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Review of Arithmetic
and Algebra
The authors understand and realize that there are wide differences in the mathematical background of readers of this book. Some of you may have taken various courses in calculus and matrix algebra, whereas others may not have taken
any mathematics courses in a long period of time. Because the emphasis of this
book is on statistical concepts and the interpretation of Microsoft Excel and statistical calculator output, no prerequisite beyond elementary algebra is needed.
To assess your arithmetic and algebraic skills, you may want to answer the following questions and then read the review that follows.
Assessment Quiz
Part 1
Fill in the correct answer.
1.
1
=
2
3
2. (0.4)2 =
236
APPENDIX B
3. 1 +
2
=
3
⎛ 1⎞
4. ⎜ ⎟
⎝ 3⎠
5.
REVIEW OF ARITHMETIC AND ALGEBRA
( 4)
=
1
= (in decimals)
5
6. 1 – (–0.3) =
7. 4 ⫻ 0.2 ⫻ (–8) =
⎛ 1 2⎞
8. ⎜ × ⎟ =
⎝ 4 3⎠
⎛ 1 ⎞ ⎛ 1 ⎞
9. ⎜
⎟ +⎜
⎟=
⎝ 100 ⎠ ⎝ 200 ⎠
10.
16 =
Part 2
Select the correct answer.
1. If a = bc, then c =
(a) ab
(b) b/a
(c) a/b
(d) None of the above
2. If x + y = z, then y
(a) z/x
(b) z + x
(c) z – x
(d) None of the above
3. (x3)(x2) =
(a) x5
(b) x6
(c) x1
(d) None of the above
4. x0 =
(a) x
(b) 1
(c) 0
(d) None of the above
ASSESSMENT QUIZ
5. x(y – z) =
(a) xy – xz
(b) xy – z
(c) (y – z)/x
(d) None of the above
6. (x + y)/z =
(a) (x/z) + y
(b) (x/z) + (y/z)
(c) x + (y/z)
(d) None of the above
7. x /(y + z) =
(a) (x/y) + (1/z)
(b) (x/y) + (x/z)
(c) (y +z)/ x
(d) None of the above
8. If x = 10, y = 5, z = 2, and w = 20, then (xy – z2)/w =
(a) 5
(b) 2.3
(c) 46
(d) None of the above
9. (8x4)/(4x2) =
(a) 2x2
(b) 2
(c) 2x
(d) None of the above
10.
X
=
Y
(a)
Y
X
(b)
1
XY
(c)
X Y
(d) None of the above
The answers to both parts of the quiz appear at the end of this appendix.
237
238
APPENDIX B
REVIEW OF ARITHMETIC AND ALGEBRA
Symbols
Each of the four basic arithmetic operations—addition, subtraction, multiplication, and division—is indicated by a symbol:
⫹ add
⫻ or ⋅ multiply
⫺ subtract
⫼ or / divide
In addition to these operations, the following symbols are used to indicate
equality or inequality:
⫽ equals
⫽ not equal
⬵ approximately equal to
⬎ greater than
⬍ less than
ⱖ greater than or equal to
ⱕ less than or equal to
Addition
Addition refers to the summation or accumulation of a set of numbers. In
adding numbers, there are two basic laws: the commutative law and the associative law.
The commutative law of addition states that the order in which numbers are
added is irrelevant. This can be seen in the following two examples:
1+2=3
2+1=3
x+y=z
y+x=z
In each example, which number was listed first and which number was listed
second did not matter.
The associative law of addition states that in adding several numbers, any
subgrouping of the numbers can be added first, last, or in the middle. You
can see this in the following examples:
2 + 3+ 6 + 7 + 4 + 1 = 23
(5) + (6 + 7) + 4 + 1 = 23
5 + 13 + 5 = 23
5 + 6 + 7 + 4 + 1 = 23
In each of these examples, the order in which the numbers have been added
has no effect on the results.
SYMBOLS
Subtraction
The process of subtraction is the opposite or inverse of addition. The operation of subtracting 1 from 2 (i.e., 2 – 1) means that one unit is to be taken
away from two units, leaving a remainder of one unit. In contrast to addition, the commutative and associative laws do not hold for subtraction.
Therefore, as indicated in the following examples:
8–4=4
but
4 – 8 = –4
3 – 6 = –3
but
6–3=3
8–3–2=3
but
3 – 2 – 8 = –7
9–4–2=3
but
2 – 4 – 9 = –11
When subtracting negative numbers, remember that that same result occurs
when subtracting a negative number as when adding a positive number.
Thus:
4 – (–3) = +7
4+3=7
8 – (–10) = +18
8 + 10 = 18
Multiplication
The operation of multiplication is a shortcut method of addition when the
same number is to be added several times. For example, if 7 is to be added 3
times (7 + 7 + 7), you could multiply 7 times 3 to obtain the product of 21.
In multiplication as in addition, the commutative laws and associative are in
operation so that:
a⫻b=b⫻a
4 ⫻ 5 = 5 ⫻ 4 = 20
(2 ⫻ 5) ⫻ 6 = 10 ⫻ 6 = 60
A third law of multiplication, the distributive law, applies to the multiplication of one number by the sum of several numbers. Here:
a(b + c) = ab + ac
2(3 + 4) = 2(7) = = 2(3) + 2(4) = 14
The resulting product is the same regardless of whether b and c are summed
and multiplied by a, or a is multiplied by b and by c and the two products
are added together.
You also need to remember that when multiplying negative numbers, a negative number multiplied by a negative number equals a positive number.
Thus:
(–a) ⫻ (–b) = ab
(–5) ⫻ (–4) = +20
239
240
APPENDIX B
REVIEW OF ARITHMETIC AND ALGEBRA
Division
Just as subtraction is the opposite of addition, division is the opposite or
inverse of multiplication. Division can be viewed as a shortcut to subtraction.
When 20 is divided by 4, you are actually determining the number of times
that 4 can be subtracted from 20. In general, however, the number of times
one number can be divided by another may not be an exact integer value,
because there could be a remainder. For example, if 21 is divided by 4, the
answer is 5 with a remainder of 1, or 5 1/4.
As in the case of subtraction, neither the commutative nor associative law of
addition and multiplication holds for division.
a⫼b⫽b⫼a
9⫼3⫽3⫼9
6 ⫼ (3 ⫼ 2) = 4
(6 ⫼ 3) ⫼ 2 = 1
The distributive law will hold only when the numbers to be added are contained in the numerator, not the denominator. Thus:
a +b a b
= +
c
c c
but
a
a a
≠ +
b +c b c
For example:
6+9 6 9
= + =2+3=5
3
3 3
1
1
=
2+3 5
but
1
1 1
≠ +
2+3 2 3
The last important property of division states that if the numerator and the
denominator are both multiplied or divided by the same number, the resulting quotient will not be affected. Therefore:
80
=2
40
then
5 ( 80 ) 400
=
=2
5 ( 40 ) 200
and
80 ÷ 5 16
=
=2
40 ÷ 5
8
FRACTIONS
Fractions
A fraction is a number that consists of a combination of whole numbers
and/or parts of whole numbers. For instance, the fraction 1/3 consists of only
one portion of a number, whereas the fraction 7/6 consists of the whole number 1 plus the fraction 1/6. Each of the operations of addition, subtraction,
multiplication, and division can be used with fractions. When adding and
subtracting fractions, you must obtain the lowest common denominator for
1 1
each fraction prior to adding or subtracting them. Thus, in adding + ,
3 5
the lowest common denominator is 15, so:
5
3
8
+
=
15 15 15
1 1
− , the same principles applies, so that the lowest
4 6
common denominator is 12, producing a result of:
In subtracting
3
2
1
−
=
12 12 12
Multiplying and dividing fractions do not have the lowest common denominator requirement associated with adding and subtracting fractions. Thus, if
ac
a/b is multiplied by c/d, the result is
.
bd
The resulting numerator, ac, is the product of the numerators a and c, whereas
the denominator, bd, is the product of the two denominators b and d. The
resulting fraction can sometimes be reduced to a lower term by dividing the
numerator and denominator by a common factor. For example, taking:
2 6 12
× =
3 7 21
4
.
7
Division of fractions can be thought of as the inverse of multiplication, so
the divisor can be inverted and multiplied by the original fraction. Thus:
and dividing the numerator and denominator by 3 produces the result
9 1 9 4 36
÷ = × =
5 4 5 1
5
The division of a fraction can also be thought of as a way of converting the
fraction to a decimal number. For example, the fraction 2/5 can be converted
to a decimal number by dividing its numerator, 2, by its denominator, 5, to
produce the decimal number 0.40.
241
242
APPENDIX B
REVIEW OF ARITHMETIC AND ALGEBRA
Exponents and Square Roots
Exponentiation (raising a number to a power) provides a shortcut in writing
numerous multiplications. For example, 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 can be written
as 25 = 32. The 5 represents the exponent (or power) of the number 2,
telling you that 2 is to multiplied by itself five times.
Several rules can be applied for multiplying or dividing numbers that contain
exponents.
Rule 1: xa ⴢ xb = x(a + b)
If two numbers involving a power of the same number are multiplied, the
product is the same number raised to the sum of the powers.
42 ⋅ 43 = (4 ⋅ 4)(4 ⋅ 4 ⋅ 4 ⋅ 4) = 45
Rule 2: (xa)b = xab
If you take the power of a number that is already taken to a power, the result
will be a number that is raised to the product of the two powers. For example,
(42)3 = (42)(42)(42) = 46
a
x
Rule 3: b = x ( a − b )
x
If a number raised to a power is divided by the same number raised to a
power, the quotient will be the number raised to the difference of the powers. Thus:
3⋅3⋅3⋅3⋅3
3
=
= 32
3
3
⋅
3
⋅
3
3
If the denominator has a higher power than the numerator, the resulting
quotient will be a negative power. Thus:
5
3
3⋅3⋅3
1
1
3
=
= 2 = 3−2 =
5
3
⋅
3
⋅
3
⋅
3
⋅
3
9
3
3
If the difference between the powers of the numerator and denominator is 1,
the result will be the number itself. In other words, x1 = x. For example:
3
3⋅3⋅3
3
=
= 31 = 3
2
3
⋅
3
3
If, however, there is no difference in the power of the numbers in the numerator and denominator, the result will be 1. Thus:
a
x
= x a−a = x 0 = 1
a
x
EQUATIONS
Therefore, any number raised to the 0 power equals 1. For example:
3
3⋅3⋅3
3
= 30 = 1
=
3
3⋅3⋅3
3
The square root, represented by the symbol
, is a special power of number, the 1/2 power. It indicates the value that when multiplied by itself, will
produce the original number.
Equations
In statistics, many formulas are expressed as equations where one unknown
value is a function of another value. Thus, it is important that you know how
to manipulate equations into various forms. The rules of addition, subtraction, multiplication, and division can be used to work with equations. For
example, the equation:
x–2=5
can be solved for x by adding 2 to each side of the equation. This results in:
x – 2 + 2 = 5 + 2. Therefore x = 7.
If x + y = z, you could solve for x by subtracting y from both sides of the
equation so that
x + y – y = z – y Therefore x = z – y.
If the product of two variables is equal to a third variable, such as:
x⋅y=z
you can solve for x by dividing both sides of the equation by y. Thus:
x⋅y z
=
y
y
x=
z
y
Conversely, if
x
= z , you can solve for x by multiplying both sides of the
y
equation by y:
xy
= zy
y
x = zy
In summary, the various operations of addition, subtraction, multiplication,
and division can be applied to equations as long as the same operation is performed on each side of the equation, thereby maintaining the equality.
243
244
APPENDIX B
REVIEW OF ARITHMETIC AND ALGEBRA
Answers to Quiz
Part 1
1. 3/2
2. 0.16
3. 5/3
4. 1/81
5. 0.20
6. 1.30
7. –6.4
8. +1/6
9. 3/200
10. 4
Part 2
1. c
2. c
3. a
4. b
5. a
6. b
7. d
8. b
9. a
10. c