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Lesson 5  page 1
Lesson 5
Arithmetic Operations and Precedence
This book assumes that you have previously learned the four basic arithmetic operations (addition, subtraction,
multiplication, and division), and that you can do these operations with small whole numbers in your head or with pencil and
paper, and with larger numbers using a calculator. In Lesson 4, you learned a fifth operation, exponentiation, or raising a
number to a power. Below is a reminder about some key points about the basic operations:
Addition and Subtraction
Addition is an operation that combines groups. Two groups that were formerly separate are now considered together.
combined with
becomes
The answer to an addition problem is called the sum or total.
Subtraction is an operation that breaks up groups. A group that was formerly together is separated.
remove
is now
The answer to a subtraction problem is called the difference.
Addition and subtraction are related, because a subtraction “undoes” an addition. We consider them opposite operations.
Example: Write the four related addition/subtraction equations using the numbers 5, 3, and 8.
5+3=8
3+5=8
8–3=5
8–5=3
Addition is commutative, meaning that 5 + 3 has the same answer as 3 + 5.
However, subtraction is not commutative, because 8 – 3 does not have the same answer as 3 – 8.
Multiplication and Division
Multiplication is an operation that combines duplicate groups.
3 copies of the group
looks like
, and combined becomes
.
The answer to a multiplication problem is called the product. Multiplication can be written with the x symbol, the • symbol,
or by putting one or both numbers in parentheses right next to the other.
Division breaks a large group into duplicate groups.
broken into 3 matching groups:
, or
The answer to a division problem is called the quotient. Division can be written with the ÷ symbol or the / symbol.
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.
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Lesson 5  page 2
Multiplication and division are related, because a division “undoes” a multiplication. We consider them opposite operations.
Example: Write the four related multiplication/division equations using the numbers 5, 3, and 15.
5 • 3 = 15
3 • 5 = 15
15 ÷ 3 = 5
15 ÷ 5 = 3
Multiplication is commutative, meaning that 5 • 3 has the same answer as 3 • 5.
However, division is not commutative, because 15 ÷ 3 does not have the same answer as 3 ÷ 15.
Division and Zero
The related multiplication/division equations explain why division by zero is not allowed in mathematics. Let’s try writing the
equations that go with the problem 11 ÷ 0 = ?.
0 • ? = 11
? • 0 = 11
11 ÷ ? = 0
11 ÷ 0 = ?
The related multiplication problems are impossible! Any number times zero is always zero, and never ever eleven. The
problem 11 ÷ 0 can’t have an answer, because there is no number we can fill in for the ?. We say division by zero is
undefined. This is not a temporary restriction that will be removed in a higher mathematics class – it is fundamentally not
possible.
On the other hand, the problem 0 ÷ 11 is possible, and the answer is 0, because the related multiplication problems are just
fine – it is true that 0 • 11 = 0. The relate
11 • 0 = 0
0 • 11 = 0
0 ÷ 0 = 11 This one is a little odd, however. Is 0 ÷ 0 really 11? NO, keep reading.
0 ÷ 11 = 0
Now 11 is an arbitrary example number, so it’s strange that it would be the answer to 0 ÷ 0. Let’s try another number – how
about 42?
42 • 0 = 0
0 • 42 = 0
0 ÷ 0 = 42 OK, wait a minute. How can the answer to the same problem
0 ÷ 42 = 0
be 11 one minute and 42 the next?
The answer is that division by zero (ANYTHING ÷ 0)
is never never allowed.
The problem 0 ÷ 0 is UNDEFINED.
Remembering the Zero Division Rule
Write the division sign as a fraction bar, like this:
0
0
11
11
not possible
0
Picture this as a stick figure of a person. 0 is the head,
and the 11 can be the legs. When 0 is on top, the
person is right-side up, and the problem has an
answer (the answer is 0). When the person is upside
down, the problem has no answer.
Some people remember it this way: 0/K and N/0.
Or try the problems on your calculator, and see the
results:
0/11= 0
© 2010 Cheryl Wilcox
11/0=ERROR
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Lesson 5  page 3
Ranking the Operations
The operations have a relative importance, based on how powerful they are.
Exponents and Roots* are the most powerful operations. They have the highest rank.
*The operation of raising a number to a power, also has an opposite operation, which is called “taking the root.” You don’t have to worry
about that now, we’ll cover it later in the course, but you should know that the operations come in pairs of opposites.
Multiplication and Division are the middle rank operations. They outrank the addition/subtraction pair, but are
lower than the exponent/root pair.
Addition and Subtraction are the least powerful operations and have the lowest rank.
When more than one operation appears in an expression, higher ranking operations are done before lower ranking
operations. This is usually called the order of operations.
Sometimes it’s necessary to do a lower-ranking operation before one that’s ranked higher. This is indicated with
parentheses around the numbers and operation.
Parentheses override rank, and operations in parentheses* are done first.
*Or other grouping symbols such as absolute value, which we’ll learn later.
You may have learned the acronym PEMDAS to help you remember the order of operations.
Court Etiquette
When you enter the court, greet each person you encounter. However, you MUST follow correct court precedence in
your greetings:
First, greet any members of the
royal family, in the order you
encounter them.
Next, greet any knights or
dragons, in the order you
encounter them.
Finally, greet any
commoners, in the order you
encounter them.
But there is one more rule to remember:
The gold medallion is a sign of great honor! Greet all medallion wearers first, in order of their rank. Then proceed to
those not wearing the medallion.
Example: Identify the operations and their rank, then evaluate 3 • 24.
The first operation you encounter is multiplication, and the second is exponentiation.
First do exponentiation, the do multiplication.
3 • 24
= 3 • 16
= 48
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Lesson 5  page 4
Example: Identify the operations and their rank, then evaluate 2 • 6 + 2 • 13.
The first operation you encounter is multiplication, next is addition, and the third is multiplication.
First, do the first multiplication, then do the second multiplication, finally do the addition.
2 • 6 + 2 • 13
= 12 + 2 • 13
= 12 + 26
= 38
You may have noticed that the evaluation we did above is just like evaluating the perimeter formula P = 2L + 2W, where the
length is 6 and the width is 13. When we derived that formula in Lesson 3, we thought about the rectangle as two lengths
and two widths, then added to find the perimeter. The order of operations follows that train of thought. First double the
length, then double the width, then add the two numbers to find the perimeter.
+
The next example is like the second version of the formula, P = 2(L + W). In deriving that formula, we thought of the
perimeter of the rectangle as two copies of the length plus width.
and
This would look like (L + W) + (L + W), or more simply 2(L + W). Since addition is ranked lower than multiplication, we
require parentheses to override the ranking and do the addition first.
Example: Identify the operations and their rank, then evaluate 2(6 + 13).
The first operation you encounter is multiplication, next is addition, BUT the addition is in parentheses.
First, do the addition in parentheses, then do multiplication.
2(6 + 13)
= 2(19)
= 38
This is like the second version of the formula, P = 2(L + W). Again the length is 6 and the width is 13. Since this version of
the formula only requires two operations instead of three, it is a bit quicker to evaluate. The answer, of course, is the same
either way.
© 2010 Cheryl Wilcox
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Lesson 5  page 5
Example: Evaluate 2 + 4 • 53
2 + 4 • 53
= 2 + 4 • 125
= 2 + 500
= 502
One of the advantages of using a scientific calculator (rather than a four-function calculator) is that the correct order of
operations is programmed into the scientific calculator. You can enter the expression in the order it is written, and so long as
you do not press = or ENTER until the end, the calculator will do the operations in the correct order.
Example: Evaluate 2 + 4 • 53 using a scientific calculator.
Enter 2 + 4 x 5 ^ 3 = and read the answer 502.
The calculator also has keys for parentheses:
Example: Evaluate 4 + 3(17 – 8)2 with pencil and paper and then check your answer using a scientific calculator.
4 + 3(17 – 8)2
= 4 + 3(9)2
= 4 + 3 • 81
= 4 + 243
= 247
Calculator: Enter 4 + 3 x ( 17 – 8 ) ^ 2 = and read the answer 247.
Example: Evaluate 6 ÷ 3(2) with pencil and paper and then check your answer using a scientific calculator.
6 ÷ 3(2)
The parentheses have no operation inside to evaluate first. It is just another way to write multiplication.
You could rewrite the expression to make it clearer:
6÷3•2
Since multiplication and division have the same rank, we do them in the order encountered.
6÷3•2
= 2•2
= 4
Calculator: Enter 6 ÷ 3 x 2 = and read the answer 4.

© 2010 Cheryl Wilcox
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Lesson 5  page 6
Lesson 5: Arithmetic Operations and Precedence
Worksheet
Name
Write each step, then check your answer with a calculator.
1. Evaluate 15 – 32
Operation Precedence
P arentheses
E xponents/Roots
M ultiplication / D ivision
A ddition / S ubtraction
2. Evaluate 2 + 5(30 – 33)
First do operations within parentheses in order of rank.
3. Evaluate 200 – 3(4 + 6 • 3)
4. Evaluate 52 – (14 + 6)(7 – 3 • 2)
5. Evaluate (4 + 6)3 – (8 + 7)2
6. Evaluate 94 – 32 ÷ 4 • 2
7. Evaluate (32 – 9) ÷ (32 – 8)
Is the answer different if you do 94 – 32 ÷ (4 • 2)?
Evaluate (32 – 8) ÷ (32 – 9)
© 2010 Cheryl Wilcox
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Lesson 5  page 7
Lesson 5: Arithmetic Operations and Precedence
Homework 5A
1. Estimate the number of marbles in the plastic box, if each
compartment contains two layers below similar to the top
layer pictured (3 layers of marbles).
Name
2. a. Write 138 as a multiplication.
b. Evaluate 138.
c. Write the number in words.
d. Round the number to 1 significant figure.
3. Find the perimeter (P) and area (A) of the rectangle with
the given dimensions, length (L) and width (W).
a. L = 19 feet, W = 12 feet.
P=
A=
4. Find the volume (V) of a box with the given dimensions.
a. L = 42 inches, W = 38 inches, H = 18 inches
V=
b. L = 90 m, W = 65 m, H = 30 m
V=
b. Subtract 1 foot from each side of rectangle a for the
dimensions of rectangle b.
L = _____ ft, W = _____ ft
c. Add 1 m to each side of box b for the dimensions of box c.
P=
L = _____ m, W = _____ m, H = _____ m
V=
A=
5. Find the area and perimeter of the figure. Grid is cm2.
6. a. Write the four related addition/subtraction equations
using the numbers 4, 7, and 11.
b. Write the four related multiplication/division equations
using the numbers 4, 7, and 28.
Area __________
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Perimeter ____________
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Lesson 5  page 8
7. Evaluate each expression.
a. 4 • 92
b. (4 • 9)2
c. 5 + 6 • 7
d. (5 + 6) • 7
e. 22 – 42 + 18 – 32
f. (22 – 4)2 + (18 – 3)2
g. 102 ÷ 5 • 2
h. 102 ÷ (5 • 2)
8. The twenty-foot equivalent unit (often TEU or teu) is an
inexact unit of cargo capacity often used to describe the
capacity of container ships and container terminals… One
TEU represents the cargo capacity of a standard intermodal
container, 20 feet (6.1 m) long and 8 feet (2.4 m) wide.
There is a lack of standardization in regards to height,
ranging between 4.25 and 9.5 feet (1.30 and 2.9 m), with
the most common height being 8.5 feet (2.6 m).
9. Round the numbers in the presentation using your own
judgment.
a. “The largest container shipping company is APM-Maersk,
which has a capacity of 2,031,886 TEU. The total world fleet
capacity is 13,108,859 TEU.”
a. Find the volume of a container with
L = 20 feet, W = 8 feet, and H = 5 feet.
b. Emma Maersk is the the largest container ship ever built.
Her cargo capacity is 14,875 TEU and the cost of building
was $144,885,000.
b. Find the volume of a container with
L = 20 feet, W = 8 feet, and H = 9 feet.
10. Draw a line 2 cm long and a line 3 cm long. Measure each to the nearest inch: _____________________________
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Lesson 5  page 9
Lesson 5: Arithmetic Operations and Precedence
Homework 5A Answers
1. Estimate the number of marbles in the plastic box, if each
compartment contains two layers below similar to the top
layer pictured (3 layers of marbles).
2. a. Write 138 as a multiplication.
13 • 13 • 13 • 13 • 13 • 13 • 13 • 13
b. Evaluate 138.
815,730,721
c. Write the number in words.
815 million, 730 thousand, 721
d. Round the number to 1 significant figure.
800,000,000
3. Find the perimeter (P) and area (A) of the rectangle with
the given dimensions, length (L) and width (W).
a. L = 19 feet, W = 12 feet.
P = 2(19 ft + 12 ft) = 2(31 ft) = 62 feet
A = (19 ft)(12 ft) = 228 ft2
b. Subtract 1 foot from each side of rectangle a for the
dimensions of rectangle b.
L = 18 ft, W = 11 ft
P = 2(18 ft + 11 ft) = 2(29 ft) = 58 ft
A = (18 ft)(11 ft) = 198 ft2
5. Find the area and perimeter of the figure. Grid is cm2.
4. Find the volume (V) of a box with the given dimensions.
a. L = 42 inches, W = 38 inches, H = 18 inches
V = (42 in)(38 in)(18 in) = 28,728 in3
28,728 cubic inches
b. L = 90 m, W = 65 m, H = 30 m
V = (90 m)(65 m)(30 m) = 175,500 m3
175,500 cubic meters
c. Add 1 m to each side of box b for the dimensions of box c.
L = 91 m, W = 66 m, H = 31 m
V = (91 m)(66 m)(31 m) = 186,186 m3
186,186 cubic meters
6. a. Write the four related addition/subtraction equations
using the numbers 4, 7, and 11.
4 + 7 = 11
11 – 7 = 4
7 + 4 = 11
11 – 4 = 7
b. Write the four related multiplication/division equations
using the numbers 4, 7, and 28.
Area 16 cm2
© 2010 Cheryl Wilcox
Perimeter 22 cm
4 • 7 = 28
28 ÷ 7 = 4
7 • 4 = 38
28 ÷ 4 = 7
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Lesson 5  page 10
7. Evaluate each expression.
a. 4 • 92
b. (4 • 9)2
4 • 92 = 4 • 81 = 324
c. 5 + 6 • 7
(4 • 9)2 = (36)2 = 1296
d. (5 + 6) • 7
5 + 6 • 7 = 5 + 42 = 47
e. 22 – 42 + 18 – 32
(5 + 6) • 7 = (11) • 7 = 77
f. (22 – 4)2 + (18 – 3)2
22 – 42 + 18 – 32 = 22 – 16 + 18 – 32
(22 – 4)2 + (18 – 3)2 = (18)2 + (18 - 3)2
= 22 – 16 + 18 – 9 = 6 + 18 – 9 = 24 – 9 = 15
= (18)2 + (15)2 = 324 + (15)2 = 324 + 225 = 549
g. 102 ÷ 5 • 2
h. 102 ÷ (5 • 2)
102 ÷ 5 • 2 = 100 ÷ 5 • 2 = 20 • 2 = 40
8. The twenty-foot equivalent unit (often TEU or teu) is an
inexact unit of cargo capacity often used to describe the
capacity of container ships and container terminals… One
TEU represents the cargo capacity of a standard intermodal
container, 20 feet (6.1 m) long and 8 feet (2.4 m) wide.
There is a lack of standardisation in regards to height,
ranging between 4.25 and 9.5 feet (1.30 and 2.9 m), with
the most common height being 8.5 feet (2.6 m).
102 ÷ (5 • 2) = 102 ÷ (10) = 100 ÷ 10 = 10
9. Round the numbers in the presentation using your own
judgment.
a. “The largest container shipping company is APM-Maersk,
which has a capacity of 2,031,886 TEU. The total world fleet
capacity is 13,108,859 TEU.”
about 2 million TEU
over 13 million TEU
a. Find the volume of a container with
L = 20 feet, W = 8 feet, and H = 5 feet.
V = (20 ft)(8 ft)(5 ft) = 800 cubic feet
b. Emma Maersk is the the largest container ship ever built.
Her cargo capacity is 14,875 TEU and the cost of building
was $144,885,000.
about 15 thousand TEU, $145 million
a. Find the volume of a container with
L = 20 feet, W = 8 feet, and H = 9 feet.
V=
(20 ft)(8 ft)(9 ft)
= 1440 cubic
feet
10. Draw a line 2 cm long and a line 3 cm long. Measure each to the nearest inch: both lines are about 1 inch long
© 2010 Cheryl Wilcox
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Lesson 5  page 11
Lesson 5: Arithmetic Operations and Precedence
Homework 5B
1. Estimate the number of cannonballs in the whole stack
(not just the ones you can see). Each layer is complete (the
stack is not hollow) and it looks the same on the other side.
Name
2. a. Write 127 as a multiplication.
b. Evaluate 127.
c. Write the number in words.
d. Round the number to 1 significant figure.
3. Find the perimeter (P) and area (A) of the rectangle with
the given dimensions, length (L) and width (W).
a. L = 48 inches W = 36 inches.
4. Find the volume (V) of a box with the given dimensions.
a. L = 2 cm, W = 3 cm, H = 1 cm
V=
P=
b. L = 20 cm, W = 30 cm, H = 10 cm
A=
V=
b. Start with rectangle a (above). Add 1 inch to the length
and subtract 1 inch from the width for the dimensions of
rectangle b.
c. L = 200 cm, W = 300 cm, H = 100 cm
V=
L = _____ in, W = _____ in
P=
A=
5. Find the area and perimeter of the figure. Grid is cm2.
6. a. Write the four related addition/subtraction equations
using the numbers 2, 6, and 8.
b. Write the four related multiplication/division equations
using the numbers 2, 6, and 12.
Area __________
© 2010 Cheryl Wilcox
Perimeter ____________
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Lesson 5  page 12
7. Evaluate each expression.
a. 35 – 33
b. (35 – 3)3
c. 6 + 10 ÷ 2
d. (6 + 10) ÷ 2
e. 3 • 54
f. (3 • 5)4
g. 85 – 92 + 12 – 32
h. (85 – 9)2 + (12 – 3)2
8. Wikipedia: For larger volumes of liquid, one measure
commonly used in the media in many countries is the
Olympic-size swimming pool. A large Olympic swimming
pool with dimensions 50 m × 25 m × 2 m (approx 164 ft × 82
ft × 6.6 ft) holds …
9. Round the numbers in the presentation using your own
judgment.
a. a. Every day, the well is leaking almost 4 Olympic-sized
swimming pools of oil, up to 105,387 gallons per hour.
a. Find the volume of the pool in cubic meters.
b. Find the volume of the pool in cubic feet. (Round 6.6 feet
to 7 feet, if you like).
b. …the combined purchases of London based
organisations signed up to the Green Procurement Code
diverted 72,490 tonnes of waste from landfill. This is the
equivalent of 187 Olympic sized swimming pools and
represents a saving of 22,643 tonnes of CO2.
c. Use this volume converter to convert the cubic meters to
gallons (U.S. liquid). How many gallons of water in an
Olympic-sized swimming pool?
10. Draw a line 11 cm long and a line 9 cm long, then measure each to the nearest inch:
© 2010 Cheryl Wilcox