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
Name: _____________________________________________________
Date: ______________
Add and Subtract Fractions
To add fractions with the same denominators, add the numerators.
+
2 ← numerator
5 ← denominator
=
1
5
2
5
3
5
To subtract fractions with different denominators, use a common denominator.
a common multiple of
each denominator
1 1
– =
2 6
Write the answer in lowest terms.
3
6
–
÷2
1
6
2 1
=
6 3
2
=
6
Multiples of 2: 2, 4, 6, 8, …
Factors of 6: 2, 3, 6
÷2
1. Add or subtract. Write your answers in lowest terms.
×2
a)
1 1
+ =
6 6
b)
6
+
1
6
4 3
−
5 10
8
10
=
1
6
?
6
÷ _____
–
3
10
=
5
10
÷ _____
6
=
5
=
10
÷ _____
÷ _____
284
MHR ● Chapter 6: Fraction Operations
4 8
=
5 10
×2
Name: _____________________________________________________
Date: ______________
Add and Subtract Mixed Numbers
mixed number
1 3⎞
⎛
• includes a whole number and a proper fraction ⎜ e.g., 1 , 2 ⎟
2 5⎠
⎝
improper fraction
10 ⎞
⎛
• a fraction in which the numerator is greater than the denominator ⎜ e.g.,
⎟
8⎠
⎝
To subtract or add mixed numbers:
Use Regrouping:
Use Improper Fractions:
2
3
1
1
4 −2
4 +2
4
4
2
4
18 11
2
1
Write as improper fractions.
=4 +2
Find a commmon denominator. = −
4 4
4
4
7
⎛2 1⎞
=
Subtract the numerators.
= ( 4 + 2 ) + ⎜ + ⎟ Add.
4
⎝4 4⎠
3
3
=1
Write as a mixed number.
=6
Write as a mixed number.
4
4
2. Add or subtract. Write your answers in lowest terms.
a) Use regrouping.
b) Use improper fractions.
3 1 −1 2
5
2
11 + 2 1
2
3
Find a common denominator.
Find a common denominator.
−1
10
10
Subtract the whole numbers.
=1
+2
6
6
Write as improper fractions.
=3
⎛
⎜⎜
⎜⎜
= (3 − 1) + ⎜⎜
⎜⎜
⎜⎜
⎜⎝
−
⎞⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎟
+
6
6
Add the numerators.
=
Subtract the fractions.
=
⎛
⎜⎜
⎜⎜
+ ⎜⎜
⎜⎜
⎜⎜
⎜⎝
−
⎞⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎟
=
6
+
6
=
6
Write in lowest terms.
=
=
6
Get Ready ● MHR 285
Name: _____________________________________________________
Canada’s Ecozones
Date: ______________
An ecozone is an area
that has similar plants
and animals.
Canada has many ecozones of different sizes.
The boundaries between ecozones depend on geography, climate,
animals, plants, and human activities.
1
1
a) The Pacific marine ecozone covers
b) About of the Prairies ecozone area is in
2
10
of Canada’s coastline. The Northwest
1
Saskatchewan, and about of the area is
1
3
Atlantic ecozone covers of Canada’s
in Alberta. What fraction of the area is in
5
Manitoba?
coastline.
Use a common denominator to find the
Add the areas in Alberta and
sum of the 2 ecozones.
Saskatchewan. Show your work.
Common denominator for
1
1
and :
10
5
1 1
+
10 5
1 1× 2
= +
10 5 × 2
=
1
+
10
Use your answer above to find the area that
covers Manitoba.
Let 1 represent the whole part of the
ecozone.
1−
=
=
=
286
MHR ● Chapter 6: Fraction Operations
6
−
6
Think 1 =
6
.
6
Name: _____________________________________________________
Date: ______________
6.1 Warm Up
1. Write each fraction.
a)
b)
improper fraction:
mixed number:
2. Draw fraction strips to solve.
1 1
a)
+
3 3
b)
1 1
+
8 2
b)
3 1
−
5 2
3. Solve. Write your answers in lowest terms.
a)
1 2
+
12 3
Find a common
denominator.
4. Multiply.
a) 2 × 4 =
b) 4 × 5 =
c) 3 × 2 =
d) 5 × 3 =
e) 3 × 3 =
f) 6 × 5 =
6.1 Warm Up ● MHR 287
Name: _____________________________________________________
Date: ______________
6.1 Multiplying a Fraction and a Whole Number
Working Example 1: Multiply Using a Model
proper fraction
5⎞
⎛
• a fraction in which the denominator is greater than the numerator ⎜ e.g., ⎟
8⎠
⎝
Find 3 ×
5
using fraction strips. Write the product in lowest terms.
6
You can model with pattern
blocks instead of fraction strips.
Solution
_1 _1 _1 _1 _1 _1
6 6 6 6 6 6
5 5 5 5
3× = + +
6 6 6 6
Count the shaded parts of the strips:
5 5 5
+ + =
6 6 6
=
=
+ _1 _1 _1 _1 _1 _1
6 6 6 6 6 6
+ _1 _1 _1 _1 _1 _1
6 6 6 6 6 6
_1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
6
Write the answer in lowest terms.
_1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1 _1
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
÷3
_1
2
15
=
6
_1
2
_1
2
_1
2
_1
2
2
÷3
So, 3 ×
5 5
= .
6 2
Draw fraction strips to find the product. Write the answer in lowest terms.
2×
÷ ______
5
6
+
=
+
=
÷ ______
288
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________ Date: ______________
Working Example 2: Multiply Using a Diagram
Find 3 ×
2
. Write the product in lowest terms.
5
Solution
3×
2 2 2
= + +
5 5 5
Model the fractions using a number line.
The denominator is 5.
So, you need 5 equal parts between each whole number.
0
1
You could draw rectangles
instead of using a number line.
2
+
=
2 2 2
+ + =
5 5 5
So, 3 ×
+
5
2 6
= .
5 5
Draw a diagram to find each product.
a) 2 ×
2
3
0
2 2
+ =
3 3
b) 3 ×
1
2
3
4
+
+
3 3 3
+ + =
4 4 4
6.1 Multiplying a Fraction and a Whole Number ● MHR 289
Name: _____________________________________________________
Date: ______________
Working Example 3: Apply Multiplication With Fractions
A spider has 8 legs.
3
An ant has as many legs as a spider.
4
How many legs does an ant have?
Solution
3
3
3
of 8 means × 8 or 8 × .
4
4
4
3
An ant has of the number of legs of a spider.
4
Use repeated addition on the number line to show 8 ×
3
.
4
You could draw rectangles.
0
1
2
3
4
The answer is
So, 8 ×
3
=
4
An ant has
5
6
+
+
+
+
+
+
=
.
.
legs.
Jenelle is making a recipe that needs 6 cups of flour.
2
She wants to make of the recipe.
3
How many cups will she need to use?
Write a multiplication statement: 6 ×
Model the multiplication on the number line.
Jenelle needs
290
cups of flour.
MHR ● Chapter 6: Fraction Operations
0
1
2
3
4
+
Name: _____________________________________________________ Date: ______________
1. The diagram models 3 ×
6
.
5
a) The diagram represents
+
+
=
+
+
=
b) Use the number line to model the same equation.
0
1
2
3
4
c) Circle the model you like to use. FRACTION STRIPS or NUMBER LINE.
Give 1 reason for your answer.
_________________________________________________________________________
_________________________________________________________________________
2. Write the multiplication statement that each diagram shows.
a)
_1 _1 _1 _1 _1
5 5 5 5 5
+ _1 _1 _1 _1 _1
5 5 5 5 5
+
+
=
______________ ×
=
+ _1 _1 _1 _1 _1
5 5 5 5 5
= _1 _1 _1 _1 _1 _1 _1 _1 _1 _1
5 5 5 5 5 5 5 5 5 5
b)
=
+
+
=
______________ ×
=
6.1 Multiplying a Fraction and a Whole Number ● MHR 291
Name: _____________________________________________________
Date: ______________
3. Write the multiplication statement that each number line shows.
a)
b)
0
1
0
4. Write the multiplication statement that each model shows.
a)
1 whole
b)
=
1
2
1
6
1
3
1
2
⫽
⫽
5. Draw a diagram to find each product.
a) 4 ×
1
2
b) 3 ×
+
+
7
10
+
Of means to multiply.
6. The width of a Canadian flag is
What is the width of the flag?
4×
=
1
of its length. The length of the flag is 4 m long.
2
0
1
2
Sentence: ____________________________________________________________________
292
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________ Date: ______________
7. A minibus has room for 12 people. The bus is
×
3
full. How many people are on the minibus?
4
3
=
4
0
1
2
3
4
5
6
7
8
9
Sentence: ____________________________________________________________________
8. a) Write a word problem for
1
× 8.
4
b) Show how to solve your problem.
_________________________________
_________________________________
_________________________________
_________________________________
_________________________________
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 =
20 ×
Number of ecozones = 20
4
=
0
Canada has
marine ecozones.
b) How many terrestrial ecozones does Canada have?
Number of ecozones = 20
1
2
3
4
5
6
Use your answer
from part a).
Marine ecozones =
20 – marine ecozones = terrestrial ecozones
20 –
Canada has
=
terrestrial ecozones.
6.1 Math Link ● MHR 293
Name: _____________________________________________________
Date: ______________
6.2 Warm Up
1. Write the multiplication statement for each diagram.
a)
1
0
+
b)
+
+
=
_______ ×
=
+
c)
_______ ×
d)
0
=
_______ ×
_______ ×
=
2. Draw a diagram to solve.
a) 4 ×
3
8
b)
7×
1
2
3. Divide.
294
=
a) 15 ÷ 5 =
b) 30 ÷ 6 =
c) 40 ÷ 2 =
d) 6 ÷ 3 =
e) 12 ÷ 4 =
f) 21 ÷ 7 =
MHR ● Chapter 6: Fraction Operations
1
2
3
= _______
Name: _____________________________________________________ Date: ______________
6.2 Dividing a Fraction by a Whole Number
Working Example 1: Divide Using a Model
Find
1
÷ 3.
4
Solution
Use a fraction strip to show
The fraction strip shows that
Each of the 3 equal parts of
1
:
4
_1
4
1
3
is equal to .
4
12
1
is
4
:
12
_1
4
_1
4
_1
4
1 1 1 1 1 1 1 1 1 1 1 1
— ——— ————————
12 12 12 12 12 12 12 12 12 12 12 12
1
into
4
3 equal parts.
Divide each
1 1 1 1 1 1 1 1 1 1 1 1
— ——— ————————
12 12 12 12 12 12 12 12 12 12 12 12
1
÷3=
4
Use fraction strips to find the answer.
3
÷3
4
3
.
4
Shade the fraction strip to show
Divide each
3
=
4
1
_
4
1
_
4
1
_
4
1
into 3 equal parts.
4
12
Each of the 3 equal parts of
3
÷3=
4
1
_
4
12
or
4
3
is
4
12
or
4
.
.
6.2 Dividing a Fraction by a Whole Number ● MHR 295
Name: _____________________________________________________
Date: ______________
Working Example 2: Divide Using Diagrams
2
÷ 4. Write your answer in lowest terms.
3
Solution
Find
Draw and label a number line that shows thirds.
0
_1
3
_2
3
1
0
_1
3
_2
3
1
_1
3
_2
3
1
To model division by 4, divide each third
into
equal parts.
There are 12 parts in the whole, so each part is
Use brackets to divide
2
into 4 equal parts.
3
Each of the 4 parts is
or
So,
2
÷4=
3
6
12
12
.
0
1
.
6
.
Use a number line to find each quotient. Write your answers in lowest terms.
1
÷5
0
1
2
1
ˆ Label the number line to show .
2
ˆ Divide each half into 5 equal parts to show division by 5.
There are
ˆ Use brackets to divide
So,
296
1
÷5=
2
parts in the whole, so each part is
1
into 5 equal parts. Each of the 5 parts is
2
.
MHR ● Chapter 6: Fraction Operations
.
.
Name: _____________________________________________________ Date: ______________
Working Example 3: Apply Division With Fractions
3
of a jar of sauce on 6 servings of pasta.
4
He used the same amount of sauce on each serving.
What fraction of the jar of sauce did he use on each serving?
Mustafa used
Solution
3
÷ 6.
4
Label the number line to show quarters.
Find
0
To show division by 6, divide each quarter into 6 parts.
There are
3
into 6 equal parts.
4
Each of the 6 parts is
3
÷6=
4
24
24
_1
4
0
parts in the whole, so each part is
Use brackets to divide
So,
1
0
_1
2
_3
4
1
_1
2
_3
4
1
.
24
_1
4
.
.
Write in lowest terms.
÷3
3
=
24
8
÷3
Mustafa used
8
of a jar of sauce on each serving.
6.2 Dividing a Fraction by a Whole Number ● MHR 297
Name: _____________________________________________________
Four students equally shared
Find
1
÷
2
Date: ______________
1
of a cake. What fraction of the cake did each student eat?
2
.
0
1
ˆ Label the number line to show halves.
ˆ Divide each half into
There are
parts in the whole. Each part is
ˆ Use brackets to divide
So,
1
÷
2
1. a) Model
equal parts to show the division.
1
into
2
.
equal parts.
=
1
÷ 2 using fraction strips or a number line.
2
1
÷ 2=
2
b) What did you use to solve the question? Circle FRACTION STRIPS or NUMBER LINE.
Give 1 reason your choice.
_________________________________________________________________________
_________________________________________________________________________
298
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________ Date: ______________
2. Find the quotient using fraction strips.
a)
1
÷ 2
4
b)
1
÷ 3
3
ˆ Divide the strip into quarters.
1
ˆ Shade .
4
ˆ Divide each quarter into 2 equal parts.
There are
Each part is
parts in the whole.
8
. So,
1
÷2=
4
1
÷3=
3
8
3. Find each quotient using a number line.
a)
3
÷2
5
b)
0
1
1
÷4
2
0
1
ˆ Label the number line to show fifths.
ˆ Divide each fifth into 2 equal parts.
There are
the whole, so each part is
So,
3
÷2=
5
.
3
into 2 equal parts.
5
ˆ Use brackets to divide
Each part is
parts in
1
.
So,
1
÷4=
2
.
6.2 Dividing a Fraction by a Whole Number ● MHR 299
Name: _____________________________________________________
Date: ______________
4. Two different fish curries, dhopa and molee curry, are made with coconut.
1
of a coconut to make
2
2 servings of dhopa.
What fraction of a coconut is in each
serving?
a) You need
b) You need
1
coconut to make 4 servings of
2
molee.
What fraction of a coconut is in each
serving?
1
÷2=?
2
Draw a fraction strip to solve.
÷
=?
Draw a number line to solve.
1
÷ _____________ =
2
1
÷2=
2
2
full.
3
Four students share the juice equally.
What fraction of the full container does each student get?
5. A container of orange juice is
÷
=
Sentence: ____________________________________________________________________
The Montane Cordillera and Boreal Cordillera ecozones have almost equal areas.
1
Together, the 2 ecozones equal about
of the area of Canada.
10
What fraction of the area of Canada does each of these ecozones cover?
1
÷2=
10
Sentence: ________________________________________________________________________
300
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________ Date: ______________
6.3 Warm Up
1. Draw diagrams to solve.
a)
1
÷3
6
c) 5 ×
b)
2
5
d) 6 ×
2. Decide whether each fraction is closer to 0,
Use the number line to help you.
a)
3
5
3
5
c)
d)
7
is closer to
8
1
2
3
.
1
6
0
.
1
1
2
is closer to
3
.
1
2
2
3
0
7
8
0
0
1
3
is closer to
5
1
3
1
, or 1.
2
b)
0
3
÷2
4
1
is closer to
6
1
.
3. Multiply.
a) 4 × 2 =
b) 5 × 3 =
c) 12 × 2 =
d) 7 × 6 =
6.3 Warm Up ● MHR 301
Name: _____________________________________________________
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.
3
Open the paper and shade of it.
5
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 3
× =
2 5
10
Find each product using paper folding.
a)
1 1
×
4 2
ˆ Fold paper into quarters along its length.
1
ˆ Shade .
4
ˆ Fold paper in half across its width.
Draw a line to show the fold.
ˆ Draw slanted lines on half of it.
There are
equal rectangles.
ˆ How many are shaded and have
slanted lines?
1 1
× =
4 2
302
MHR ● Chapter 6: Fraction Operations
b)
2 2
×
3 3
Date: ______________
Name: _____________________________________________________ Date: ______________
Working Example 2: Multiply Using Diagrams
Find
2 1
× . Write the product in lowest terms.
3 2
Solution
Draw a rectangle.
Draw lines to divide its length into thirds.
_2
3
Draw a line to divide the width of the rectangle in half.
2
of the rectangle is grey
6
and has slanted lines.
2 1 2
× =
3 2 6
÷ ____
1_
2
_2
3
2 1
=
6 3
÷ ____
So,
2 1 1
× = .
3 2 3
Find each product using diagrams.
Write your answers in lowest terms.
a) Find
1 1
× .
2 2
b) Find
1 3
× .
3 4
ˆ Divide the length in half.
1
ˆ Shade .
2
ˆ Divide the width in half.
ˆ Draw slanted lines on the top half.
1 1 number of shaded parts with lines
× =
total number of parts
2 2
Write in lowest terms.
=
6.3 Multiplying Proper Fractions ● MHR 303
Name: _____________________________________________________
Date: ______________
Working Example 3: Multiply Using a Rule
Estimate and calculate
8 5
× .
15 6
Solution
Calculate:
Estimate:
Is
8
1
closer to 0, , or 1?
15
2
8
15
0
1
8 1
≈
15 2
=
5
1
Is closer to 0, , or 1?
6
2
5
6
0
8 5
×
15 6
8 × 5 ← Multiply the numerators
=
15 × 6 ← Multiply the denominators
90
Write in lowest terms.
÷ 10
1
5
≈1
6
=
8 5
×
15 6
1
≈ ×1
2
1
≈
2
9
÷ 10
Plot on a number line.
0
4
9
1
The answer is reasonable because it is close to
the estimate of
2
.
An estimate is reasonable if it
is close to the actual answer.
304
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
Estimate and calculate. Write your answers in lowest terms.
3 2
×
5 9
Calculate:
Estimate:
Is
3
1
closer to 0, , or 1?
5
2
3
≈
5
Is
3 2
×
5 9
×
=
×
2
1
closer to 0, , or 1?
9
2
2
≈ _____________
9
← Multiply the numerators
← Multiply the denominators
=
÷3
3 2
× ≈
5 9
× ___________
=
Write in lowest terms.
≈ _____________
÷3
1. Brendan calculated
3 2 6
× = . Brendan made a mistake.
5 5 5
a) What mistake did he make?
_________________________________________________________________________
b) Find the correct answer.
6.3 Multiplying Proper Fractions ● MHR 305
Name: _____________________________________________________
Date: ______________
Use paper folding
to help you.
2. Find the products by drawing diagrams.
a)
5 1
×
6 2
b)
3 5
×
4 6
ˆ Divide the length in sixths.
5
ˆ Shade .
6
ˆ Divide the width in half.
ˆ Draw slanted lines on the top half.
5 1 number of shaded parts with lines
× =
6 2
total number of parts
=
3. Estimate and calculate
3 2
× . Write your answers in lowest terms.
8 3
Calculate:
Estimate:
3
≈
8
2
≈
3
3 2
×
8 3
×
=
3 2
× ≈
8 3
×
× _____________
=
≈
=
306
MHR ● Chapter 6: Fraction Operations
Write in lowest terms.
Name: _____________________________________________________
Date: ______________
4. Calculate. Write your answers in lowest terms.
a)
3 3
×
4 4
b)
5 3
×
6 8
1
1
of an apple pie in her fridge. She ate of it.
2
4
What fraction of the whole pie did she eat?
5. Tamar had
1
×
2
1
Sentence: ____________________________________________________________________
1
of the people in the world live in Canada or the United States.
20
1
live in Canada.
Of the people who live in Canada or the United States, about
10
What fraction of people in the world live in Canada? Show your work.
6. About
×
Sentence: ____________________________________________________________________
6.3 Multiplying Proper Fractions ● MHR 307
Name: _____________________________________________________
Date: ______________
1
of his time sleeping.
3
1
While he is asleep, he dreams for of the time.
4
7. Marius spends
a) What fraction of the time is Marius
dreaming?
b) How many hours a day is Marius dreaming?
Show your work.
Fraction from part a) × number of hours
×
__________________________________
__________________________________
__________________________________
__________________________________
1
of the area of Canada.
10
1
The Pacific Maritime ecozone covers about of British Columbia.
5
What fraction of the area of Canada does the Pacific Maritime
ecozone cover? Show your work.
British Columbia is about
×
Sentence: _______________________________________________________________________
308
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6.4 Warm Up
1 2 3 4
, , , , …=1
1 2 3 4
1. Change the improper fraction to a mixed fraction.
a)
5
2
=
1
1
1
1
1
+ + + +
2
2
2
2
2
=
2
+
2
+
4
3
d)
6
4
1
2
1
2
=
c)
b)
5
3
2. Change the mixed number to an improper fraction.
2
a)
=
b)
1
d) 3
1
4
1
2
3 3 2
+ +
3 3 3
=
c) 1
2
3
3
3
4
6.4 Warm Up ● MHR 309
Name: _____________________________________________________
Date: ______________
6.4 Multiplying Improper Fractions and Mixed Numbers
Working Example 1: Multiply Mixed Numbers Using a Model
Find 2
1
3
×1 .
2
4
A mixed number is made up of
a whole number and a fraction.
Solution
Draw lines to separate each
mixed fraction into a whole
number and a proper fraction.
Draw a rectangle.
1
2_
2
2
Show the area of each
part in the diagram.
_1
2
1
3
1_
4
_1
2
2
_3
4
1
2¥1
_3
4
2 ¥ 3_
4
1_ ¥ 1
2
1_ ¥ 3_
2 4
Calculate the area of each part.
2×
2×1=
=
1
×1
2
3
4
4
=
1 3
×
2 4
=
8
3
=
2
1
=1
2
1 1 3
+ +
2 2 8
⎛ 1 1 3⎞
= (2 + 1) + ⎜ + + ⎟
⎝ 2 2 8⎠
Add the areas together: 2 + 1
=
+
=3+
=3+1
=4
8
3
8
3
8
So, 2
1
3
3
×1 = 4 .
2
4
8
310
MHR ● Chapter 6: Fraction Operations
4
4 3
+ +
8
8 8
Add the whole numbers.
Find a common denominator.
Add the numerators.
Change to a mixed fraction.
Name: _____________________________________________________
Date: ______________
Find each product using a model.
1
1
a) 1 × 1
2
3
b) 2
ˆ Label the outer edges of the diagram.
ˆ Label the inside of the diagram.
ˆ Calculate the area of each part.
1
1
×1
4
4
ˆ Label the outer edges of the diagram.
ˆ Label the inside of the diagram.
ˆ Calculate the area of each part.
1×1=
1×
1
=
2
1×
=
1 1
× =
2 3
ˆ Add all the areas together.
+
+
ˆ Add all the areas together.
+
6.4 Multiplying Improper Fractions and Mixed Numbers ● MHR 311
Name: _____________________________________________________
Date: ______________
Working Example 2: Multiply Mixed Numbers Using a Rule
Estimate and calculate 4
1
1
× 2 . Write the product in lowest terms.
2
3
Solution
Estimate:
Calculate:
Round each mixed number to the
closest whole number.
Write the mixed numbers as improper fractions.
4
1
4 ≈5
2
1
2 ≈
3
1 2 3
, , ... = 1
1 2 3
1 2 2 2 2 1
= + + + +
2 2 2 2 2 2
=
2
5×2=
So, 4
1
1
×2 ≈
2
3
.
2
1
=
3
3
=
+
3
+
1
3
3
Multiply the improper fractions.
4
1
1 9 7
×2 = ×
2
3 2 3
=
×
2×3
← Multiply the numerators
← Multiply the denominators
=
6
Write in lowest terms.
÷3
63
=
6
2
÷3
= 10
312
MHR ● Chapter 6: Fraction Operations
1
2
20
= 10 .
2
21
1
So,
= 10 + .
2
2
Name: _____________________________________________________
Date: ______________
Estimate and calculate.
1
1
1 ×3
10
2
Estimate:
1
Calculate:
Change to improper fractions.
1
≈ _____________
10
1
3 ≈ _____________
2
×
So, 1
1
1
=
10
+
=
=
1
1
× 3 ≈ _____________ .
10
2
1
3 =
2
+
+
+
=
Multiply the improper fractions.
1
=
1
1
×3
10
2
×
×
=
×
← Multiply the numerators
← Multiply the denominators
=
6.4 Multiplying Improper Fractions and Mixed Numbers ● MHR 313
Name: _____________________________________________________
Date: ______________
1
1
× 3 like this:
2
4
1 1 1
1
1
1
2 × 3 = 6 and × = , so 2 × 3 = 6 .
2 4 8
2
4
8
1. Henri multiplied 2
a) What mistake did Henri make?
_________________________________________________________________________
_________________________________________________________________________
b) What is the correct answer?
2. Write each improper fraction as a mixed number.
a)
11
3
3
= +
3
+
+
=3
314
17
6
b)
MHR ● Chapter 6: Fraction Operations
=
+
= _____________
+
5
6
5
6
Name: _____________________________________________________
Date: ______________
3. Write each mixed number as an improper fraction.
4
a)
=
=
3
4
2
b)
4 4
+ +
4 4
+
+
3
4
7
8
=
+
+
7
8
=
4
4. Use a model to find each product.
1
1
a) 1 × 1
5
2
1
1
b) 1 × 2
2
3
ˆ Draw rectangle to solve.
ˆ Draw rectangle to solve.
ˆ Label the diagram parts.
ˆ Multiply each area.
ˆ Label the diagram parts.
ˆ Multiply each area.
ˆ Add all the areas together.
ˆ Add all the areas together.
6.4 Multiplying Improper Fractions and Mixed Numbers ● MHR
315
Name: _____________________________________________________
Date: ______________
5. Estimate and calculate. Write your answers in lowest terms.
a)
4 10
×
5 7
b) 2
1
2
×1
5
3
Estimate:
Estimate:
10 3
=1
7
7
4
≈ _____________
5
10
≈ _____________
7
×
So,
=
4 10
×
≈ ________ .
5 7
Calculate:
Calculate:
4 10
×
5
7
×
=
×
← Multiply the numerators
← Multiply the denominators
=
Write in lowest terms.
÷ _____
=
Is your
calculation close
to your estimate?
= _________
÷ _____
316
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6. Two and a half laps of a running track equal 1 km.
How many laps equal 3 km?
3×
Write the mixed number as an improper fraction.
Multiply the improper fractions.
Write in lowest terms.
Sentence: __________________________________________________________________
1
1
hours of daylight, it was sunny for of the time.
2
3
How many hours was it sunny?
7. On a day in Winnipeg with 10
Sentence: __________________________________________________________________
6.4 Multiplying Improper Fractions and Mixed Numbers ● MHR 317
Name: _____________________________________________________
Date: ______________
8. Andreas has $18.
2
times as much as Andreas.
3
How much money does Bonnie have?
3
times as much as Bonnie.
5
How much money does Cheryl have?
a) Bonnie has 1
b) Cheryl has 1
c) How much money do they have altogether?
Sentence: ________________________________________________________________________
1
of the area of Canada.
26
9
The Northern Arctic ecozone is about 3
times as big as the Hudson Plains ecozone.
10
What fraction of the area of Canada does the Northern Arctic ecozone cover?
The Hudson Plains ecozone covers
1
9
×3
26
10
Write the mixed number as an improper fraction.
Multiply the fractions.
Sentence: _______________________________________________________________________
318
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6.5 Warm Up
1. Write each fraction as a mixed number.
a)
18
5
b)
23
3
2. Write each mixed number as an improper fraction.
a) 3
3
7
b) 1
3
11
3. Write each set of fractions with a common denominator.
a)
12 1
, =
15 3
,
b)
5 7
, =
2 8
d)
12 9
, =
9 6
,
Multiples of 15: 15, 30, 45, 60, …
Multiples of 3: 3, 6, 9, 12, 15, 18, …
The lowest common multiple is
c)
3 1
, =
4 6
,
.
,
4. Divide.
a) 12 ÷ 2 =
b) 40 ÷ 8 =
c) 13 ÷ 1 =
d) 24 ÷ 3 =
6.5 Warm Up ● MHR 319
Name: _____________________________________________________
Date: ______________
6.5 Dividing Fractions and Mixed Numbers
Working Example 1: Divide Using Diagrams
Find
2 1
÷ .
3 4
Solution
1
2
s are in .
4
3
ˆ Divide 1 rectangle into 3 parts. Shade 2 of the parts.
ˆ Divide another rectangle into 4 parts.
1
2
The diagrams show that the number of s in is between 2 and 3.
4
3
1
2
A common denominator of and is
.
4
3
So, divide a rectangle into twelfths.
ˆ Shade the parts up to the dotted line.
8
In , there are 2 whole groups of
12
2 1
2
3
2
÷ = 2 or
, plus of another group.
3
3 4
3
12
3
ˆ Draw diagrams to see how many
Find
3 1
÷ using diagrams.
4 3
ˆ Shade 3 parts of the first rectangle.
ˆ Draw a dotted line from the end of the shaded part
through the other 2 rectangles.
1
3
and is
3
4
ˆ Divide the third rectangle into twelfths.
ˆ Shade the parts up to the dotted line.
.
A common denominator of
ˆ Count the number of shaded parts:
of
4
, plus
12
So,
320
4
12
. In
9
, there are
12
of another group. There are
3 1
÷ = _____________
4 3
MHR ● Chapter 6: Fraction Operations
or
4
whole groups
whole groups of
.
3
.
12
Name: _____________________________________________________
Date: ______________
Working Example 2: Divide Using a Rule
Estimate and calculate.
a)
7 1
÷
8
4
Solution
Estimate:
7
.
8
ˆ Draw a diagram divided into quarters.
7
ˆ Draw a dotted line from to the next diagram.
8
ˆ Draw a diagram showing
The number of quarters in
7
is between 3 and
8
. So,
7 1
1
÷ ≈3 .
8
4
2
reciprocal
• flip the fraction to switch the numerator and denominator
2
3
• example: the reciprocal of is
3
2
Calculate:
Method 1: Divide Using a Common
Denominator
Method 2: Divide Using Multiplication
To divide by a fraction, multiply by its reciprocal.
7 1
÷
8 4
Find a common denominator.
=
7
÷
8
8
Divide the numerators.
7
=
2
Write in lowest terms.
1
=3
2
7 1
÷
8 4
7 4
= ×
8 1
=
flip the numerator
and denominator
8
÷4
28 7
=
8 2
÷4
1
=3
2
6.5 Dividing Fractions and Mixed Numbers ● MHR 321
Name: _____________________________________________________
b) 2
Date: ______________
1
3
÷3
2
4
Solution
Estimate:
1
3
2 ≈ 3 and 3 ≈ 4 .
2
4
1
3
2 ÷ 3 ≈3 ÷ 4
2
4
≈
4
Calculate:
Method 1: Divide Using a Common
Denominator
2
1
3
2 ÷3
2
4
2
Method 2: Divide Using Multiplication
Write as improper fractions.
1 2 2 1
= + +
2 2 2 2
=
2
3 4 4 4 3
3 = + + +
4 4 4 4 4
1
3
÷3
2
4
Write as improper fractions.
=
÷
2
5 4
= ×
2 15
=
15
.
4
Write as a reciprocal.
Write in lowest terms.
30
÷ 10
=
4
5 15
÷
2 4
10 15
=
÷
4
4
=
=
322
15
Find a common denominator.
30
=
Divide the numerators.
÷ 10
Write in lowest terms.
3
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
Estimate and calculate.
a)
4
3
÷
5 10
b) 3
1
2
÷1
6
3
Estimate:
Estimate:
Calculate:
Calculate:
6.5 Dividing Fractions and Mixed Numbers ● MHR 323
Name: _____________________________________________________
Date: ______________
Working Example 3: Apply Division With Fractions
Baby teeth are replaced by adult teeth as people get older.
5
Children have as many teeth as adults do. Children have 20 teeth.
8
How many teeth do adults have?
You could also divide using
common denominators.
Solution
5 20 5
20 ÷ =
÷
5
8 1 8
Divide 20 by to find the number of adult teeth.
8
160 5
=
÷
5
8
8
20 ÷
Multiply by the reciprocal.
8
=
20
×
1
Check:
Use multiplication to check the division.
5
× 32
8
5 32 ← Multiply the numerators
= ×
8 1 ← Multiply the denominators
8
160
=
= 32
Adults have
teeth.
=
8
= 20
1
of the tray.
6
How many servings are in 3 trays of lasagna?
One serving of lasagna is
3÷
6
Multiply by the reciprocal.
Check your answer.
1
×
6
=
1
×
6
1
=
There are
servings
of lasagna in 3 trays of lasagna.
= _____________
324
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
3 2
÷ .
4 3
Is Mike’s method correct? Circle YES or NO.
Give 1 reason for your answer.
1. Mike solved
Mike’s Work:
3 2
÷
4 3
4 2
= ×
3 3
8
=
9
__________________________________________________________________________
2. Find
5 1
÷ using diagrams.
8 4
ˆ Divide 1 rectangle into 8 parts.
Shade 5 parts.
ˆ Draw a dotted line from the end of the
shaded part through the other 2 rectangles.
ˆ Divide the second rectangle into 4 parts.
ˆ A common denominator of 8 and 4 is
ˆ Divide the third rectangle into eighths.
ˆ Shade the parts up to the dotted line.
.
5 1
÷
8 4
=
=
In
1
2
5
, there are _____ whole groups
8
2
1
of , plus of another group.
8
2
Write as an improper fraction.
6.5 Dividing Fractions and Mixed Numbers ● MHR 325
Name: _____________________________________________________
Date: ______________
3. Divide using a common denominator.
a)
3 9
÷
5 10
=
1 5
b) 1 ÷
2 6
Find a common
denominator.
÷
1
1 =
2
=
Divide the
numerators.
=
+
2
2
Write as an
improper fraction.
2
÷
Find a common
denominator for
2 and 6.
÷
Divide the
numerators.
Write in lowest
terms.
=
=
=
4. Divide using multiplication.
a)
3 4
÷
4 5
=
3
×
4
2
5
b) 1 ÷ 2
3
6
Write as a reciprocal.
=
326
MHR ● Chapter 6: Fraction Operations
Write as
improper
fractions.
Name: _____________________________________________________
1
of an hour to perform.
4
How many performers are there in a 2-h show?
5. In a comedy show, each performer has
2÷
Date: ______________
2=
2
1
4
There are
6. It takes 2
performers in 2 h.
1
cups of flour to make 1 cake. How many cakes can you make with 15 cups of flour?
2
Sentence: ____________________________________________________________________
The wettest part of the Prairies ecozone is the Manitoba Plain.
The average yearly amount of precipitation is about 70 cm.
4
This amount of precipitation is 2 of the amount in the dry grasslands.
5
What is the average annual precipitation in the grasslands?
Precipitation is water
that falls as rain, snow,
4
sleet, or hail.
70 ÷ 2
5
Sentence: _________________________________________________________________
6.5 Math Link ● MHR 327
Name: _____________________________________________________
6.6 Warm Up
Brackets, multiply or divide,
then add or subtract.
1. Use the order of operations to calculate.
4 × 7 – 10
a)
– 10
=
Multiply.
=7–
Multiply.
Subtract.
=
Subtract.
8 × (3 – 1)
c)
7–2×3
b)
=
Date: ______________
10 – (2 + 6) ÷ 2
d)
=8×
Brackets.
= 10 –
=
Multiply.
= 10 –
÷2
Brackets.
Divide.
=
1
÷ 2 using a common
5
denominator.
b) Divide 5 ÷
2. a) Divide
reciprocal.
3. Change mixed numbers to improper fractions.
a) 3
1
3
=
b) 2
+
+
=
328
MHR ● Chapter 6: Fraction Operations
+
2
5
4
by multiplying by the
9
Name: _____________________________________________________
Date: ______________
6.6 Applying Fraction Operations
Working Example 1: Use the Order of Operations
Calculate using the order of operations.
Brackets, multiply or divide,
then add or subtract.
1 1
a) 2 ÷ ×
4 2
Solution
1
×
4
2 4
= × ×
1 1
2÷
=
=
1
2
1
2
×
Divide by multiplying the reciprocal.
1
2
Multiply.
8
2
=
b)
Write in lowest terms.
1
× (9 – 2)
3
Solution
1
× (9 – 2)
3
1
= ×
3
=
=2
Brackets.
Multiply.
7=
7
1
3
1
3
Write as a mixed fraction.
6.6 Applying Fraction Operations ● MHR 329
Name: _____________________________________________________
c) 2
1 ⎛ 3
1⎞
÷ ⎜1 + 1 ⎟
4 ⎝ 4
4⎠
Date: ______________
1+1+
Solution
3 1
+
4 4
4
4
=2+1
=3
=2+
2
=2
1 ⎛ 3
1⎞
÷ ⎜1 + 1 ⎟
4 ⎝ 4
4⎠
1
÷
4
=
=
÷
×
4
=
4 4 1 9
+ + =
4 4 4 4
1
4
=
Add the whole numbers.
Add the fractions.
1
1
3
Write the mixed number as an improper fraction.
Multiply by the reciprocal.
Write in lowest terms.
12
3
4
a) 4 – 2 ÷
= 4−
=
=
3
5
b) 2
2
×
1
Multiply by
the reciprocal.
4
−
1
−
Find a common
denominator.
=
330
MHR ● Chapter 6: Fraction Operations
1
1 5
× –
4
2 8
Name: _____________________________________________________
Date: ______________
Working Example 2: Apply Fraction Operations
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
1
h when you work for 1 h.
2
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
time-and-a-half = 1 1
2
1
6×1
2
Write as an improper fraction:
2 1 3
+ =
2 2 2
=6×
=
6
3
×
1
2
=
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.
6.6 Applying Fraction Operations ● MHR 331
Name: _____________________________________________________
Date: ______________
Method 2: Evaluate One Expression
Bev’s regular rate of pay is $25/h. For 6 h at time-and-a-half, Bev is paid 1
1
×
2
h.
1
⎛
⎞
An expression of her total earnings is: 25 × ⎜ 40 + 1 × 6 ⎟
2
⎝
⎠
1
⎛
⎞
25 × ⎜ 40 + 1 × 6 ⎟
2
⎝
⎠
⎛
⎞
⎜
⎟
× 6⎟
= 25 × ⎜ 40 +
⎜
⎟
⎜
⎟
⎝
⎠
3 6⎞
⎛
= 25 × ⎜ 40 + × ⎟
2 1⎠
⎝
⎛
⎜
= 25 × ⎜ 40 +
⎜
⎜
⎝
= 25 × 49
= 1225
Bev earns $
2
⎞
⎟
⎟
⎟
⎟
⎠
2 1 3
+ =
2 2 2
Brackets first.
Write the mixed number as an improper fraction.
Multiply the numbers in the bracket.
18
= 18 ÷ 2 = 9
2
40 + 9 = 49
Add.
Multiply.
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
1
2
Amount earned at time-and-a-half:
Total earnings =
=
332
MHR ● Chapter 6: Fraction Operations
× 15 =
+
Name: _____________________________________________________
Date: ______________
⎛1 1⎞ 2
1. Mia missed the lesson on how to find the answer for ⎜ + ⎟ × .
⎝2 4⎠ 3
a) List the steps to solve this question.
b) Find the answer.
____________________________
____________________________
____________________________
2. Calculate.
3 1
2
– ×
4 2
3
a)
3
= –
4
=
=
Multiply.
Find a common
denominator.
–
b)
1 ⎛ 1
÷ ⎜1
2 ⎝ 4
⎛
1 ⎜
=3 ÷ ⎜
2 ⎜
⎜
⎝
⎛
⎜
=⎜
⎜
⎜
⎝
3
=
−
3⎞
⎟
4⎠
Write as an
improper fraction.
⎞
⎟
⎟ Brackets first.
⎟
⎟
⎠
−
⎞ ⎛
⎟ ⎜
⎟÷⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
×
⎞
⎟
⎟
⎟
⎟
⎠
Write as an
improper fraction.
Multiply by
the reciprocal.
=
=
6.6 Applying Fraction Operations ● MHR 333
Name: _____________________________________________________
3. Calculate.
a)
5 1 3
− ×
6 3 4
Multiply.
Find a common denominator.
Subtract.
b) 3
1 3 5
÷ −
2 4 6
Change to improper fraction.
Divide.
Subtract.
c)
334
7 2 1
+ −
8 3 4
MHR ● Chapter 6: Fraction Operations
1 1 2
d) 1 × ÷
2 3 3
Date: ______________
Name: _____________________________________________________
Date: ______________
4. Leo earns $16/h as a gardener.
When he works more than 35 h in 1 week, he earns time-and-a-half.
How much does he earn for working 36 h in a week?
Hours worked at regular pay =
Amount earned at regular pay: 35 ×
=
Hours worked at time-and-a-half: 36 – 35 =
Time worked over 35 hours:
×1
Amount earned at time-and-a-half: 1
1
=
2
1
× 16
2
16 =
Total earnings =
16
1
+
=
Sentence: ___________________________________________________________________
6.6 Applying Fraction Operations ● MHR 335
Name: _____________________________________________________
Date: ______________
5. Two thirds of the land on a farm is used for beef cattle.
The rest of the land is used to grow crops.
a) How much land is used to grow crops?
Draw a diagram to help you.
1−
=
Let all the land on
the farm equal 1.
2
3
−
b) Half of the land for crops is used to grow
corn. What fraction of the land is used to
grow corn?
×
2
3
1
2
=
Sentence: ________________________
Sentence: ________________________
_________________________________
_________________________________
1
of the species of mammals that live in Canada can
4
be found in the Taiga Shield ecozone.
About 50 species of mammals can be found in this ecozone.
About
a) How many species of mammals live in
Canada?
50 ÷
336
b) How many species of mammals in Canada
live outside the Taiga Shield ecozone?
1
4
Sentence: ___________________________
Sentence: ___________________________
___________________________________
___________________________________
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6 Chapter Review
Key Words
For #1 to #3, write the number that matches the description.
1. 3
1
4
improper fraction
8
9
11
3.
3
2.
mixed number
proper fraction
4. a) Unscramble the letters to make a key word.
CIRCLOPERA: R
L
b) What does this word mean?
_______________________________________________________________________
6.1 Multiplying a Fraction and a Whole Number, pages 288–293
5. Find the product using a diagram.
1
5×
4
ˆ Divide each rectangle
into 4 parts.
ˆ Shade 1 part in
each rectangle.
ˆ Add the shaded parts.
1
5×
4
=
1
+
4
+
+
+
+
+
+
=
+
=
=
Chapter Review ● MHR 337
Name: _____________________________________________________
Date: ______________
6. Use a number line to multiply.
4×
=
2
3
0
1
+
+
2
3
+
=
=
7. The average mass of a porcupine is about 12 kg.
3
The average mass of a raccoon is about of a porcupine’s mass.
4
What is the average mass of a raccoon?
Use diagrams or a
number line to help you.
×
Sentence: __________________________________________________________________
8. The length of a rectangle is 6 cm. The width is
What is the width?
2
of the length.
3
Sentence: __________________________________________________________________
338
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6.2 Dividing a Fraction by a Whole Number, pages 295–300
9. Use a diagram to divide.
a)
3
÷2
4
2
÷4
3
b)
ˆ Divide and shade the fraction strip to
3
show .
4
ˆ Label the number line to show thirds.
0
1
ˆ Divide each third into 4 equal parts.
ˆ Divide each quarter into 2 equal parts.
There are
There are
parts in a whole, so
each part is
.
parts in the whole,
.
so each part is
ˆ Use brackets to divide
parts.
So,
3
÷ 2=
4
.
.
Each part is
So,
10. A recipe for making 6 servings of potato salad includes
What fraction of an onion is in each serving?
÷
2
into 4 equal
3
0
2
÷ 4=
3
.
1
an onion.
2
1
Sentence: __________________________________________________________________
Chapter Review ● MHR 339
Name: _____________________________________________________
Date: ______________
6.3 Multiplying Proper Fractions, pages 302–308
11. Use a diagram to solve.
a)
1 3
×
2 4
b)
ˆ Divide the length in half.
1
ˆ Shade .
2
ˆ Divide the width into quarters.
3
ˆ Draw slanted lines on of it.
4
ˆ Divide the length in thirds.
2
ˆ Shade .
3
ˆ Divide the width into quarters.
1
ˆ Draw slanted lines on of it.
4
# of shaded parts with lines
=
total # of parts
# of shaded parts with lines
=
total # of parts
So,
1 3
× =
2 4
.
12. Estimate and calculate
So,
2 1
× =
3 4
3 3
× .
5 5
Estimate:
3
1
Is closer to 0, or 1?
5
2
Calculate:
3 3
×
5 5
3
≈
5
=
3 3
× ≈
5 5
×
≈
340
2 1
×
3 4
MHR ● Chapter 6: Fraction Operations
.
Name: _____________________________________________________
Date: ______________
6.4 Multiplying Improper Fractions and Mixed Numbers, pages 310–318
13. Estimate and calculate
8 6
× . Write your answers in lowest terms.
3 5
Estimate:
Calculate:
Change to mixed numbers:
8 6
×
3 5
8
=
3
+
+
2
3
=
2
3
=
÷3
6
5
= +
5
5
=
=
÷3
8
6
≈ _____________ and ≈ _____________
3
5
×
So,
=
=
8 6
× ≈ ______________ .
3 5
14. The distance from Winnipeg to Regina is 570 km.
1
The distance from Winnipeg to Calgary is 2 times the distance from Winnipeg to Regina.
3
What is the distance from Winnipeg to Calgary?
×2
1
3
Sentence: ____________________________________________________________________
Chapter Review ● MHR 341
Name: _____________________________________________________
Date: ______________
6.5 Dividing Fractions and Mixed Numbers, pages 320–327
15. Divide.
2 5
÷
3 6
a)
b) 3
2
= ×
3
Multiply by the reciprocal.
=
Write in lowest terms.
1
1
÷2
2
4
ˆ Write the mixed numbers as improper
fractions.
ˆ Multiply by the reciprocal.
ˆ Write as a mixed number.
=
1
of a bale of hay per day.
2
How long will 15 bales of hay last?
16. A horse eats
15 ÷
Sentence: __________________________________________________________________
342
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
6.6 Applying Fraction Operations, pages 329–336
17. Calculate.
1 3 1
+ ×
3 2 3
a)
=
b)
1
+
3
1
=1
1 ⎛7 5⎞
÷⎜ − ⎟
2 ⎝8 8⎠
Brackets first.
1
÷
2
Find a common denominator.
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:
Time worked over 32 h:
=
–
×1
1
2
=
Amount earned at time-and-a-half:
×
=
Total earnings =
+
=
Sentence: ____________________________________________________________________
Chapter Review ● MHR 343
Name: _____________________________________________________
Date: ______________
6 Practice Test
For #1 to #5, choose the correct answer.
1. Which expression does not equal 4 ×
A 1
C
1
3
1
×4
3
2. Which expression equals
B
4
3
D
1 1 1 1
× × ×
3 3 3 3
4 2
÷ ?
5 3
A
4
2
×
5
3
B
5
3
×
4
2
C
4
3
×
5
2
D
5
2
×
4
3
3. What is the reciprocal of
2
?
3
A
4
3
B
3
2
C
3
1
D
4
6
4. What is the value of the expression
A
12
24
B
5
8
C
23
24
D
5
4
1
1
3
+ × ?
2
6
4
5. What is
3
5
÷
in lowest terms?
4 12
A
9
5
B
5
16
C
36
20
D
15
48
344
1
?
3
MHR ● Chapter 6: Fraction Operations
Brackets, multiply or divide,
then add or subtract.
Name: _____________________________________________________
Date: ______________
Short Answer
6. Calculate.
3 5
×
a)
8 6
c) 3
3 3
×
5 8
b)
Change the mixed
number into an
improper fraction.
6
7
÷
5 10
1
⎛ 1 3⎞
d) ⎜1 + ⎟ ÷ 1
2
⎝ 4 4⎠
Brackets, multiply
or divide, then add
or subtract.
1
h for $14/h.
2
How much did she earn?
7. Leisha worked 6
Sentence: __________________________________________________________________
Practice Test ● MHR 345
Name: _____________________________________________________
Date: ______________
1
of a byte.
8
How many bits are needed to make 16 bytes?
8. In computer terminology, a bit is
Sentence: __________________________________________________________________
9. Printer paper is sold in packages of 500 sheets.
3
If a printing job uses 1 packages of paper, how many sheets were used?
4
Sentence: ___________________________________________________________________
346
MHR ● Chapter 6: Fraction Operations
Name: _____________________________________________________
Date: ______________
Most of the Boreal Plains ecozone is covered by woods and forests.
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.
Province/
Territory
Fraction of Boreal Plains
in the Province/Territory
Alberta
13
25
British
Columbia
1
20
Manitoba
17
100
Northwest
Territories
1
50
Saskatchewan
6
25
Examples:
Multiplication question:
How many square kilometres does the Boreal
Plains ecozone cover in Saskatchewan?
6
Answer: 750 000 ×
25
4 500 000
=
25
= 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?
1
1
÷
Answer:
20 50
1
50
×
=
20
1
50
=
20
1
=2
2
a) Your question:
_________________________________________________________________________
b) Your answer:
Practice Test ● MHR 347
Name: _____________________________________________________
Date: ______________
Unscramble the letters of each term. Use the clues to help you.
A
1.
3
4
ONODTRIEMAN
2. For
3.
11
12
4.
2
3
5
6
, this would be .
6
5
AOPCELRCIR
EANTRMROU
ORERPP ATCFINRO
5. answer to a division question
6. answer to a multiplication question
7. This would include the following list:
• brackets
• multiply and divide in order
• add and subtract in order
4
5
8.
2
9.
15
6
348
B
MHR ● Chapter 6: Fraction Operations
OETNQITU
DURTOPC
RDEOR FO EAOPINOSR
IDXME BREUNM
OMIRREPP ATINFORC
Name: _____________________________________________________
Date: ______________
Math Games
Fabulous Fractions
8
9
1
2
7
6
3
4
5
1. Play Fabulous Fractions with a partner.
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
I spun 6, 7, 1, and 4.
6 1
My fractions are × .
7 4
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
• spinner with 9 sections
numbered 1 to 9 per pair
of students
• paper clip per pair of students
• Fabulous Fractions
Multiplication BLM per
student
• Fabulous Fractions Division
BLM per student
I spun 6, 7, 1, and 4.
6 1
My fractions are ÷ .
7 4
Player 2
Math Games ● MHR 349
Name: _____________________________________________________
Date: ______________
Challenge in Real Life
Create a Class Flag
Mr. Jansen’s grade 8 class is made up of 10 boys and 14 girls.
• Organize Mr. Jansen’s Class
Flag BLM
• Organize Your Class Flag BLM
• 2 different coloured pencils
You are a student in Mr. Jansen’s class.
Create a class flag that will show the students’ interests.
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
× 10
5
1 10
= ×
5 1
=
d) One half of the boys play hockey. None
of the hockey players are in the band.
How many boys play hockey?
1
× 10
2
5
= ______________
There are
band.
boys in the
• Draw an instrument in
of the boys’ boxes.
• Draw a hockey puck in
of the boys’ boxes.
e) One half of the girls play volleyball.
How many girls play volleyball?
Draw a volleyball in
350
MHR ● Chapter 6: Fraction Operations
of the girls’ boxes.
Name: _____________________________________________________
Date: ______________
2. Now, create your own class flag.
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
Example: Are you in the band?
# of Boys
Yes = 2
# of Girls
Yes = 3
b) Design a flag that shows the class information. Show your calculations.
Challenge in Real Life ● MHR 351
Answers
Get Ready, pages 284–285
Math Link
a) 5 b) 15
6.2 Warm Up, page 294
1 6
2 8
8 16
3
1. a) 6 × =
b) 4 × =
c) 2 × =
d) 4 × = 3
7 7
3 3
5 5
4
12
+
+
+
=
2. a)
8
1
1
b)
1. a)
3
2
1
5
2. a) 2
b) 3
10
6
Math Link, page 286
a)
3
1
b)
10
6
b)
6.1 Warm Up, page 287
1. a)
+
b)
=
3
1
b)
4
10
4. a) 8 b) 20 c) 6 d) 15 e) 9 f) 30
3. a)
1
10
6.1 Multiplying a Fraction and a Whole Number, pages 288–293
Working Example 1: Show You Know
+
Working Example 2: Show You Know
b)
0
1
2
9
4
+
+
=
4
6 6 6 18
+ + =
5 5 5 5
3. a)
3
10
b)
0
1
2
3
4
c) Answers may vary. Example: I prefer fraction strips because I find it
easier to see the answer.
Practise
2. a) 3 ×
2 6
5 10
b) 2 × =
=
5 5
4
4
3. a) 3 ×
1 3
2 6
b) 3 × =
=
6 6
5 5
4. a) 3 ×
1 3
1 4
b) 4 × =
=
2 2
6 6
+
+
+
+
+
=
1
8
1
9
0 1 2 3 4 5 6 7 8 9 1
10 10 10 10 10 10 10 10 10
1
4
0
1
2
2
3
1
Apply
1
1
4. a)
of a coconut b)
of a coconut
4
8
1
of a full container
5.
6
Math Link
1
20
1. a)
b)
6. 2 m
7. 9 people
8. a) Answers will vary. Example: There are 8 seats at a restaurant counter.
1
full. How many people are at the counter?
The seats are
4
352
b)
b)
6.3 Warm Up, page 301
=
Apply
b) 8 ×
=
b) Answers may vary. Example: I prefer fraction strips because they make
it easier to visualize the fractions.
Practise
1
8
2
10
+
1
4
2. a)
b)
+
0 1 2 3 4 5 6 7 8 9 1
10 10 10 10 10 10 10 10 10
Communicate the Ideas
5. a) 2
+
Working Example 3: Show You Know
1
1
÷4=
2
8
Communicate the Ideas
1. a)
Working Example 3: Show You Know
1. a)
+
Working Example 1: Show You Know
1
4
Working Example 2: Show You Know
=
+
4
3
+
6.2 Dividing a Fraction by a Whole Number, pages 295–300
2. a)
a)
+
3. a) 3 b) 5 c) 20 d) 2 e) 3 f) 3
2
2
b) 2
5
3
5
3
7
2
1
=2
4
MHR ● Chapter 6: Fraction Operations
1
18
3
8
c) 2
d) 2
+
+
+
+
+
+
+
+
=
+
=
2. Estimates may vary. a) 1 b)
Practise
1
c) 1 d) 0
2
2. a) 3
3. a) 8 b) 15 c) 24 d) 42
6.3 Multiplying Proper Fractions, pages 302–308
3. a)
Working Example 1: Show You Know
a)
1
8
b)
4
9
2
5
b) 2
3
6
19
23
11
13
b)
c)
d)
4
8
4
7
4. a) 1
Working Example 2: Show You Know
a)
1
4
b)
1
4
_1
2
1
1¥1
_1 ¥ 1
2
_1
5
1
1¥ _
5
_1 ¥ _1
2 5
5. a) Estimate: 1; Calculate: 1
Working Example 3: Show You Know
Estimate: 0; Calculate:
1
4
5
2
15
b) 3
1
_1
2
2
1¥1
_1 ¥ 1
2
_1
3
1
1¥ _
3
_1 ¥ _1
2 3
1
2
1
2
b) Estimate: 4; Calculate: 3
7
3
Apply
Communicate the Ideas
1. a) Brendan should multiply the denominators. b)
6
25
Practise
6. 7
1
laps
2
7. 3
1
hours
2
8. a) $30 b) $48 c) $96
5
2. a)
12
5
b)
8
Math Link
3
of the area of Canada
20
6.5 Warm Up, page 319
3. Estimate:
1
1
; Calculate:
4
4
1. a) 3
Apply
4. a)
5.
9
5
b)
16
16
1
of the pie
8
3
2
b) 7
5
3
2. a)
24
14
b)
7
11
3. a)
12 5
20 7
9 2
24 27
b)
c)
d)
,
,
,
,
15 15
8 8
12 12
18 18
4. a) 6 b) 5 c) 13 d) 8
1
of the world’s people
6.
200
6.5 Dividing Fractions and Mixed Numbers, pages 320–327
1
7. a)
of his time b) 2 hours
12
2
Working Example 1: Show You Know
1
4
Math Link
1
of the area of Canada
50
6.4 Warm Up, page 309
Working Example 2: Show You Know
1
1
2
1
b) 1 c) 1 d) 1
1. a) 2
2
3
3
2
a) Estimate: 3; Calculate: 2
2. a)
8
3
7
13
b)
c)
d)
3
2
4
4
2
9
b) Estimate: 2; Calculate: 1
3
10
Working Example 3: Show You Know
18
6.4 Multiplying Improper Fractions and Mixed Numbers,
pages 310–318
Communicate the Ideas
Working Example 1: Show You Know
1. NO. Mike needs to multiply by the reciprocal of
1
_1
2
1
1×1
_1 × 1
2
_1
3
1
1× _
3
_1 × _1
2 3
a) 2
b) 2
2
_1
4
1
2×1
_1 × 1
4
_1
4
1
2× _
4
_1 × _1
4 4
13
16
Practise
2. 2
17
20
Communicate the Ideas
1. a) Henri forgot to multiply 2 ×
1
2
3. a)
2
9
b)
3
5
4. a)
15
10
b)
16
17
Working Example 2: Show You Know
Estimate: 4; Calculate: 3
2
.
3
5 1
1
– ÷ – = 2–
8 4
2
1
1
1
and 3 × . b) 8
4
2
8
Answers ● MHR 353
Apply
Math Link
5. 8 performers
a) 200 species b) 150 species
6. 6 cakes
Chapter Review, pages 337–343
Math Link
1. mixed number 2. proper fraction 3. improper fraction
25 cm
4. a) reciprocal b) the inverse of a fraction
6.6 Warm Up, page 328
1. a) 18 b) 1 c) 16 d) 6
2. a)
1
1
b) 11
10
4
3. a)
10
12
b)
3
5
6.6 Applying Fraction Operations, pages 329–336
Working Example 1: Show You Know
a)
2
1
b)
3
2
Working Example 2: Show You Know
$705
5. 1
1
4
6. 2
2
3
+
Practise
2. a)
5
b) 7
12
7
5
7
3
3. a)
b) 3
c) 1
d)
12
6
24
4
Apply
4. $584
1
1
of the land b)
of the land
5. a)
3
6
+
1
=
+
2
3
2
3
1
7. 9 kg
8. 4 cm
3
8
9. a)
1
0
6
1
10.
of an onion
12
b)
Communicate the Ideas
1
1
and . Step 2: Add the
1. a) Step 1: Find the common denominator for
2
4
2
1
fractions in the brackets. Step 3: Multiply by . b)
3
2
0
+
11. a)
3
8
b)
1
6
1
9
; Calculate:
4
25
1
13. Estimate: 3; Calculate: 3
5
14. 1330 km
12. Estimate:
15. a)
4
5
b) 1
5
9
16. 30 days
5
17. a)
b) 6
6
18. $528
Practice Test, pages 344–346
1. D 2. C 3. B 4. B 5. A
6. a)
5
5
7
1
b) 1
c) 1
d) 1
16
7
20
3
7. $91
8. 128 bits
9. 875 sheets
Wrap It Up!, page 347
Answers will vary. Example:
a) How many square km does the Boreal Plains ecozone cover in Manitoba?
b) 127 500 km2
Key Word Builder, page 348
1. denominator 2. reciprocal 3. numerator 4. proper fraction 5. quotient
6. product 7. order of operations 8. mixed number 9. improper fraction
Challenge in Real Life, pages 350–351
1. a) and b) Answers will vary. c) 2 d) 5 e) 7
2. Answers will vary.
354
MHR ● Chapter 6: Fraction Operations