1.4 | Order of Operations (BEDMAS)

Transcription

Chapter 1 | Review of Basic Arithmetic
29. Cheng walked 5 41 km in 1 21 hours. How many kilometres did he walk in 1 hour?
30. It took 15 41 hours to complete three-fourths of a project. How long did it take to complete the entire project?
31. The product of two numbers was 9. If one number is 3 43 , what was the other number?
32. If a wire that is 43 43 cm long is cut into several 1 41 cm equal pieces, how many pieces were there?
33. A stack of plywood sheets measures 49 21 inches high. If each plywood sheet is three-fourths of an inch thick, how many
sheets of plywood are in the stack?
34. A garment factory has 40 41 metres of cotton fabric. If 1 43 metre of the fabric is required for a dress pattern, how many
dresses can be made?
35. A bottle contained 80 mg of medicine. Each dose of the medicine is 2 21 mg. How many doses were there in the bottle?
36. It took two-thirds of an hour for a machine to make one component. How many components can be made in 40 hours?
37. A company identified one-twentieth of the 320 bulbs that it received from a supplier as being defective. How many bulbs
were not defective?
38. Matthew received a bonus of $6850. He spent two-thirds of this amount on a vacation. How much did he have left?
39. Three software programmers worked 17 21 hours, 25 43 hours, and 11 41 hours each, to develop an e-commerce site. If each
of them was paid $18 per hour, how much did they receive in total?
40. It took three consultants 27 43 hours, 21 41 hours, and 18 21 hours each, to design a product. If each of them was paid $55
per hour, what was the total amount paid to them?
1.4 |
Order of Operations (BEDMAS)
When arithmetic expressions contain multiple operations with brackets, exponents, divisions,
multiplications, additions, and subtractions, the arithmetic operation is performed in the following
sequence:
1. Perform all operations within the brackets. If there is more than one bracket, start with the
innermost bracket and move outwards to complete all the brackets.
2. Perform operations with exponents.
3. Perform the necessary divisions and multiplications in the order in which they appear from
left to right.
4. Complete the operation by performing the necessary additions and subtractions in the order
in which they appear.
The order of operations: Brackets, Exponents, Divisions, Multiplications, Additions, Subtractions
can be remembered by the acronym, BEDMAS.
Example 1.4(a)
Computing Arithmetic Expressions by Following the Order of Operations.
Compute the following arithmetic expressions:
(i) 6 + 4 # 50 ' (8 - 3)2 - 1
(ii) 12 + 32 [(8 # 5) / 5] - 7 + 2
23
24
6 + 4 # 50 ' (8 - 3)2 - 1
= 6 + 4 # 50 ' (8 - 3)2 - 1 Work on the operations in the bracket
2
= 6 + 4 # 50 ' (5) - 1
Perform the operation with the exponent
= 6 + 4 # 50 ' 25 - 1Perform multiplication
= 6 + 200 ' 25 - 1
Perform division
= 6 + 8 - 1
Perform addition
= 14 - 1 Perform subtraction
= 13
(ii) 12 + 32 [(8 # 5) / 5] - 7 + 2
= 12 + 32 [(8 # 5) / 5] - 7 + 2
Work on the operations in the inner bracket
2
= 12 + 3 (40 / 5) - 7 + 2
Work on the operations in the outer bracket
2
= 12 + 3 (8) - 7 + 2
Perform the operation with the exponent
= 12 + 9 (8) - 7 + 2
Perform multiplication
= 12 + 72 - 7 + 2
Perform addition and subtraction
= 79
(i)
Order of arithmetic
operations:
Brackets
Exponents
Divisions
Multiplications
Additions
Subtractions

A number with no sign
is considered to be
positive, (+).
For example, 5 = +5
Signed Numbers
Signed numbers are either positive numbers, numbers greater than zero (for example, +6, +15) or
negative numbers, numbers less than zero (for example, -4, -12).
Positive numbers may or may not have a positive (plus, "+") sign. When signed numbers are added,
subtracted, multiplied, or divided, the result will be a number with a sign.
Addition and Subtraction of Signed Numbers
The following are rules to be followed while adding or subtracting signed numbers.
Adding Two Signed Numbers
a. If the the signs of the two numbers are the same: disregard the sign of the numbers, add the
numbers, and keep the common sign.
For example,
■ Adding +7 and +3
Two signs should not be
written next to each other.
Brackets should be used to
separate the two signs.
For example,
–4 – (+2) instead of –4 – +2
+8 + (–5) instead of +8 + –5
+10(–2) instead of +10 × –2
■ Adding -6 and -2
= +(7 + 3)
= -(6 + 2)
= +10
= -8
b. If the signs of the two numbers are different: disregard the sign of the numbers, subtract the
smaller number from the larger number, and keep the sign of the large number.
For example,
■ Adding +8 and -15
= +8 + (-15)
= -(15 - 8)
= -7
■ Adding -3 and +7
= -3 + (+7)
= +(7 - 3)
= +4
Subtracting Two Signed Numbers
Change the subtraction (minus) sign to addition (plus) sign and change the sign of the number being
subtracted. Then follow the above addition rules.
For example,
■ Subtracting -12 from +18
= +18 - (-12)
= +18 + (+12)
= +(18 + 12)
= +30
■ Subtracting -5 from -7
= -7 - (-5)
= -7 + (+5)
= -(7 - 5)
= -2
■ Subtracting -6 from -2
= -2 - (-6)
= -2 + (+6)
= +(6 - 2)
= +4
Multiplication and Division of Signed Numbers
The following are rules to be followed while multiplying or dividing two signed numbers:
Multiplying two
signed numbers
a. If the sign of the two numbers are the same: the final answer will be positive.
(+) (+) = (+)
(−) (−) = (+)
(+) (−) = (−)
(−) (+) = (−)
For example,
■
■
■(-5)(-4) = +20
■ --123 = +4
(+5)(+4) = +20
+ 12
= +4
+3
b. If the sign of the two numbers are different: the final answer will be negative.
For example,
■
■
Dividing two
signed numbers
■
+ 12
= -4
-3
■ (-5)(+4) = -20
■ -+123 = -4
- 25 = - 5 or - 5 3
3
+ 15
■
(+5)(-4) = -20
+ 30 = + 15 = - 15 or - 15 2
2
-4
-2
Note: When multiplying or dividing more than two signed numbers, group them into pairs and
determine the sign using the signed number rules.
For example,
■
■■ (-3) (-2) (+4) (-1) (-5)
= (6) (-4) (-5)
= (-24) (-5) = 120
^- 15h^+ 8h^- 50h
^- 25h^14h
=
-^15 # 8) (- 50h
+ 15 # 8 # 50
=
- 25 # 14
- ^25 # 14h
22
11
15
15##88##50
50
15
15##88##22
-====
25
25##14
14
14
14
11
77
120
15 # 8
==7
7
Exponential Notation

Exponents provide a shorter way of representing the products of repeated numbers.
For example, when 2 is multiplied 5 times, in standard notation, it is represented as:
2#2#2#2#2
However, when 2 is multiplied 100 times, it would be tedious to represent it using the standard
notation. The shorter way to represent this repeated multiplication is by using exponents.
exponent
25
base
When 2 is multiplied 5 times, it is represented as 25 using exponents.
Similarly if 'a' is multiplied 'n' times it would be represented as an, where 'n' is a positive integer.
a # a # a # a # a # f # a = an
1 444444 2 444444 3
n factors of a
To convert an exponential notation to a standard notation, expand the notation to show the
repeated multiplication. The number in the exponent shows the number of times the base is multiplied.
25
26
For example, 84 = 8 # 8 # 8 # 8
83 = 8 # 8 # 8
82 = 8 # 8
When the exponent is 1, the result is the number itself.
For example, 81 = 8.
This is represented by the formula a1 = a.
When the exponent is 0, the result is 1.
For example, 30 = 1.
This is represented by the formula a0 = 1.
When the exponent is negative, it is represented by a-n =
Negative exponent
means dividing by
that many number
of factors instead of
multiplying.
n
a = a # a # a # a # a #…# a
1
1
=
a # a # a # a # a # ... # a
an
a-n =
1
, where 'n' is a positive integer.
an
(multiplication of 'n' factors of 'a')
(division of 'n' factors of 'a')
Therefore, an and a-n are reciprocals.
For example, ■■ 8
■■ 8
■■ 8
-1
=
1
1
=
1
8
=
1
8#8#8
-2
8
1
1
= 2 =
8#8
8
-3
=
1
8
3
A positive number with a
negative exponent will not result
in a negative answer.

Fractions with Exponents

When a fraction has a positive exponent, the number in the exponent indicates the number of
times the numerator is multiplied by itself and the number of times the denominator is multiplied
by itself.
For example,
■■
2#2#2
2 3
2 2 2
c m = ` 5j` 5j` 5j = 5 # 5 # 5
5
■■
2
3
4
5
=
2#2#2#2
3#3#3#3#3
When a fraction has a negative exponent, change the fraction to its reciprocal and drop the sign.
After this change, the number in the exponent indicates the number of times the numerator and
denominator is multiplied.
For example,
5 3
2 -3
5#5#5
c m = b l = ` 5j` 5j` 5j =
5
2
2#2#2
2 2 2
Note: The reciprocal of
2
5
is .
5
2
Calculator Method to Solve Problems

The exponent key on different calculators can be identified by symbols such as yx, xy, , etc. The sequence
of operations to calculate the exponents also depends on the calculator. In this section, you will learn to use
the Texas Instruments BA II Plus calculator to solve exponents and order of operations problems.
^
Example 1.4(b)
Calculating Exponents using Texas Instruments BA II Plus Calculator
Calculate: (i) 164 (ii) 5-4
(iii) (1.04)4 - 1
(iv) 1 - (1.005)-4
(i) 164
Solution

(ii) 5-4
Enter 16
Enter 5
x
x
Press the y key
Press the y key
Enter the exponent value 4
Enter the exponent value - 4
Press the equal key
Press the equal key
(iii) (1.04)4 - 1
(iv) 1 - (1.005)-4
Solve using BEDMAS order of operations
Enter 1.04
x
Press the y key
Enter 1.005
x
Press the y key
Enter the exponent value 4
Enter the exponent value −4
Press the minus key
Press the equal key
Enter 1
Press the plus/minus key to
make the result negative
Press the addition key
Press the equal key
Enter 1
Press the equal key
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28
Example 1.4(c)
Using the Store and Recall Functions to Solve Order of Operation Problems
RS
-12 R
X SS
SS1 - S 1 + 0.03
SS
4
(ii) 2000SS
SS
0.03
S
X
S
SS
4
T
T
R
RS
36
X - 1 SS
SS S 1 + 0.05
12
SS
Solve: (i) 1380SS
SS
S 0.05 X
SS
S
12
T
T
Solution
1.4 |
RS
-12 R
X SS
S1 - S 1 + 0.03
SS
4
(ii) 2000SS
SS
SS
0.03 X
S
S
S
4
T
T
R
RS
36
X - 1 SS
S S 1 + 0.05
12
SS
(i) 1380SS
SS
SS
0.05 X
S
S
S
12
T
T
Exercises Answers to the odd-numbered problems are available at the end of the textbook
Evaluate Problems 1 to 18 by following the order of operations (BEDMAS).
1. a. 45 - (5 + 7)2
b. 18 - 8 ÷ 2 + 6
3
2. a. 32 + (9 - 5) b. 4 × 6 ÷ 2 - 5
3. a. 128 ÷ (-16) ÷ (-2)
b. 4(-32) + 4
4. a. 200 ÷ (-10) ÷ 5
b. 20(5 - 22) + 5
5. a. (-5)(-3)(12)(-3)(-1) b. 31 - [(15 ÷ 3) # 32 ÷ 22(26 ÷ 13)]
6. a. (-1)(-3)(2)(-5)(-6) b. 112 - 4 # 54 ÷ (5 - 2)3 - 3
7. a. [(1 + 12)(1 - 5)]2 ÷ (5 + 3 # 22 - 4) b. (-8)(2)(3)(-4)(-1)
8. a. 3(72 + 2 # 15 ÷ 3) - (1 + 3 # 4)2
b. (12)(1)(-7)(4)(-2)
2
9.
8 # 7 - 16 + 3 # 2
2 - _59 - 7 i ' 2 # 6
5
2
2
11.
10.
2
3 # ^6 # 4 ' 8h + 157 - 11
2
3
2
6 + (8 - 5) - (6 + 13) 2
60 ' 12 + 95 - ^- 2h # ^- 11hC
- 52 + ^56 ' 8h # 9 + 6
12.
3
32 ' (- 8) - 9 + (127 - 5 ) # 2
6^- 54h ' 6 + 33@ ' 6^- 8h # ^- 3h@
6^- 6h # 7 + ^- 12h ' ^- 4h + 3@ ' 6
6^- 24h ' 8 + 4@ ' 614 - 16 + 3@
16 + 4^ 3h
^16 + 4h 3
+
15.
10 - 4 + 1 10 - ^4 + 1h
13.
17. 14 - 3 "6^6 - 9h^- 4h + 12@ ^- 2h, 14.
16.
8 "9 - ^- 2h - 7 64 + ^- 1h@,
- 12 + 4 6- 4 - ^- 3h@ ' 7
2^6 + 4h
2 ^ 6h + 4
+
24 - ^7 + 3h 24 - 7 + 3
18. 5^- 4h - 3 "^- 9 + 6h + ^- 3h - 4 62^- 1h - 7@,
Evaluate the following exponents:
-2
2
3 3
3 -3
b. c m
c. 8-2
20. a. c m b. c 4 m 19. a. b 2 l 5
4
3
7
Evaluate the following problems and round your answer to two decimal places:
240
7
m
b. 1500 b1 + 0.04 # l
21. a. 4500 c1 + 0.005 #
365
12
22. a. 600 c1 + 0.002 #
23. a.
24. a.
180
m
365
2500
150 c1 + 0.001 #
m
365
1250
160 c1 + 0.02 #
m
365
5
B
8
25. a. 3000 ^1.06h - 1 0.06
10
26. a. 8008^1.04h - 1B 0.04
27. 5400(1.005)8
b. 960 b1 + 0.05 # 3 l
12
b.
b.
1800
b1 + 0.05 #
5
l
12
2700
b1 + 0.04 #
7
l
12
b.
1400 8^1.03h20 - 1B
0.03
b.
7508^1.02h25 - 1B
0.02
28. 1275(1.03)7
29. 2400(1.02)-10
30. 4650(1.04)-6
31.
240081 - ^1.02h 8B
0.02
32.
400 81 - ^1.05h
0.05
33.
950 81 - ^1.03h
0.03
34.
1200 81 - ^1.04h
0.04
- 15
3000 #
1 - Q1 + 0.01 V-32
#(1 + 0.01)
0.01
43
1.5 |
B
B
24
36. 1800# Q1 + 0.075 V - 1 # (1 + 0.075)
0.075
38. 160 # 1 - Q1 + 0.0625 V
0.0625
-12
#(1 + 0.0625)
55
40. 1355(1 + 0.055) 6
39. 80,000(1 + 0.02) 3
41. 275,000(1 + 0.01)-
- 12
- 20
B
20
35. 500 # Q1 + 0.025 V - 1 #(1 + 0.025)
0.025
37.
c. 6-3
43
3
42. 2650(1 + 0.035)-
29
6
Averages
Simple Arithmetic Average (Arithmetic Mean)

The simple arithmetic average of numbers is also often called the arithmetic mean (or mean) of numbers. It is the
sum of all the values of the terms divided by the number of terms added. The answer will always be larger than
the lowest value amongst the terms and smaller than the highest value amongst the terms.
Changing the order of the numbers does not change the average of the numbers and the average of
a list of integers is not necessarily an integer.
29