Chapter 9: Fraction Operations

Transcription

1 Chapter 9: Fraction Operations Getting Started, p.. a) For example, = 6 6 and = 9 b) For example, written as an improper fraction is, which is equivalent to 0 6 and 9. c) For example, = and = 6. d) For example, 9 = and 9 = 7 6. To compare the fractions, write them with a common base. This common base is 0 since 6 = = 6 0 and Since = 0, = 0 0. Finally, since 7 0 = 0, = = 0 0. Now order the fractions from least to greatest. 0 0, 6 0, ,and. This is equivalent to 0,, 6 7, and.. blue parts + yellow parts = coloured parts + =. a) Write the fractions with a common denominator of. + = 9 + = b) = = 6. a) + = = 7 0 b) = 0 0 = Therefore, of the rectangle is modelled by blue and yellow parts.. a) 6 = 6 6 c) = 9 So three groups of make 9 or. = 6 = There is half a circle more yellow than red. b) = 9 = There are of a strip more yellow than red. d) 9 0 = 0 So two groups of 9 make 0 0 or. Nelson Mathematics Solutions 9-

2 7. Write all mixed numbers as improper fractions, and all fractions with a common denominator. Then solve for. a) + = + = 6 + = 0 0 = 9 0 b) + = + = + 6 = 0 6 = 7 6, or 6 c) + =. + = 6 = or Repeated addition a) Eating Multiplication Result or, or 7 of the pizzas means that Aaron and his friends ate one whole pizza and of the second pizza. Since =, it means that of the slices of the second pizza were eaten. Therefore out of pizza slices/pizza = 6 slices, 6 = 6 slices are left. b) If of the pizzas are gone, then = 9 =, or. are left. 0. Thea poured 7 =, or glasses. She could have poured full glasses, and one glass that was full. 9. Adding and Subtracting Fractions Less Than, pp. 9. a) 6 + = = 7 6 or 6 b) 7 = 7 6 =. a) 7 = 7 = For example, For example, b) + = = is equivalent to, and is equivalent to. Since 0 is greater than, is greater than. More students are wearing T-shirts than long-sleeved shirts. To find out by how much more, subtract. = 0 = 7 is 7 greater than. 7. For example, the blue counters represent since they take up one row out of two. The yellow counters of the pizzas are left. represent since they take up one column out of five. Therefore, the diagram represents Chapter 9: Fraction Operations

3 . The first bar represents 7 represents. 7 = 0 7 = The first shaded fraction is second one. and the second bar greater than the 9. To complete the equation, write and with a common denominator of. 0 is equivalent to and 9 is equivalent to. Therefore the complete equation is + = a) + 6 = a) For example, + 6 = + 0 =, or = 9, or 7 Everyone is right because they stated equivalent fractions to. b) 7 is in simplest form since you can't simplify it any more.. a) + 7 = + 9 d) = 9 =, or = b) + 7 = + 0 =, or 6 e) = 0 0 = 7 0 = 6 b) + = = 0, or 0 c) 7 = 7 6 = d) = c) = =, or f) 6 = =. If of the container is empty, then of the container must be full. Since of the container is already filled, you can write this as = 9 = 7 7 more of the container must be filled so that is empty. =, or Nelson Mathematics Solutions 9-

4 . a) The fraction of students participating in one or more programs is everyone except the students who are not participating in anything. = = b) The fraction of students who participate in more than one program can be found by adding the fractions for programs and or more.. To see how much she added, subtract how much water is in the pail now from how much there was in the pail originally. She added 6 a) n + n = = of a pail of water. = 7, or = 7. First, evaluate all expressions. A. 0 = 0 0 = 0 B. + = + = Two-thirds of the students participate in one or more programs. + 6 = + = = = b) a b = 6 = 0 = 9 C = = 9 0 D = = Expression D is more than since is equivalent to 9. Expression C is 0 or more than since is equivalent to 0. Expression A is 0 and that s 0 more than, since 0 6 is equivalent to. Expression B is or 0 0, which is only 0 more than. Since is less than 0 0,, and, expression B is closest to. D is 0 away from.. For example, you used measuring cups, one holding cup, one holding cup and one holding cup. How much flour did you measure? 9. Use guess and test. Try: = = too high 0 Try: = = 9 It works! 0 0. To determine how many people never read the news online, subtract the fraction of people who read the news regularly and the fraction of people who read the news rarely from the entire group of surveyed people. (% can written as, which is 00 equivalent to.) + = = 7 7 = 7 Of the people surveyed, do not read the news 7 online. 9- Chapter 9: Fraction Operations

5 . of the class are girls, and of these wear braces. = are girls who wear braces. of the class are boys, and of these wear braces. = are boys who wear braces. + = of the students wear braces. 9. Adding and Subtracting Fractions Greater Than, pp.9 9. Write the fraction of the cups of sugar needed for chocolate chip cookies as. Since he needs 6. a) = + 0 = 7 = 7 7. a) + = = 0 = 6 0 b) = = = 0 batches, you need = 0 cups of sugar. For the cherry cookies, you need = cups of sugar. To see the difference in the amounts of sugar, subtract these amounts. 0 = 0 6 Therefore, Reilly will 6 need more cups of = 6, or 6 sugar to make chocolate 6 chip cookies.. a) + = + = b) = = 0 = b) = 0 0 = 0 0 = 7 0. a) + = 6 = b) + = + 9 = c) 9 + = = 6 or 7 6 d) + = 0 + = 6 = 7 7 e) + 6 = 9 + = 7 f) = = = 9 0 = 0 0 Nelson Mathematics Solutions 9-

6 9. a) = 7 = b) 7 = 7 = 0. The time Jasleen spent doing homework and talking on the phone is: Subtract this amount from the hours she had after dinner, to determine how much time she had before bed. c) = = 7 0 d) = 7 = = 0 or 0 + = + = = 6, or 6 6 = 6 6 = 6 Jasleen had h before bed. 6. walls left = total walls walls painted. First evaluate each expression. a) = 6 6 = 6 6 b) 6 9 = 6 9 = = 6 Since is greater than, the difference in b) is 6 greater.. a) b) + 6 = = 6 = 9 = Anita spent 9 hours on = the piano and at soccer practice. Anita spent more hours at soccer practice.. For example, I had cups of flour in a bowl, but I only needed cups. How much extra flour did I have? 6. pages pages = = = = Jeff has walls left to paint.. a) She is correct because the first distance is, then she went another farther and she 6 landed on 6, which is. b) She is correct because the distance from = The non-advertising parts of the newspaper would fill pages if Aviv put them together. 7. The whole number part could be 7 or. If the sum of the two fraction parts is greater than, then the total will be greater than. If the sum of the fraction parts is less than, the total will between 7 and.. The second fraction must be half of, or. The first fraction could be any fraction. For example, the fractions could be and. to is units. 9-6 Chapter 9: Fraction Operations

7 9. The closest whole number to is. Lee subtracted from 7, giving. Then, she had to add the difference between and, which is. 0. Since each of the cars hold 0 people, and they are 90% full, 90 can be written as, to represent 00 0 people in the car. Similarly, in the other cars, 6 can be written as. If you rearrange the 00 0 people, you can fill some empty seats = = 9 0 = cars would be filled completely, and there would be one more car with 9 people in it.. a) There are small bags, and each large bag holds times as much as a small bag. To calculate how many small bags there are, multiply the number of large bags by and add to it the number of small bags left over. + = 0 + = = Therefore, the leftover popcorn would fill large bags.. You want to make the first whole number as great as possible, and the second whole number as small as possible. So put 9 in the first blank and in the third blank. For the proper fractions part of the numbers, you want the first and last ones to be as great as possible and the middle one as small as possible. Since the denominator of the first fraction is, and it must be proper, the greatest the numerator can be is. For the second fraction, the denominator is 6, but we want it to be as small as possible. You have already used digits and, so put the next smallest digit,, in the place. For the final fraction, you have only two digits left, and. The fraction must be proper, so put in the numerator and in the denominator. Evaluate the expression = = + 7 = The greatest value is. 9. Fractions of Fractions, pp a) b) 7 9. First evaluate each one. a) of 9 0 is 6 0 = 7 or The leftover popcorn would fill small bags. b) Since large bags hold times as much, multiply the b) of 9 is 9 answer in a) by. = 7 = 7 = c) of 6 7 is 7 Compare the fraction strips to see that the order is b), c), a). Nelson Mathematics Solutions 9-7

8 6. There are two strips of coloured, so we are e) of 9 is 9 = looking at. Of that, a quarter of it has coloured fraction strips beneath it. So the picture models of, which is = a) of the coloured strips are coloured in the second strip, so the picture represents of. These two coloured strips are, so the relationship is of, which is =. b) One fourth of the coloured larger strip is shaded, and the larger strips represent. So the relationship is of, which is 0. c) Of the total circle, is shaded, and of that amount f) of is 0 = 9. a) of each of the three fourths is coloured and this is out of equal pieces. b) of is c) of and of both equal. 0. For example, a) of is another is shaded. So the relationship is of is 9.. a) of 6 is b) of is b) of is c) 6 of is. The is divided into sections and of them are coloured, so that is of. But when you look at the whole picture, this is sections out of 6, which is 6, which is equivalent to. d) 6 of is. a) is of 9- Chapter 9: Fraction Operations

9 b) Draw the diagram using the fact that =. is 9 of 6. a) For example, piece out of could be half of pieces out of. So, is of. b) For example, pieces out of could be a third of pieces out of, since pieces out of is the same c) 0 is of. a) For example, is twice as much as and so as piece out of. So, is of. c) For example, pieces out of could be half of pieces out of 6. So, is 6 of. d) For example, pieces out of 7 could be pieces out of from pieces out of 7. So, 7 is of 7. it is like taking two groups of of, which would be just of. b) For example, is twice as much as. So it is like taking the same portion out of only half. If you take that portion twice, it would just be of. c) For example, of is twice as much as of, so of would be half of half as much, which is a quarter of of. Therefore, of is four times as much as of.. For example, a fraction that has a 6 in the numerator can be evenly split into sixths easily. Thus, 6 of 6 7 has to be 7. 6 of is more difficult to 7 determine since cannot be evenly split into sixths 7 easily.. To figure out of, divide each section into three equal pieces and shade one of each of those pieces. This is. 7. For example, 0 of the students wear glasses. of these are far-sighted. What fraction are farsighted?. Half of half of the circle is one quarter of the circle. Half of this is of the circle. 9. a) 9 is of b) 7 is of 7 c) is of, since = d) 6 is of 0. For example, if each separated box represents whole then: Therefore, of the students will be permitted to participate in the first day of the program. of is Nelson Mathematics Solutions 9-9

10 . a) 60% expressed as a fraction is 6. We need to 0 determine of 6. Divide the 6 pieces in four parts 0 by making each part sections of 0, and shade of these parts for a total of 9 shaded sections out of 0. The group of girls who use instant messaging 9 represents of the class. 0 b) Since the fraction in simplest form is 9 0, there would have to be a multiple of 0 students in the class, and 0 is too many. So, there are likely 0 students in the class.. For example, a diagram representing of is simpler, quicker, and more accurate thatn multiplying decimals. of is 9. Multiplying Fractions, pp.0-0. a) 9 b) 7 of all Canadians are downhill skiers are from British Columbia. 7. a) b) c) 7. a) = For example, 6 b) = c) 6 = 0 or For example, d) 6 = 6 or For example, For example,. = 6 0 = 0 For example, 9. a) = 60 = 7 b) 9 = 9 = = 0 = c) = = 60 = 0 0. = = 6 d) 7 = 7 = 6 0 = = Therefore, the bed takes up of the floor space. 9-0 Chapter 9: Fraction Operations

11 . a) = 0 = Jessica is awake at home b) hours/day of the day. day = 0 hours = 0 hours Jessica is awake for 0 h at home.. a) i) a = = = 9 ii) a = = = 9 = 7. a) = = b) For example, to continue the pattern should be half of iii) a = = = =. Since =, it should be =. = 6 6 =. a) For example, the rectangle has total dimensions of by, and the area is 0 units. The part of the rectangle that is shaded is units by units, for an area of 6 units out of the 0 total. So = 6 0. b) Examine the factors of 6 and the factors of 0, and make fractions with factor pairs. For example, 6 0 or 0 or all have a product of a c, since you multiply numerators to see how b d many pieces are shaded and denominators to know the section size for each piece. = 7 = 6 b) Multiply by higher powers of multiplying by more times. Since multiplying by corresponds to is less than, results in a smaller product. So, multiplying by each time results in lesser and lesser products. 6. The product is less than each fraction because you're only taking a part of either fraction. 7. a) = 0. b) = 0 is the same as 0., so 0 00 = if you multiply the fractions 0 0 or the decimals you get the = same answer, only in a different form. 00. a) 0 = 9 0 About 9 Americans eat vanilla 0 ice cream once a month. b) = About 67. million Americans eat vanilla ice cream each month. Nelson Mathematics Solutions 9-

12 9. a) There are six parts in total, and only one is red and has the letter A on it. Therefore, the probability is 6. b) Since you have a probability of 6 = for landing on red and a probability of for landing on letter A after you have landed on red, the probability of landing on the red A is. 0. You want the product to be in simplest form. 9 That would happen if the numerator in the other fraction were a = 7. This would give the other 7 fraction as 7 + = 7 9, and =, as desired. 9 So, a = 7.. Follow the pattern: = = = The numerator of the product is and the denominator is the same as the last fraction multiplied. So, the product will be 00. Mid-Chapter Review, pp a) + = = 6 b) 6 = 6 6 = 6 =. a) + 6 = + 6 = = 9, or 7 b) 6 = 6 = 9 =. a) Less than. Since is less than which is less than and + =, the sum + must be less than. b) Greater than. Since is greater than, which is equivalent to, the sum + must be greater than. c) Greater than. Since is greater than 0 which is equivalent to, the sum + must be greater 0 than. d) Greater than. Since and are both greater than, the sum + must be greater than. e) Less than. Since = and is less than 0 6, the sum 6 + must be less than. 0 f) Greater than. Since + = and <, the sum + must be greater than.. a) + = + d) = = + = 0 0 = 7 = Chapter 9: Fraction Operations

13 b) 7 0 = = = 0 c) = = = 9 e) = = = 6 f) 7 0 = = = a) Since are shaded in the first part and in the second, the fraction addition is +. The sum is + = + 9 = 7 or b) Since the first arrow is and the second arrow 0 is 0, the fraction addition is 0 +. The sum is = 0 6 a) + 7 = = or b) + = = 6 or c) = = d) 9 0 = 0 0 = 0 e) = 0 + = or 7 f) 6 = 6 = 9 = 7. a) For example, use a common denominator of. The two numerators will add to. So one pair could be +. Using a common denominator of 0, we need the sum to be 6 0, since it is equivalent to. Two possible pairs are and b) For example, use a common denominator of. The difference of the numerators will be. So three possible pairs are,, and 6.. For example, the denominator is, so choose fractions that have denominators that are multiples of. For the first set, choose as the first fraction. The total of the three fractions has to be 7, so the remaining two fractions need to add to 7 = =. If you choose the second fraction to be, then the third fraction must be =. So the first set of three fractions is + +. In a similar manner you can choose the other three possible sets. For example, , , and The fraction of his project that Stephen has completed is + = + = 7 So, he has 7 = of his project left to complete. 0. For example, In the first hour of the book sale, of the books were sold. What fraction was left? Answer: = =. Add the two numbers to find the total. + = + =, or In total, the training is days. Nelson Mathematics Solutions 9-

14 . a) + = 9 and + is approximately. So So, you are adding + to find the difference + is about 0. between and. b) + is approximately, so + c) + = and about. 0 + is about. is about, so 0 + is d) Since is about 0, is about 0. e) Since 0 is about 0, is about. 0 f) 7 6 is less than, or about.. a) 6 + = 6 + = 9 b) = + 9 = = 7 9 d) 6 = 6 6 e) = 6 = 7 = 7 = = c) f) + = = 0 9 = =. This happens if the fractions add to exactly. For example, + = or 6.. The model shows the difference by counting up 6. a) 6 = 6 6 = 6 = 6 b) The answer will be less than when the 000 denominator is greater than 000. The pattern is that in each line the denominator of the answer is doubled. So, in the next line, the seventh line, the denominator will be 6 =. In the eighth line, the denominator will be = 6. In the ninth line, the denominator will be 6 =. In the tenth line, the denominator will be = 0. This is more than 000, so the answer of 0 is less than. It will be the difference For example, use fraction strips to model. Shade of the sections of the eighths strip. Keep of these sections to show that of is, which is equivalent to.. of is =. from to, so first add another to get to. to get to and then of 0 is 6 0 = 0 =. So, of is the same as of Chapter 9: Fraction Operations

15 9. a) of 7 is b) is of. Yes, the denominator could be any multiple of 0 and then you could take a common factor out of the numerator and denominator to simplify it to 0. For example, = 0 which is equivalent to 0.. No. is greater than 7 and when you multiply c) of is by a fraction less than, the answer has to be less than either of the multiplicands.. a) 7 = 7 = 0. a) 6 = b) 7 = The first product is = greater. b) 9 = 7. a) 6 = 6 0 b) 6 7 = c) = 0. a) 7 = 7 = 6 b) = = 6 d) 9 6 = 9 6 = = 7 e) 0 = 0 = 0 = 6 = 6 = = 6 = = The first product is c) = 6 = greater than the second. = 0 c) 6 = 6 = = f) = = 00 Subtract to determine how much greater the first product is than the second: 0 = 0 0 = 7 0 The first product is 7 greater than the second. 0 Nelson Mathematics Solutions 9-

16 6. For example, Jane's room is as long as Andy's and only as wide. What fraction of the area of Andy's room is Jane's room? 9.6 Multiplying Fractions Greater Than, pp. 0-. a) 6 is about 6, and is about or. 6=. So an estimate of 6 b) 7 9 is about 7 and 6 estimate of a) = 9 = 7 or is about. is about 7. Therefore, an is about 7 7 = 9. The shaded area is 7 = or. b) 7 = 7 = = 6 7 = 7 6. a) 9 = 9 7 = 6 = 7 b) = = 96 or 7. To calculate the number of dozen cookies with icing, multiply the total number of dozens of cookies by the fraction of them that are iced. 7 = 7 7 = = Therefore, dozen cookies have icing.. a) = 9 = b) = = 6 c) = = = or of is. d) 6 = 7 6 = of 7 is 7 = 7 or 9-6 Chapter 9: Fraction Operations

17 e) 6 7 = 6 7 = 60 c) 6 = 9 6 = 09 0 or 69 0 = 0 7 or 7 f) 9 6 = = = a) = 6 = Each by rectangle is one whole. Four thirds rows are shaded, and three halves columns. There are a total of two whole by figures shaded, if you rearrange the partially shaded pieces. b) 7 = 7 = = 7 or 7 Each column represents whole. There are whole Area = = = = a) = 9 0 = 90 b) = =,or 7 columns, and 7 of the last column are shaded. One = or fourth of this is whole column and 7 column, which is 7. of the last c) = 9 = 7 =, or d) = 6 9 = 0 = 6, or 7 Nelson Mathematics Solutions 9-7

18 e) 6 = 9 6 = = 76 9, or 9 b) For example, you could use a model of. f) 6 = 7 6 =, or. a) = =, or You would need cups of oatmeal for batches of muesli. b) Tai has only calculated the areas of the shaded rectangles. He needs to find the areas of all rectangles and for the correct answer.. Area = length width = = 0 = 0 = 6, or 6 = = 0 9 or 9 Andrea s bedroom is 9 the area of Kit s bedroom. You would need 6 cups of oatmeal for batches of muesli.. To determine how much money Zoë has compared to her brother, subtract how much she spent from what she had ( times as much money). = 0 0 = 0 0 = 0 = 6 or Zoë now has times as much money as her brother.. a) Estimating would not help him realize his mistake because most likely his estimate would be =. Therefore. his answer would not be far off of his estimate.. To compare the height of Mount Columbia and the highest point on Prince Edward Island, multiply together how many times as high as Mount Columbia is to Mount Carleton and Mount Carleton is to the highest point in Prince Edward Island. = = 9 0 = Therefore Mount Columbia is 6 9 times as high as 0 the highest point in Prince Edward Island. 6. a) 0 0 = 0 0 = 7 00 = 9 or Chapter 9: Fraction Operations

19 b) =.0 as a decimal, and 0 0 =.0.. = 7. c) The answers were the same; both times you had to multiply by and adjust it to make it hundredths instead of ones.. 7 = 7. For example, Mark has times as many You can fit of the blocks into the shaded marbles as I have and Kyle has times as many as Mark has. How many times as many marbles does Kyle have as I have?. Improper means that the numerator is greater than the denominator in each fraction. The fraction must be in lower terms, so the product has a denominator that is a multiple of and a numerator that is a multiple of. For example: choose the denominators to be,, and, which have a product of 60. We need an equivalent fraction for with a denominator of 60, so the numerator must be 60 = 0 = 0. Three numbers that have this product are,, and 7. Since the fractions must be improper, pair the numerator with the denominator, the numerator with the denominator, and the numerator 7 with the denominator. This gives that a =, b =, and c = For example, rename as an improper fraction, 6. For a whole number result, the other fraction to multiply this by could have in the numerator. Then, the other fraction would need a number that divides 6 evenly as its denominator. Two numbers that do this are and. So two possible fractions are and, resulting in the mixed numbers of and. 9.7 Dividing Fractions I, pp. -. There are shaded yellow, and 9 shaded pink. They are trying to figure out how many times 9 goes region on the top row. 6. a) You can solve the problem with common denominators. A common denominator for and is. = 6 = 6 b) As an improper fraction, =. Now solve the problem using common denominators. A common denominator for and = = 6 6 =,or is cups cup measure = 0 Use a common denominator of = 6, to write this as: 0 = = 0 = 6 Craig needs to fill the measuring cup 6 times.. a) There are out of pieces coloured in the first strip, so is being divided by the blue coloured section of the bottom strip, representing out of pieces. So, the division is. into. So the division being represented is 9. Nelson Mathematics Solutions 9-9

20 b) There are two full strips coloured yellow, so is being divided by a fraction. That fraction is represented by out of pieces in the bottom strip being coloured blue. So, the division is. 9. a) For equivalent fractions, use a common denominator of. = b) = = 6 = 7 6 = 0 = 0,or 0 c) = = 0 = 0,or6 d) 6 = 0 0 = 0. =. Use equivalent fractions with a denominator of 0 and divide the numerators. = = 6. Use equivalent fractions to determine the quotient. a) 6 = 0 = 0 b) 7 0 = = 0 7,or 7 c) 6 = = = d) = 6 = 6 =. cups cup measure = 9. Use a common denominator of =, to write this as: = 9 =,or It will take Fredreka current rate.. = 9 = h to finish her report at her = 7 = 7,or7 Craig needs to fill the measuring cup 7 times. = 7,or So, Alana checks the turkey or times. 9-0 Chapter 9: Fraction Operations

21 . Yes, the order you divide fractions does matter. For example, if the fractions are different lengths, the shorter one will fit into the longer one more than once, but the longer one will fit into the shorter one less than once. = 0 = 0 = 0 = 0 6. For example, if the sections are the same size, it s just finding out how many groups of sections fit into 6 sections and it s always. The one exception is the denominator must not be zero. 7. For example, there are 6 sixths in. So if you are trying to figure out how many sixths fit into piece, you will get six. Finding out how many sixths fit inside another number is the same as counting how many units of 6 pieces can fit inside. One is division by and the other is multiplication by 6, so they are 6 the same thing.. a) = = 6 6 =,or7 Therefore, there are about 7 sections in the movie. b) Since you got a fraction when you divided the length of the movie by the length of each section, you know that each section is not exactly a of an hour. If each section was hour long, the quotient would be a whole number. 9. a) Since is half the size of, there will be twice as many sections fitting into. b) Since 6 is twice as long as, will fit into it twice as many times. 0. If a store discounts an item by 0% 0 00 =, then the discounted price is of the original price. If another store discounts an item by, the discounted price is of the original price. The first store is selling the item for a higher price than the second store. To determine what fraction the lower sale price is of the higher one, divide the lower price by the higher. = 0 = 0 = 6 Therefore the lower sales price is of the higher 6 sales price.. The question is asking what fraction divided by another fraction gives. Since multiplication is the opposite of division, if you choose one fraction you can find its pair by multiplying. For example, choose. To determine what fraction to divide by to get, multiply the two values. = = To determine another pair, multiply by. = = 0 = Nelson Mathematics Solutions 9-

22 Therefore, two possible pairs are and. 9. Dividing Fractions II, p Evaluate each expression. a) = = 0, or. a) = = 6 b) 7 = 7 =,or c) = = = 6. To determine how many small cans Lynnsie will be able to fill, divide by. = b) = = =,or = d) = = 6,or = Lynnsie could fill large cans of paint. small cans of paint with her Therefore, b) and d) have quotients of. 0. a) Evaluate each expression. i) ii) 9 6 iii) 7 7. a) 9 9 = 9 9 = 7 d) = = 0 = = 9 0 = 9 6 = 0 = 7 = 7,or 7 =,or = 6,or = 7,or b) = =,or e) = = 0, or c) 7 = f ) = 9 0 = 6 or = 7 60 or. To divide, multiply by the reciprocal. So, you would be multiplying 7 by, which is more than, and you would get a product greater than 7. Therefore, ii) and iii) have quotients greater than. b) If the first fraction is greater than the second, then the answer is greater than.. a) 9 = 9 = 0 9 or 9 b) = 0 = 0 = 0 = 6 or 6 9- Chapter 9: Fraction Operations

23 c) The quotient is not greater than. c) 0 = 0 d) 7 = 7 = or. a) Yes. Reciprocal method: 6 = 6 = 60 = b) Yes. For example, a b c d = a b d c = a c d b Timo s method: 6 = 6 = = 0 = 0 = pages/min. To determine how many pitchers Miri could fill, divide the number of pitchers she has filled by the fraction of punch she has used. = = = = pitchers = (a c) d b = (a c) b d = (a c) (b d) = a c b d Or, if a b =, then a =, so a = and 6 b = 6, so b = 6. If didn t divide evenly into and evenly into 6, you would have to use an equivalent fraction before you divide.. a) Miri would have filled punch.. a) 0 = 0 9 b) = 0 9 = 60 9 = 0,or6 laps pitchers with all of the 0 = 0 0 = 0 9 = 0 = 0 9 = 0 = pages/min c) = 0 9,or 9 laps b) = 9 0 = 0 = 9 = 0 = 60 = pages/min = 0 9 = 0, or laps Nelson Mathematics Solutions 9-

24 6. a) For example, a small glass of juice holds as much as a large glass. How many small glasses can you fill by pouring in juice from b) For example, it takes pitcher. If you have room to fill fraction of the large glasses? glasses of juice to fill a pitchers, what glasses of juice can you use? 7. Since half of is, and the quotient must be less than. is less than this,. Express as an improper fraction,. We get a whole number as an answer after dividing by a fraction less than. Since multiplication is the opposite of division, you can determine what fractions to divide dividing For example, Use the whole number. by to get whole numbers by by whole numbers. Use the whole number 7. 7 = 7 Use the whole number. = = = = = Three possible fractions are,, and. 9. a) = = 9 6 b) 6 6 = = = 0. The product of one fraction and the reciprocal of the other must equal the sum of the two fractions. This will be easiest to do if the fractions have a common denominator. Using guess and check, it can be seen that if the denominator is, then + = and = =. Using a similar method, it can be found that 6, and, are pairs of fractions that satisfy both of the required properties. 9.9 Communicating about Multiplication and Division, p... is 0. This is the same as. So, I need. and another of.. To calculate can take of each hundredths grid. of., you of the first hundredths grid is 0 hundredths. Similarly, of the second and third grids are also 0 hundredths each. of the last grid is = hundredths. This gives a total of = 7 hundredths. Express 7 hundredths as a decimal to finish the calculation, that is =.6.. For example, this is true because in multiplication, order does not matter... = 00 0 You get the answer by multiplying for the numerator and using 000 for the denominator.. 0. = 0 00 You still have the same numerator of and the same denominator of In the first diagram, rows. represents the first two of each row is shaded. This gives 6 shaded boxes out of. In the second diagram, represents the first rows. of each row is shaded. This also gives 6 shaded boxes out of. 9- Chapter 9: Fraction Operations

25 6. a) is the same as 6 You can show the number of sets of thirds in 6 thirds is. b) For example, means that you covered of a wall with cans of paint and you want to know how much paint is needed for full wall. You know that if cans covers, then cans covers of 0 = 0 0 = = + 0 = 0 So 6 blue circles would be worth 0 0. and another a wall. But a full wall is three thirds, so the amount of paint is. Since is a half of, multiplying by means multiplying by and then take half. 7. Another name for is always n. It doesn't matter n what value you use for n, as long as it is not zero. When you multiply a b n, you end up multiplying n the numerator by n and the denominator by n. Multiplying by does not change anything.. a) Since 60% = = and. =, you can calculate to get 60% of.. b) For example, it's easier to multiply fractions since the numbers and smaller and they are all integers. 9. You know that 6 is and 6 is. Since these numbers are less than or equal to the given numbers, the product of and 6 must be greater than =, so 6 of something is of that thing and another fifth of it. So if 0 were split among blue circles, there would be in each circle 0... means sets of.. That s. +.6 =. and that s only one decimal place. If you multiply it as 0, that would be or Therefore, the answer has only one decimal place.. means how many sets of are in. means how many sets of are in. Since is exactly twice as great as, it will fit in exactly half as many times.. No, since is less than, will be less than. Since 0. is greater than 0.0, will be greater than. So will be much greater than, not of it. In fact, of is Order of Operations, pp. -6. a) + = + = + = + 6 = 9 b) + 6 = + 6 = = 6 Nelson Mathematics Solutions 9-

26 . For example, + = + = + = = 79 or = + = 6 + = = 79, or = + = + = = 6, or. a) For example, 9 = 9 = = 6 = 9 9 = 9 = 9 = 6 0 = 6 9 = 9 = 9 = 7 0 = 6, or 7 Tamara could calculate 9,6 6, and without using brackets. b) Yes, for example, 9 = = 0 9 = 7 0, or a) + 6 = + 6 = = + 6 = = 97, or b) + 6 = = = = = 9 0 c) + 6 = + 6 = = = 7 0, or Chapter 9: Fraction Operations

27 d) = = 6 + = 7 + = = e) + = + f) + 6 = = + = + = 6, or 6 7. First, evaluate all expressions. A = = 0 + 9, or 0 0 B = = = = 0 + 9, or 0 0 C = 7 + = 7 = 7, or or 9 D = + 7 = = 7 6, or 7 9 Expressions A and B have the same value.. a) = = = = 6 7 = 0, or 9 b) 0 + = 0 + = + = 7 + = 79,or 6 7 c) + = + = + = + = 0 =,or d) 7 + = 7 + = + = = 0 0 Nelson Mathematics Solutions 9-7

28 e) = = = = 9, or 7 9 f) = = = 0 9 = 9 7 = 6 6 = 6 6, or 6 9. The missing digit is, as the calculation shows. + = + 9 = + = + = = = 6 = = 0. By trial and error, = = = + = + = 7, or. Using 6 as a common denominator, you can let a = 6,b = 6,c = 6, then a b + c is, = = 9, or 6 6 Another possible combination is to let a = 9, b = 9, and c = 6 7. Then a b + c is, = = or 9 9. Evaluate each expression. A. + = + = + = B. 7 0 = 0 7 = 0 = 0 9- Chapter 9: Fraction Operations

29 C. 7 D. + = 7 = 6 + = 6 = + = 60 = =, or 7 Expressions A, B, and D have values less than.. a) Using guess and check: For example, a = and b = a + b + a b = + = + = + = + = 7 7 < = = 6 =. For example, +. + = + + = + = + = 7 =, or = 7. a) By guessing and checking, + = = = = = 9 0 Therefore, m =. b) By guessing and checking, 6 = = b) For example, c = c = = = + c = + = > c) For example, d = 6 d = 7 6 = 7 d = 6 = 7 > = = Therefore, n =. Chapter Self-Test, p. 7. a) + = + =,or b) + 9 = = 7 9 c) + = 9 + = d) 7 0 = = 0 e) = =, or 6 f) = = Nelson Mathematics Solutions 9-9

30 . a) 0 = 0 0 = 9 or 0 60 c) 6 = = 60 b) + 6 = 6 + d) = = 0 or = From least to greatest is c), a), b), d).. For example,. a) + = 6 = b) + = 6 + = 9 c) 6 0 = = 0 d) = 6 6 = 6 6 = 6 = 6. After dinner on Monday, = or 7 7 of the pizzas are left. 7 = 9 9 = 0 or 6 of the pizzas are left after Tuesday a) of =, or 6 b) of 6 9 = 6, or c) 7. For example, multiplying by 6 means taking 6 of something; that's only part of it, not all of it, so the answer is less than the number you start with.. a) = 0 = c) 7 6 = = b) 6 6 = 0 6 = 9. a) For example, 0. For example, If you have d) 7 = 6 = of a can of paint and only need to use of it for a project, what fraction of the can will you use?. For example, = + OR = + 9 = =,or. a) = =,or 0 0 b) = = 9,or c) = 9 = =,or = 7 7 = 9 0, or 9 0 d) 9 = 9 6 = 7, or 7 of 7 = Chapter 9: Fraction Operations

31 . a) For example, to divide by, ask how many are in? Use fraction strips. There is whole piece and another of that piece, so is. b) = = 0 = 6,or. a) =, or c) =, or b) = =, or. For example,. = 0 multiply by 0 0 so the last decimal can appear there. 6. a) 6 = = = 7 = 9 6 d) = 7 = 0, or and. =. When you 0, the denominator is hundredths, b) 6 = 6 c) = 6 = 6 = = = = 7 = 9 6 or Chapter Review, pp. 0-. a) + 7 = + 0 = b) 7 = 0 =. a) + = = 9 or b) = 0 9 = Nelson Mathematics Solutions 9-

32 c) 9 = = 6 or Therefore, the sum is greater than the difference.. a) + = + d) 0 = 0 0 = = 0 b) + 6 = e) 9 7 = 6 6 = 9 or 7 = 7 6 c) + 7 = + f) 7 = 6 = 9 =. How much Jake mowed each day will be subtracted from, the whole lawn. + = = 0 = 6 0 c) = 7 d) = = = There was of the lawn left to mow after yesterday. = 0 0 = 0 of the lawn is left to mow. 0. a) 7 0 = = 0 6. For example: For a = and b = 9, = = 7, or 7 For a = 7 and b =, = = 6, or 6 + = b) 7. a) + 6 = = 0 b) = 6 6 = 6 c) = 7 = 6 d) 6 6 = 6 0 = 0 = 9- Chapter 9: Fraction Operations

33 . To find the total measure of the dry ingredients, add the measures of flour and sugar. + = + 9 = 7 = There are cups of dry ingredients. 9. a) = =, = = = = 0 b) The numerator is always, and the denominator always doubles. 0 doubled is 60 (term 7); 60 doubled is 0 (term ); 0 doubled is 60 (term 9); 60 doubled is 0 (term 0). The tenth term in the second row is. 0 c) The second fraction is half of the first. To get a common denominator, I have to double the first fraction ( = ). I then add the numerator of the second fraction to get + =. 0. a) of is = 0 c) of 6 is 6 = c), since of is. d) 6, since of 6 is.. a), since of is. b) 0, since 0 of 6 is. c), since of 9 is. d), since of 7 is 7... Only A is greater than. A. 6 = = > B. 6 7 = 7 < C. 9 9 = = 7 < D. = 6 = < b) of 9 is 9 = 7 =. a), since of is. b), or, since of is. = d) 6 of is 6 = =. a) 6 = 6 = = 0, or The first product is greater than the second. b) 6 = =, or 9 9 = 6, or The first product is greater than the second. Nelson Mathematics Solutions 9-

34 6. a) 9 7 = 6 b) = 9 d) = 0 = e) 7 = = 7 c) = f) 7 7 = 6 7. = 9 6, so 9 of the drama club are 6 grade girls.. = 7 = 0 = 7 or Eileen is now on the phone times as much as her sister. 0. For example, I can fit of the rectangles from the bottom row into the in the top row. 6. Since each set of numbers has a common denominator, you only need to compare the numerators. So 6 6 = and = as well.. a) 6 6 = c) 6 = 6 = 0 6 b) = = 0 = 0, or d) 9 = 9 = 7, or a) 7 = 7 7 = b) = = 6 =,or c) = = 0 9, or 9 d) 6 6 = 6 6 = 0 0 = 7,or e) = = 6,or f) 6 = 6 = 6 0 = 0,or 0 =, or. $.0 is dollars or 9. 9 = 9 = 6 = Therefore, there are quarters in $.0.. To determine the amount of sugar needed for a whole batch of cookies, divide the amount of sugar used by the fraction of a batch made. = = 9 She needs of her sugar to make the whole batch Chapter 9: Fraction Operations

35 . a) For example, 6 and : 6 + = = 6 =, or b) For example, and : + = + = < c) For example, 6 and 0 : 6 0 = 6 0 = 0 6 = or > 6 = > = 6 = 6 or = = 6 60 = 6. For example,. is or equivalently, =, so multiply 9 = 9. The product is =.. 7. A. = 0 = = 9 0 B. = 0 = 7 = 7 = = 0, or 7 0 C. = = = = 0, or A is the greatest. You can tell which fraction is the largest by comparing the numerator if they all have the same denominator = = = = 0 + = = 9 0 Cumulative Review: Chapters 7 9, pp.. Quadrant is the quadrant in the bottom right portion of the graph. The coordinates of the point in this portion of the graph are (, ). The answer is A.. The coordinates of A are (, ). When a point is reflected in the x-axis, the x-coordinate stays the same and the y-coordinate becomes the opposite. So, the coordinates of the image of A will be (, ). The answer is A.. The coordinates of A are (, ). When a point is rotated 70 clockwise about the origin, the y-coordinate becomes the opposite and then the x- and y-coordinates are switched. So, the image of A will have coordinates (, ). The answer is D.. Let n be the number of players signing up for a sport. If the number of players signing up is multiplied by 7, then the expression will be 7n. Since three players leave, the new expression is 7n. With 60 players left, the equation will be 7n = 60. the answer is C.. x + 7 = x = 7 x = 6 x = The answer is A. Nelson Mathematics Solutions 9-

36 6. t. = 6. t = t = 6.6 t = 6. The answer is B. 7. Let t be the amount of time Toma spent riding her bike. Since she spent h watching a movie and she was out for h in total, the equation for the amount of time she spent biking is t + = t = b) Let x be the mean space between the centre of one pier to the centre of the next pier. Since there are piers, there are spaces in between them, so the distance between the centre of adjacent piers can be expressed by x = 000 m x = 000 m x 6 m c) Since Joe spent a total of h crossing the bridge and it took him each time, the number of times he 6 crossed the bridge is, 6 = 6 t = 9 t = t = 6 or The answer is B. 6. Since Sheree has time as much as Rishi, multiply the two values together to find the number of bags of peanuts Sheree has. = = or 7 The answer is D = + = + 0 = + = 9 + = or The answer is B. 0. a) The two emergency shoulders and the two lanes must add up to the width of the bridge. Let x be the width of one lane on the bridge. Then x +.7 m =.00 m x +.0 m =.00 m x =.00 m.0 m x = 7.0 ` x =.7 m The width of each traffic lane is.7 m. = = Therefore, Joe makes = round trips. d) Let a represent the number of automobiles, r represent the number of recreational vehicles, t represent the number of trucks, m represent the number of motorcycles, and b represent the number of buses. The total amount of money collected is 9.9a +.r +.7m +.t b. The total amount of money collected in that one hour is: 9.9(7) +.(9) +.7() +.(6) () = = The total amount collected was $ e) The number of vehicles that were neither trucks nor automobiles can be represented as = = 0 of the vehicles were neither automobiles nor 0 trucks. f) For example, since the two reflections have to move the truck from the third quadrant to the first quadrant, one possible set of reflections would be first reflect on the x-axis and then reflect on the y-axis to move the red truck to the position of the green truck. 9-6 Chapter 9: Fraction Operations