CHAPTER 6 Fraction Operations GET READY 284 Math Link 286 6.1 Warm Up 287 6.1 Multiplying a Fraction and a Whole Number 288 6.2 Warm Up 294 6.2 Dividing a Fraction by a Whole Number 295 6.3 Warm Up 301 6.3 Multiplying Proper Fractions 302 6.4 Warm Up 309 6.4 Multiplying Improper Fractions and Mixed Numbers 310 6.5 Warm Up 319 6.5 Dividing Fractions and Mixed Numbers 320 6.6 Warm Up 328 6.6 Applying Fraction Operations 329 Chapter Review 337 Practice Test 344 Wrap It Up! 347 Key Word Builder 348 Math Games 349 Challenge in Real Life 350 Answers 352
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CHAPTER
6 Fraction Operations
GET READY 284
Math Link 286
6.1 Warm Up 287
6.1 Multiplying a Fraction and a Whole Number 288
6.2 Warm Up 294
6.2 Dividing a Fraction by a Whole Number 295
6.3 Warm Up 301
6.3 Multiplying Proper Fractions 302
6.4 Warm Up 309
6.4 Multiplying Improper Fractions and Mixed Numbers 310
Canada’s Ecozones Canada has many ecozones of different sizes. The boundaries between ecozones depend on geography, climate, animals, plants, and human activities.
a) The Pacific marine ecozone covers 110
of Canada’s coastline. The Northwest
Atlantic ecozone covers 15
of Canada’s
coastline. Use a common denominator to find the
sum of the 2 ecozones.
Common denominator for 110
and 15
:
1 110 51 1 2
10 5 2
110
+
×= +
×
= +
=
b) About 12
of the Prairies ecozone area is in
Saskatchewan, and about 13
of the area is
in Alberta. What fraction of the area is in Manitoba?
Add the areas in Alberta and Saskatchewan. Show your work.
Use your answer above to find the area that
covers Manitoba. Let 1 represent the whole part of the ecozone.
One quarter of Canada’s 20 ecozones are marine ecozones. They include parts of the oceans. The rest of Canada’s ecozones are terrestrial ecozones (land, rivers, lakes, and wetlands).
a) How many marine ecozones does Canada have?
One quarter = 4
Number of ecozones = 20
20 × =
Canada has marine ecozones. b) How many terrestrial ecozones does Canada have?
of a cake. What fraction of the cake did each student eat?
Find 12
÷ .
Label the number line to show halves.
Divide each half into equal parts to show the division.
There are parts in the whole. Each part is .
Use brackets to divide 12
into equal parts.
So, 12
÷ =
1. a) Model 12
÷ 2 using fraction strips or a number line.
12
÷ 2 =
b) What did you use to solve the question? Circle FRACTION STRIPS or NUMBER LINE. Give 1 reason your choice. _________________________________________________________________________ _________________________________________________________________________
0 1
6.2 Dividing a Fraction by a Whole Number ● MHR 299
6.3 Multiplying Proper Fractions Working Example 1: Multiply Using Paper Folding
Find 1 3 .2 5
×
Solution Fold a rectangular piece of paper into fifths along its length.
Open the paper and shade 35
of it.
Fold the paper in half across its width. Open the paper and draw slanted lines on half of it. Count the rectangles. There are equal rectangles. How many are shaded and have slanted lines?
1 32 5 10
× =
Find each product using paper folding.
a) 1 14 2
× b) 2 23 3
×
Fold paper into quarters along its length.
Shade 14
.
Fold paper in half across its width. Draw a line to show the fold. Draw slanted lines on half of it. There are equal rectangles. How many are shaded and have slanted lines?
_________________________________________________________________________ _________________________________________________________________________ b) What is the correct answer?
2. Write each improper fraction as a mixed number.
a) 113
33
3
= + + +
=
b) 176
56
5_____________6
= + +
=
6.4 Multiplying Improper Fractions and Mixed Numbers ● MHR 315
c) How much money do they have altogether? Sentence: ________________________________________________________________________
The Hudson Plains ecozone covers 126
of the area of Canada.
The Northern Arctic ecozone is about 3 910
times as big as the Hudson Plains ecozone.
What fraction of the area of Canada does the Northern Arctic ecozone cover? 126
× 3 910
Write the mixed number as an improper fraction. Multiply the fractions. Sentence: _______________________________________________________________________
Bev earns $25/h as a machine operator. When she works more than 40 h in a week, she earns time-and-a-half. How much does Bev earn for working 46 h in a week?
To earn time-and-a-half means to be paid for 1 12
h when you work for 1 h.
Solution Method 1: Calculate in Stages Bev works 40 h at her regular pay of $25/h. Amount earned at regular rate: 40 × 25 = How many hours does she work at time-and-a-half? 46 – 40 = 6 h at time-and-a-half = ? h at regular rate
6 × 1 12
= 6 ×
= 61
× 32
= 2
= 9 6 h at time-and-a-half = 9 h at regular rate Amount earned at time-and-a-half: 9 × 25 = Total earnings = amount earned at regular rate + amount earned at time-and-a-half = + 225 = Bev earns $ for working 46 h in a week.
Bev’s regular rate of pay is $25/h. For 6 h at time-and-a-half, Bev is paid 1 12
× h.
An expression of her total earnings is: 125 40 1 62
⎛ ⎞× + ×⎜ ⎟⎝ ⎠
125 40 1 62
⎛ ⎞× + ×⎜ ⎟⎝ ⎠
Brackets first.
= 25 40 6
⎛ ⎞⎜ ⎟⎜ ⎟× + ×⎜ ⎟⎜ ⎟⎝ ⎠
Write the mixed number as an improper fraction.
= 3 625 402 1
⎛ ⎞× + ×⎜ ⎟⎝ ⎠
Multiply the numbers in the bracket.
= 25 402
⎛ ⎞⎜ ⎟⎜ ⎟× +⎜ ⎟⎜ ⎟⎝ ⎠
Add.
= 25 × 49 Multiply. = 1225 Bev earns $ for working 46 h in a week.
Ron earns $15/h as a security guard. When he works more than 35 h in 1 week, he earns time-and-a-half. How much does Ron earn for working 43 h in a week? Ron worked 35 h at regular pay. Amount earned at regular pay: 35 × 15 = Hours worked at time-and-a-half: 43 – 35 =
8 × 1 12
Amount earned at time-and-a-half: × 15 = Total earnings = + =
18. Tracy earns $12/h as a cashier. When she works more than 32 h in 1 week, she earns time-and-a-half. How much does Tracy earn for working 40 h in 1 week?
Amount earned at regular pay: × = Hours worked at time-and-a-half: – =
Most of the Boreal Plains ecozone is covered by woods and forests.
Province/ Territory
Fraction of Boreal Plains in the Province/Territory
Alberta 1325
British Columbia
120
Manitoba 17100
Northwest Territories
150
The total area of the Boreal Plains ecozone is about 750 000 km2. The table shows the approximate fraction of this ecozone found in different locations. Using the information in the table, write a word problem that can be answered using division of fractions or multiplication of fractions.
Saskatchewan625
Examples: Multiplication question: How many square kilometres does the Boreal Plains ecozone cover in Saskatchewan?
Answer: 750 000 × 625
= 4 500 00025
= 180 000 km2
Division question: For this ecozone, how many times larger is the area in British Columbia than the area in the Northwest Territories?
Answer: 120
÷ 150
= 120
× 501
= 5020
= 2 12
a) Your question: _________________________________________________________________________
Rules: • Each player spins the spinner once to decide who will play first.
If there is a tie, spin again. The player with the highest number goes first.
• For each turn, spin the spinner 4 times. Write down the 4 numbers.
• Use the 4 numbers to create 2 fractions with the greatest product. Write the fractions on your multiplication sheet. • Write the answer in lowest terms. • The player with the greater answer scores a point. • If the answers are equal, each player scores a point. • The first player with 10 points wins.
Use the tally chart to keep track.
Player 1 Player 2
2. Repeat #1, but use the 4 results to create 2 fractions with the
greatest quotient. Record the fractions on your division sheet.
Player 1 Player 2
• spinner with 9 sections
numbered 1 to 9 per pair of students
• paper clip per pair of students • Fabulous Fractions
Create a Class Flag Mr. Jansen’s grade 8 class is made up of 10 boys and 14 girls. You are a student in Mr. Jansen’s class. Create a class flag that will show the students’ interests.
• Organize Mr. Jansen’s Class
Flag BLM • Organize Your Class Flag BLM • 2 different coloured pencils
1. The flag is divided into 24 equal parts to show the number of students in the class. a) Colour 10 parts in the flag with
1 colour to show the number of boys. b) Colour 14 parts in the flag with a
different colour to show the number of girls.
c) One fifth of the boys are in the band.
How many boys are in the band?
1 1051 105 1
5
______________
×
= ×
=
=
There are boys in the band. • Draw an instrument in
of the boys’ boxes.
d) One half of the boys play hockey. None of the hockey players are in the band. How many boys play hockey?
1 102
×
• Draw a hockey puck in of the boys’ boxes.
e) One half of the girls play volleyball. How many girls play volleyball?
a) Collect information about your own class. You could ask: • Are you in the band? • Is math your favourite subject? • Do you like rock music? • Do you love pizza?
Record your information in the chart.
Question # of Boys # of Girls
Example: Are you in the band? Yes = 2 Yes = 3
b) Design a flag that shows the class information. Show your calculations.
352 MHR ● Chapter 6: Fraction Operations
Answers Get Ready, pages 284–285
1. a) 13
b) 12
2. a) 1210
b) 536
Math Link, page 286
a) 310
b) 16
6.1 Warm Up, page 287
1. a) 25
b) 223
2. a) b)
3. a) 34
b) 110
4. a) 8 b) 20 c) 6 d) 15 e) 9 f) 30 6.1 Multiplying a Fraction and a Whole Number, pages 288–293
Working Example 1: Show You Know
53
Working Example 2: Show You Know
a) 43
b) 94
Working Example 3: Show You Know
4 Communicate the Ideas
1. a) 6 6 6 185 5 5 5
+ + =
b) c) Answers may vary. Example: I prefer fraction strips because I find it
easier to see the answer. Practise
2. a) 2 635 5
× = b) 5 1024 4
× =
3. a) 1 336 6
× = b) 2 635 5
× =
4. a) 1 332 2
× = b) 1 446 6
× =
5. a) 2
b) 210
Apply
6. 2 m 7. 9 people 8. a) Answers will vary. Example: There are 8 seats at a restaurant counter.
The seats are 14
full. How many people are at the counter?
b) 8 × 14
= 2
Math Link a) 5 b) 15 6.2 Warm Up, page 294
1. a) 1 667 7
× = b) 2 843 3
× = c) 8 1625 5
× = d) 34 34
× =
2. a) 128
b) 72
3. a) 3 b) 5 c) 20 d) 2 e) 3 f) 3 6.2 Dividing a Fraction by a Whole Number, pages 295–300
Working Example 1: Show You Know 14
Working Example 2: Show You Know
110
Working Example 3: Show You Know 1 142 8
÷ =
Communicate the Ideas
1. a) 14
b) Answers may vary. Example: I prefer fraction strips because they make it easier to visualize the fractions.
Practise
2. a) 18
b) 19
3. a) 310
b) 18
Apply
4. a) 14
of a coconut b) 18
of a coconut
5. 16
of a full container
Math Link 120
6.3 Warm Up, page 301
1. a) 118
b) 38
c) 2
d) 2
+ + + =
+ =
+ =
+
1 0 2
10 42 3
+ + =+
+ + + + + + =
0 1510
610
710
810
910
410
310
210
110
+ +
=
+ + =
0 1410
510
610
710
810
910
110
210
310
0 114
12
23
+ + ++ =
+ ++ + + =
Answers ● MHR 353
2. Estimates may vary. a) 1 b) 12
c) 1 d) 0
3. a) 8 b) 15 c) 24 d) 42 6.3 Multiplying Proper Fractions, pages 302–308
Working Example 1: Show You Know
a) 18
b) 49
Working Example 2: Show You Know
a) 14
b) 14
Working Example 3: Show You Know
Estimate: 0; Calculate: 215
Communicate the Ideas
1. a) Brendan should multiply the denominators. b) 625
Practise
2. a) 512
b) 58
3. Estimate: 14
; Calculate: 14
Apply
4. a) 916
b) 516
5. 18
of the pie
6. 1200
of the world’s people
7. a) 112
of his time b) 2 hours
Math Link
150
of the area of Canada
6.4 Warm Up, page 309
1. a) 122
b) 113
c) 213
d) 112
2. a) 83
b) 32
c) 74
d) 134
6.4 Multiplying Improper Fractions and Mixed Numbers, pages 310–318
Working Example 1: Show You Know
a) 2 b) 13216
Working Example 2: Show You Know
Estimate: 4; Calculate: 17320
Communicate the Ideas
1. a) Henri forgot to multiply 2 × 14
and 3 × 12
. b) 188
Practise
2. a) 233
b) 526
3. a) 194
b) 238
c) 114
d) 137
4. a) 415
b) 132
5. a) Estimate: 1; Calculate: 117
b) Estimate: 4; Calculate: 233
Apply
6. 7 12
laps
7. 132
hours
8. a) $30 b) $48 c) $96 Math Link
320
of the area of Canada
6.5 Warm Up, page 319
1. a) 335
b) 273
2. a) 247
b) 1411
3. a) 12 5,15 15
b) 20 7,8 8
c) 9 2,12 12
d) 24 27,18 18
4. a) 6 b) 5 c) 13 d) 8 6.5 Dividing Fractions and Mixed Numbers, pages 320–327
Working Example 1: Show You Know
124
Working Example 2: Show You Know
a) Estimate: 3; Calculate: 223
b) Estimate: 2; Calculate: 9110
Working Example 3: Show You Know
18 Communicate the Ideas
1. NO. Mike needs to multiply by the reciprocal of 23
.
Practise
2. 122
3. a) 23
b) 95
4. a) 1516
b) 1017
12_
1
15_
1 1 ¥ 1 ¥ 112_
1 ¥ 15_ ¥ 1
5_1
2_
12_
1
13_
2 1 ¥ 1 ¥ 112_
1 ¥ 13_ ¥ 1
3_1
2_
12_
1
13_
1 1 × 1 × 112_
1 × 13_ × 1
3_1
2_
14_
2
14_
1 2 × 1 × 114_
2 × 14_ × 1
4_1
4_
5–8
1–4
1–2
2÷ =
354 MHR ● Chapter 6: Fraction Operations
Apply
5. 8 performers 6. 6 cakes Math Link
25 cm 6.6 Warm Up, page 328
1. a) 18 b) 1 c) 16 d) 6
2. a) 110
b) 1114
3. a) 103
b) 125
6.6 Applying Fraction Operations, pages 329–336
Working Example 1: Show You Know
a) 23
b) 12
Working Example 2: Show You Know
$705 Communicate the Ideas
1. a) Step 1: Find the common denominator for 12
and 14
. Step 2: Add the
fractions in the brackets. Step 3: Multiply by 23
. b) 12
Practise
2. a) 512
b) 7
3. a) 712
b) 536
c) 7124
d) 34
Apply
4. $584
5. a) 13
of the land b) 16
of the land
Math Link
a) 200 species b) 150 species Chapter Review, pages 337–343
1. mixed number 2. proper fraction 3. improper fraction 4. a) reciprocal b) the inverse of a fraction