton of dirt. To fill the truck completely, parts, each containing of a ton of dirt are needed. Therefore the truck takes = tons of dirt to fill it completely, and so =. the ton of dirt is divided equally among parts truck bed parts are needed to fill the truck; each part is / of a ton, so / tons of dirt are needed to fill the truck Figure 5: Showing Why = by Considering How Many Tons of Dirt it Takes to Fill a Truck if Ton Fills it Full. Annie s diagram shows that she can make full batch of cookies from
her of a cup of chocolate chips and that cup of chocolate chips will be left over. Because cups of chocolate chips are left over, it would be easy for Annie to misinterpret her picture as showing =. But the answer to the problem is supposed to be the number of batches Annie can make. In terms of batches, the remaining cup of chocolate chips makes of a batch of cookies. We can see this because quartercup sections make a full batch, so each quarter-cup section makes of a batch of cookies. So by interpreting the remaining cup of chocolate chips in terms of batches, we see that Annie can make batches of chocolate chips, thereby showing that =, not.. (a) This problem can be rephrased as if of a cup of jelly worms fill of a bag, then how many cups fill a whole bag?, therefore this is a how many in one group? division problem illustrating, not. Since = =, there are of a cup of jelly worms in a whole bag. (b) This problem is solved by, according to the how many in each group? interpretation. A group is a cup and each object is a bag of jelly worms. (c) This problem can t be solved because you don t know how many cups of jelly worms are in of a bag. (d) This problem is solved by, according to the how many groups? interpretation. Each group consists of of a cup of jelly worms. (e) This problem is solved by, according to the how many in one group? interpretation. This is because you can think of the problem as saying that of a pound of candy fills of a group and you want to know how many pounds fills whole group. (f) This problem is solved by, not. It is dividing in half, not dividing by. (g) This problem is solved by, according to the how many groups? interpretation because you want to know how many pounds are in of a pound. Each group consists of of a pound of candy. 5. If we consider Frank and John as each representing one group, and David as representing half of a group, then the $ should be dis-
0.. DIVIDING FRACTIONS 5 tributed equally among groups. Therefore, this is a how many in one group division problem. Each group should get = 5 = 5 = 8 5 = 5 5 = 5 6 0 = 5.60 dollars. Therefore Frank and John should each get $5.60 and David should get half of that, which is $.80. 6. According to the how many in one group? interpretation, the problem is solved by $.50 0.5 because $.50 fills 0.5 of a group and we want to know how much is in whole group. So the meal cost $.50 0.5 = $.50 5 00 = $.50 00 5 = $50 5 = $0 7. Each problem, except for the first and last, requires different arithmetic to solve it. (a) This is asking: equals what times? We solve this by calculating, which is. We can also think of this as a division problem with the how many groups? interpretation because we want to know how many of a cup are in of a cup. According to the meaning of division, this is. (b) This is asking: what is of? We solve this by calculating =. (c) This is asking: what is? The answer is which happens to be the same answer as in part (b), but the arithmetic to solve it is different. (d) Since cup of water fills of a plastic container, the full container will hold times as much water, or = of a cup. We can also think of this as a division problem with the how many in one group? interpretation. cup of water is put into of a group. We want to know how much is in one group. According to the meaning of division it s, which again is equal to. 8. (a) The number of years in years is. There will be that many groups of 500 miles driven. So after years you will have
6 driven miles. ( ) 500 = (5 ) 500 = 5 500 =, 750 (b) Since one year is 5 weeks there are 5 6 groups of 6 weeks in a year. Mo will use 8 ounces for each of those groups, so Mo will use (5 6 ) 8 = (5 ) 8 = 0 8 =, 0 ounces of detergent in a year. (c) There are groups of ounces in ounces. Each of those groups makes gallon. So the bottle makes = gallons 5 of weed killer. 9. One way to solve the problem is to determine how many units are in 5 units. This will tell us how many of the unit long segments to lay end to end in order to get the 5 unit long segment. Since 5 = 5 =, there are segments of length units in a segment of length 5 units. So you need to form a line segment that is times as long as the one pictured, plus another as long: Problems for Section 0. on Dividing Fractions. A bread problem: If one loaf of bread requires cups of flour, then how many loaves of bread can you make with 0 cups of flour? (Assume that you have enough of all other ingredients on hand.) (a) Solve the bread problem by drawing a diagram. reasoning. Explain your
0.. DIVIDING FRACTIONS 7 (b) Write a division problem that corresponds to the bread problem. Solve the division problem by inverting and multiplying. Verify that your solution agrees with your solution in part (a).. A measuring problem: You are making a recipe that calls for cup of water, but you can t find your cup measure. You can, however, find your cup measure. How many times should you fill your cup measure in order to measure of a cup of water? (a) Solve the measuring problem by drawing a diagram. Explain your reasoning. (b) Write a division problem that corresponds to the measuring problem. Solve the division problem by inverting and multiplying. Verify that your solution agrees with your solution in part (a).. Write a how many groups? story problem for and solve your problem in a simple and concrete way without using the invert and multiply procedure. Explain your reasoning. Verify that your solution agrees with the solution you obtain by using the invert and multiply procedure.. Write a how many groups? story problem for 5 and solve your problem in a simple and concrete way without using the invert and multiply procedure. Explain your reasoning. Verify that your solution agrees with the solution you obtain by using the invert and multiply procedure. 5. Jose and Mark are making cookies for a bake sale. Their recipe calls for cups of flour for each batch. They have 5 cups of flour. Jose and Mark realize that they can make two batches of cookies and that there will be some flour left. Since the recipe doesn t call for eggs, and since they have plenty of the other ingredients on hand, they decide they can make a fraction of a batch in addition to the two whole batches. But Jose and Mark have a difference of opinion. Jose says that 5 = 9 and so he says that they can make batches of cookies. Mark says 9 that two batches of cookies will use up cups of flour, leaving left,
8 so they should be able to make batches. Mark draws the picture in Figure 6 to explain his thinking to Jose. Discuss the boys mathematics: Figure 6: Representing 5 by Considering How Many are in 5 Cups of Flour Cups of Flour what s right, what s not right, and why? If anything is incorrect, how could you modify it to make it correct? 6. Marvin has yards of cloth to makes costumes for a play. Each costume requires yards of cloth. (a) Solve the following two problems: i. How many costumes can Marvin make and how much cloth will be left over? ii. What is? (b) Compare and contrast your answers in part (a). 7. A laundry problem: You need of a cup of laundry detergent to wash one full load of laundry. How many loads of laundry can you wash with 5 cups of laundry detergent? (Assume that you can wash fractional loads of laundry.) (a) Solve the laundry problem by drawing a diagram. Explain your reasoning. (b) Write a division problem that corresponds to the laundry problem. Solve the division problem by inverting and multiplying. Verify that your solution agrees with your solution in part (a). 8. Write a how many groups? story problem for and solve your problem in a simple and concrete way without using the invert and multiply procedure. Explain your reasoning. Verify that your solution agrees with the solution you obtain by using the invert and multiply procedure.
0.. DIVIDING FRACTIONS 9 9. Write a how many groups? story problem for and solve your problem in a simple and concrete way without using the invert and multiply procedure. Explain your reasoning. Verify that your solution agrees with the solution you obtain by using the invert and multiply procedure. 0. Write a how many groups? story problem for and solve your problem in a simple and concrete way without using the invert and multiply procedure. Explain your reasoning. Verify that your solution agrees with the solution you obtain by using the invert and multiply procedure.. Fraction division story problems involve the simultaneous use of different wholes. Solve the following paint problem in a simple and concrete way without using the invert and multiply procedure. Describe how you must work simultaneously with different wholes in solving the problem. A paint problem: You need a poster board. You have poster boards can you paint? of a bottle of paint to paint bottles of paint. How many. An article by Dina Tirosh, [?], discusses some common errors in division. The following problems are based on some of the findings of this article. (a) Tyrone says that 5 doesn t make sense because 5 is bigger than and you can t divide a smaller number by a bigger number. Give Tyrone an example of a sensible story problem for 5. Solve your problem and explain your solution. (b) Kim says that can t be equal to because when you divide, the answer should be smaller. Kim thinks the answer should be because that is less than. Give Kim an example of a story problem for and explain why it makes sense that the answer really is, not.. Write a story problem for and another story problem for (make clear which is which). In each case, use elementary reasoning about the story situation to solve your problem. Explain your reasoning.
0. Sam picked of a gallon of blueberries. Sam poured the blueberries into one of his plastic containers and noticed that the berries filled the container full. Solve the following problems in any way you like without using a calculator. Explain your reasoning in detail. (a) How many of Sam s containers will gallon of blueberries fill? (Assume Sam has a number of containers of the same size.) (b) How many gallons of blueberries does it take to fill Sam s container completely full? 5. A road crew is building a road. So far, of the road has been completed and this portion of the road is of a mile long. Solve the following problems in any way you like without using a calculator. Explain your reasoning in detail. (a) How long will the road be when it is completed? (b) When the road is mile long, what fraction of the road will be completed? 6. Will has mowed of his lawn and so far it s taken him 5 minutes. For each of the following problems, solve the problem in two ways: ) by using elementary reasoning about the story situation and ) by interpreting the problem as a division problem (say whether it is a how many groups? or a how many in one group? type of problem) and by solving the division problem using standard paper and pencil methods. Do not use a calculator. Verify that you get the same answer both ways. (a) How long will it take Will to mow the entire lawn (all together)? (b) What fraction of the lawn can Will mow in an hour? 7. Grandma s favorite muffin recipe uses cups of flour for one batch of muffins. For each of the following problems, solve the problem in two ways: ) by using elementary reasoning about the story situation and ) by interpreting the problem as a division problem (say whether it is a how many groups? or a how many in one group? type of problem) and by solving the division problem using standard paper and pencil methods. Do not use a calculator. Verify that you get the same answer both ways.
0.. DIVIDING FRACTIONS (a) How many cups of flour are in one muffin? (b) How many muffins does cup of flour make? (c) If you have cups of flour, then how many batches of muffins can you make? (Assume that you can make fractional batches of muffins and that you have enough of all the ingredients.) 8. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by. 9. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by. 0. Write a how many in one group? story problem for 9 and use your story problem to explain why it makes sense to solve 9 by inverting and multiplying, in other words by multiplying 9 by.. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by.. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying.. Give an example of either a hands-on activity or a story problem for elementary school children that is related to a fraction division problem (even if the children wouldn t think of the activity or problem as fraction division). Write the fraction division problem that is related to your activity or story problem. Describe how the children could solve the problem by using logical thinking aided by physical actions or by drawing pictures.. Buttercup the gerbil drank of a bottle of water in days. Assuming Buttercup continues to drink water at the same rate, how many bottles of water will Buttercup drink in 5 days? Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do.
5. If you used truck loads of mulch for a garden that covers of an acre, then how many truck loads of mulch should you order for a garden that covers acres? (Assume that you will spread the mulch at the same rate as before.) Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do. 6. If pints of jelly filled jars, then how many jars will you need for pints of jelly? Will the last jar of jelly be completely full? If not, how full will it be? (Assume that all jars are the same size.) Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do.