Download Unit Two - Math 10C

Mathematics 10C
Student Workbook
R
Q
W
Q
N
I
2
Lesson 1: Number Sets
Approximate Completion Time: 1 Day
12
4
2
3
12 = 2 × 2 × 3
Lesson 2: Primes, LCM, and GCF
Approximate Completion Time: 2 Days
2
52 = 25
53 = 125
Unit
Lesson 3: Squares, Cubes, and Roots
Approximate Completion Time: 2 Days
Lesson 4: Radicals
Approximate Completion Time: 2 Days
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Lesson 5: Exponents I
Approximate Completion Time: 1 Day
Lesson 6: Exponents II
Approximate Completion Time: 2 Days
UNIT TWO
Numbers, Radicals, and Exponents
Mathematics 10C
Unit
Student Workbook
2
Complete this workbook by watching the videos on www.math10.ca.
Work neatly and use proper mathematical form in your notes.
UNIT TWO
Numbers, Radicals, and Exponents
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Numbers, Radicals, and Exponents
LESSON ONE - Number Sets
Lesson Notes
N
Introduction
Define each of the following sets of numbers and
fill in the graphic organizer on the right.
a) Natural Numbers
b) Whole Numbers
c) Integers
d) Rational Numbers
e) Irrational Numbers
f) Real Numbers
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Numbers, Radicals, and Exponents
LESSON ONE - Number Sets
Lesson Notes
Example 1
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b) 0
c) 1.273958...
e) 7.4
f) 4.93
g) -
2
3
d) 7
h) π
For each statement, circle true or false.
a) All natural numbers are whole numbers.
b) All rational numbers are integers.
T
c) Some rational numbers are integers.
T
F
F
T
F
d) Some whole numbers are irrational numbers.
T
F
e) Rational numbers are real numbers, but irrational numbers are not.
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Q
N
I
Determine which sets each number belongs to.
In the graphic organizer, shade in the sets.
a) -4
Example 2
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T
F
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N
I
Example 3
Q
Numbers, Radicals, and Exponents
LESSON ONE - Number Sets
Lesson Notes
Sort the following numbers as rational, irrational, or neither.
You may use a calculator.
Rational
Irrational
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Neither
Numbers, Radicals, and Exponents
LESSON ONE - Number Sets
Lesson Notes
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W
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Order the numbers from least to greatest on a number line.
You may use a calculator.
Example 4
a)
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
b)
c)
-4
-3.75
-3.5
-3.25
-3
-2.75
-2.5
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
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1.5
1.75
Q
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
Numbers, Radicals, and Exponents
12
4
2
3
12 = 2 × 2 rise
×3
2
Introduction
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
run
Prime Numbers, Least Common Multiple, and Greatest Common Factor.
a) What is a prime number?
b) What is a composite number?
c) Why are 0 and 1 not considered prime numbers?
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Numbers, Radicals, and Exponents
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
12
4
2
3
2
d) What is prime factorization? Find the prime factorization of 12.
e) What is the LCM? Find the LCM for 9 and 12 using two different methods.
f) What is the GCF? Find the GCF for 16 and 24 using two different methods.
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12 = 2 × 2 × 3
Numbers, Radicals, and Exponents
12
4
2
3
12 = 2 × 2 × 3
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
2
Example 1
a) 1
Example 2
Determine if each number is prime, composite, or neither.
b) 14
c) 13
d) 0
Find the least common multiple for each set of numbers.
a) 6, 8
b) 7, 14
c) 48, 180
d) 8, 9, 21
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Numbers, Radicals, and Exponents
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
Example 3
12
4
2
3
12 = 2 × 2 × 3
2
Find the greatest common factor for each set of numbers.
a) 30, 42
b) 13, 39
c) 52, 78
d) 54, 81, 135
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Numbers, Radicals, and Exponents
12
4
2
3
12 = 2 × 2 × 3
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
2
Example 4
Problem solving with LCM
a) A fence is being constructed with posts that are 12 cm wide.
A second fence is being constructed with posts that are 15 cm wide.
If each fence is to be the same length, what is the shortest fence
that can be constructed?
b) Stephanie can run one lap around a track in 4 minutes. Lisa can run one lap in 6 minutes.
If they start running at the same time, how long will it be until they complete a lap together?
c) There is a stack of rectangular tiles, with each tile having a length of 84 cm and a width of 63 cm.
If some of these tiles are arranged into a square, what is the smallest side length the square can have?
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Numbers, Radicals, and Exponents
LESSON TWO - Primes, LCM, and GCF
Lesson Notes
Example 5
12
4
2
3
12 = 2 × 2 × 3
2
Problem solving with GCF
a) A fruit basket contains apples and oranges. Each basket will have the same
quantity of apples, and the same quantity of oranges. If there are 10 apples
and 15 oranges available, how many fruit baskets can be made?
How many apples and oranges are in each basket?
b) There are 8 toonies and 20 loonies scattered on a table. If these coins are organized into
groups such that each group has the same quantity of toonies and the same quantity of loonies,
what is the maximum number of groups that can be made? How many loonies and toonies are
in each group?
c) A box of sugar cubes has a length of 156 mm, a width of 104 mm,
and a height of 39 mm. What is the edge length of one sugar cube?
Assume the box is completely full and the manufacturer uses sugar
cubes with the largest possible volume.
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SUGAR CU
BES
SUGAR
CUBES
52 = 25
53 = 125
Numbers, Radicals, and Exponents
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Introduction
Perfect Squares, Perfect Cubes, and Roots.
a) What is a perfect square? Draw the first three perfect squares.
b) What is a perfect cube? Draw the first three perfect cubes.
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Numbers, Radicals, and Exponents
52 = 25
53 = 125
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
c) Complete the table showing all perfect squares and perfect cubes up to 10. The first three
are completed for you.
Number
Perfect Square
Perfect Cube
1
12 = 1
13 = 1
2
22 = 4
23 = 8
3
32 = 9
33 = 27
d) What is a square root? Find the square root of 36.
i) Using a geometric square.
ii) Using the formula A = s2
e) What is a cube root? Find the cube root of 125.
i) Using a geometric cube.
ii) Using the formula V = s3
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52 = 25
Numbers, Radicals, and Exponents
53 = 125
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Example 1
Evaluate each power, without using a calculator.
a) 32
b) (-3)2
c) -32
d) 33
e) (-3)3
f) -33
Example 2
a) 2(2)3
d)
1
43
Evaluate each expression, without using a calculator.
b) -2(-4)2
e)
1
22 + 2 3
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c) 1 - 52
f)
5(-2)3
-22
Numbers, Radicals, and Exponents
53 = 125
52 = 25
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Example 3
a)
Evaluate each root using a calculator.
b)
c)
d)
e) What happens when you evaluate
and
?
Is there a pattern as to when you can evaluate the root of a negative number?
Example 4
Evaluate each expression, without using a calculator.
a)
b)
c)
d)
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52 = 25
53 = 125
Numbers, Radicals, and Exponents
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Example 5
The area of Edmonton is 684 km2
a) If the shape of Edmonton is approximated to be a square, how wide is the city?
Edmonton
b) If the shape of Edmonton is approximated to be a circle, how wide is the city?
Example 6
The formula for the volume of a sphere is V =
4
πr3
3
a) If a sphere has a radius of 9 cm, what is the volume?
r = 9 cm
b) If a sphere has a volume of approximately 5000 cm3, what is the radius?
V = 5000 cm3
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Numbers, Radicals, and Exponents
52 = 25
53 = 125
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Example 7
The amount of time, T, it takes for a pendulum to swing back and forth is called the period.
The period of a pendulum can be calculated with the formula: T = 2 π
a) What is the period of the pendulum if the length, l, is 1.8 m?
b) What is the length of the pendulum if the period is 2.4 s?
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l
9.8
52 = 25
53 = 125
Numbers, Radicals, and Exponents
LESSON THREE - Squares, Cubes, and Roots
Lesson Notes
Example 8
The total volume of gold mined throughout
history is approximately 8340 m3.
a) If all the gold was collected, melted down, and recast as a cube,
what would be the edge length?
b) If the density of gold is 19300 kg/m3, what is the mass of the cube?
mass
The density formula is density =
volume
c) In 2011, 1 kg of gold costs about $54 000. What is the value of all the gold ever extracted?
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Introduction
Understanding Radicals
a) Label each of the following parts of a radical.
3
8
b) What is the index of
5 ?
c) What is the difference between an entire radical and a mixed radical?
d) Is it possible to write a radical without using the radical symbol
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?
Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 1
Convert each entire radical to a mixed radical.
a)
Prime Factorization Method
Perfect Square Method
b)
Prime Factorization Method
Perfect Square Method
c)
Prime Factorization Method
Perfect Cube Method
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 2
Convert each entire radical to a mixed
radical using the method of your choice.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 3
Convert each mixed radical to an entire radical.
a)
Reverse Factorization Method
Perfect Square Method
b)
Reverse Factorization Method
Perfect Square Method
c)
Reverse Factorization Method
Perfect Cube Method
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 4
Convert each mixed radical to an entire
radical using the method of your choice.
a)
b)
c)
d)
Example 5
Estimate each radical and order them on a number line.
a)
0
5
10
0
5
10
b)
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 6
Simplify each expression without using a calculator.
a)
b)
c)
d)
e)
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Numbers, Radicals, and Exponents
LESSON FOUR - Radicals
Lesson Notes
Example 7
Write each power as a radical.
a)
b)
c)
d)
e)
f)
Example 8
Write each radical as a power.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON FIVE - Exponents I
Lesson Notes
Introduction
Exponent Laws I
a) Product of Powers
General Rule:
b) Quotient of Powers
General Rule:
c) Power of a Power
General Rule:
d) Power of a Product
General Rule:
e) Power of a Quotient
General Rule:
f) Exponent of Zero
General Rule:
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Numbers, Radicals, and Exponents
LESSON FIVE - Exponents I
Lesson Notes
Example 1
Simplify each of the following expressions.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON FIVE - Exponents I
Lesson Notes
Example 2
Simplify each of the following expressions.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON FIVE - Exponents I
Lesson Notes
Example 3
Simplify each of the following expressions.
a)
b)
c)
d)
e)
f)
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am + n
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Numbers, Radicals, and Exponents
LESSON FIVE - Exponents I
Lesson Notes
Example 4
For each of the following, find a value for m that satisfies the equation.
a)
b)
c)
d)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Introduction
Exponent Laws II
a) Negative Exponents
General Rule:
b) Rational Exponents
General Rule:
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 1
Simplify each of the following expressions. Any variables in your final
answer should be written with positive exponents.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 2
Simplify. Any variables in your final answer should be written with positive
exponents.
a)
b)
c)
d)
e)
f)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 3
Simplify each of the following expressions. Any variables in your final
answer should be written with positive exponents.
a)
b)
c)
d)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 4
Simplify. Any variables in your final answer should be written with positive
exponents. Fractional exponents should be converted to a radical.
a)
b)
c)
d)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 5
Simplify. Any variables in your final answer should be written with positive
exponents. Fractional exponents should be converted to a radical.
a)
b)
c)
d)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 6
Write each of the following radical expressions with
rational exponents and simplify.
a)
b)
c)
d)
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 7
A culture of bacteria contains 5000 bacterium cells. This particular type
of bacteria doubles every 8 hours. If the amount of bacteria is represented
by the letter A, and the elapsed time (in hours) is represented by the letter t,
the formula used to find the amount of bacteria as time passes is:
a) How many bacteria will be in the culture in 8 hours?
b) How many bacteria will be in the culture in 16 hours?
c) How many bacteria were in the sample 8 hours ago?
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Numbers, Radicals, and Exponents
LESSON SIX - Exponents II
Lesson Notes
Example 8
Over time, a sample of a radioactive isotope will lose its mass.
The length of time for the sample to lose half of its mass is called
the half-life of the isotope. Carbon-14 is a radioactive isotope
commonly used to date archaeological finds. It has a half-life
of 5730 years.
If the initial mass of a Carbon-14 sample is 88 g, the formula used
to find the mass remaining as time passes is given by:
In this formula, A is the mass, and t is time (in years) since the mass of the sample was measured.
a) What will be the mass of the Carbon-14 sample in 2000 years?
b) What will be the mass of the Carbon-14 sample in 5730 years?
c) If the mass of the sample is measured 10000 years in the future, what
percentage of the original mass remains?
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Answer Key
Numbers, Radicals, and Exponents Lesson One: Number Sets
Reals
Rationals
Integers
Wholes
Introduction: a) The set of natural numbers (N) can be thought of as the counting numbers.
b) The whole numbers (W) include all of the natural numbers plus one additional number - zero.
c) The set of integers (I) includes negative numbers, zero, and positive numbers.
d) The set of rational numbers (Q) includes all integers, plus terminating and repeating decimals.
e) Irrational numbers (Q) are non-terminating and non-repeating decimals.
f) Real numbers (R) includes all natural numbers, whole numbers, integers, rationals, and irrationals.
Example 1: a) I, Q, R b) W, I, Q, R c) Q, R d) N W I Q R e) Q R f) Q R g) Q R h) Q R
Example 2: a) true b) false c) true d) false e) false
Example 3: Rational:
Irrational:
Example 4: a)
Naturals
Irrationals
Neither:
b)
c)
Numbers, Radicals, and Exponents Lesson Two: Primes, LCM, and GCF
Introduction: a) A prime number is a natural number that has exactly two distinct natural number factors: 1 and itself.
b) A composite number is a natural number that has a positive factor other than one or itself.
c) 0 is not a prime number because it has infinite factors. 1 is not a prime number because it has only one factor - itself.
d) Prime Factorization is the process of breaking a composite number into its primes. 12 = 2 × 2 × 3
e) The LCM is the smallest number that is a multiple of two given numbers. LCM of 9 & 12 is 36.
f) The GCF is the largest natural number that will divide two given numbers without a remainder. GCF of 16 & 24 is 8.
Example 1: a) neither b) composite c) prime d) neither
Example 2: a) 24 b) 14 c) 720 d) 504
Example 3: a) 6 b) 13 c) 26 d) 27
Example 4: a) 60 cm b) 12 minutes c) 252 cm
Example 5: a) 5 baskets, with 3 oranges and 2 apples in each. b) 4 groups, with 5 loonies and 2 toonies in each. c) cube edge = 13 mm
Numbers, Radicals, and Exponents Lesson Three: Squares, Cubes, and Roots
Introduction:
a) A perfect square is a
number that can be expressed
as the product of two equal factors.
First three perfect squares: 1, 4, 9
d) A square root is one of two
equal factors of a number.
The square root of 36 is 6.
c)
b) A perfect cube is a number
that can be expressed as the product
of three equal factors.
First three perfect cubes: 1, 8, 27
e) A cube root is one of three
equal factors of a number.
The cube root of 125 is 5.
Number
Perfect Square
Perfect Cube
1
12 = 1
13 = 1
2
22 = 4
23 = 8
3
32 = 9
33 = 27
4
42 = 16
43 = 64
5
52 = 25
53 = 125
6
62 = 36
63 = 216
7
72 = 49
73 = 343
8
82 = 64
83 = 512
9
92 = 81
93 = 729
10
102 = 100
103 = 1000
Example 1: a) 9 b) 9 c) -9 d) 27 e) -27 f) -27
Example 2: a) 16 b) -32 c) -24 d) 1/64 e) 1/12 f) 10
Example 3: a) 2.8284... b) error c) 2 d) -2 e) error, -1.5157...
Example 4: a) 20 b) 1/3 c) -1/4 d) 7/10
The odd root of a negative number can be calculated,
Example 5: a) 26.2 km b) 29.5 km
but the even root of a negative number is not calculable.
Example 6: a) 3054 cm3 b) 10.61 cm
Example 7: a) 2.7 s b) 1.4 m
Example 8: a) 20.28 m b) 160 962 000 kg c) 8.7 trillion dollars
Numbers, Radicals, and Exponents Lesson Four: Radicals
Introduction:
a)
index
3
radical symbol
radicand
8
radical
Example
Example
Example
Example
1:
2:
3:
4:
a)
a)
a)
a)
b)
b)
b)
b)
c)
c)
c)
c)
Example 5: a)
Example 6: a)
b)
c)
c) an entire radical does not
have a coefficient, but a
mixed radical does.
Example 7: a)
b)
c)
d)
Example 8: a)
e)
f)
d)
b)
b) the index is 2
d) Yes. Radicals can be
represented with fractional
exponents.
d)
d)
e)
b)
e)
f)
c)
d)
e)
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f)
Answer Key
Numbers, Radicals, and Exponents Lesson Five: Exponents I
Introduction:
a)
,
,
b)
,
c)
,
d)
b)
c)
d)
e)
f)
Example 2: a)
b)
c)
d)
e)
f)
Example 3: a)
b)
c)
d)
e)
f)
Example 4: a) 5
b) 3
c) 2
d) 7
,
,
e)
f)
,
Example 1: a)
,
,
1,
,
1,
a0 = 1
Numbers, Radicals, and Exponents Lesson Six: Exponents II
Introduction:
a)
,
Example 1: a)
Example 3: a)
Example 5: a)
,
,
,
b)
c)
b)
b)
d)
e)
c)
b)
Example 7: a) 10 000 bacteria
c)
,
f)
d)
d)
,
,
,
Example 2: a)
b)
Example 4: a)
Example 6: a)
b) 20 000 bacteria c) 2500 bacteria
c)
b)
b)
c)
c)
Example 8: a) 69 g
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d)
e)
d)
d)
b) 44 g c) 30%
f)