Math Review
Multiplication1
The following ways of symbolizing multiplication are all exactly the same. They
all mean to multiply one number by the other.
axb
2x3=6
ab
23=6
a*b
2*3=6
ab
ab
a(b)
2(3) = 6
(a)(b)
(2)(3) = 6
(a)(b)(c)
(2)(3)(5) = 30
Most calculators and computer programs, including SPSS, use the asterisk (*) to
symbolize multiplication.
Division
The following ways of symbolizing division are all exactly the
same. They all mean to divide the top number (the numerator)
by the bottom number (the denominator).
a/b
a
b or
1
a b
10/2 = 5
10
2 =5
This math review is drawn from math reviews in: Garner, Roberta (2005) The Joy of Stats. Peterborough, Ontario:
Broadview Press; Healey, Joseph H. (2005) Statistics. A tool for Social Research. 7th edition. Belmont, CA: Thomson
Wadsworth; Howell, David C. (2004) Fundamental Statistics for the Social Sciences. 5th edition. Belmont, CA: BrooksCole.
Fractions & Percents
Fractions are proportions: 36/100 could mean you got 36 correct answers on an
exam out of a total of 100 questions.
To turn this into a percent, do the division, then multiply by 100 (move the decimal
two places to the right):
36/100 = .36
.36 * 100 = 36 percent or 36%
All fractions can be expressed as decimals. If you can’t do it in your head, do the
division on a calculator. To turn a fraction into decimal, divide the top number
(numerator) by the bottom number (denominator) (this is waaaay easier with a
calculator!):
¾ = 34
= .75
You should know these common fractions and decimal places, and be able to go
back and forth easily:
½ = .5 = 50%
¼ = .25 = 25%
1/5 = .2 = 20%
2/4 = .5 = 50%
¾ = .75 = 75%
1/10 = .1 = 10%
To multiply a fraction
Multiply the numerators together, and the denominators together:
3 2 6
 
5 3 15
Remember, a whole number can be expressed as a fraction – any number divided
by 1 is itself:
4
4 is the same as
1
To divide a fraction,
invert (turn upside down) the second fraction, and mutiply them together:
2 3 2 4
  
5 4 5 3
8

15
2
Percentaging a distribution
First, add up the numbers to get the total. Then use the total as the denominator,
and divide the parts of the distribution into it.
In one week at the Muddy Waters Pub, customers drank 340 Kokanees, 213 Buds,
170 Coors, and 110 Coronas. Calculate the percentage of each beer that customers
drank that week. (Hint: eyeball the numbers first, so you know if your calculations
are correct!)
1. Get the total number of beers those drunks drank:
340 + 213 + 170 + 110 = 833
2. Now, divide the total for each beer into the overall total:
a. Kokanee: 340/833 = .40816, rounded to .41 = 41%
b. Bud: 213/833 = .2557, rounded to .26 = 26%
c. Coors: 170/833 = .2040 = .20 = 20%
d. Corona: 110/833 = .1320 = .13 = 13%
3
Squaring numbers
To square a number is to multiply it by itself.
X2 (reads “X squared”)
If X=2, then
2
X = 2x2 = 4
(2)(2) = 4
(X)(X)
Square roots
A square root of a number is another number, that when
multiplied by itself, gives you back your original number.
For example, the square root of 16 is 4, because 4x4=16.
The symbol for a square root is:
X
25  5
because 5x5 = 25.
For everything but simple numbers, you pretty much need a
calculator to do square roots.
4
Order of operations
Basic rules:
1) Do all squares and square roots first
2) Then do all multiplication and division
3) Then complete the addition and subtraction
4) These rules are overridden when there are operations in parentheses – then,
do the things in the parentheses ( ) first.
8 + 2 x 22/2 = ?
Do the squares first:
=8 + 2 x 4/2
then multiply:
= 8 + 8/2
then divide:
=8 + 4
=12
a+b(c)
2 + 3(4) (do multiplication
first)
= 2+12
=14
When there are parentheses, do the operations in them first:
(8 + 2) – 4(3)2/(8-6)
= (10) – 4*9/2
= 10 – 36/2
= 10 – 18
= –8
5
Positive & Negative Numbers
A number can be positive if it has no sign (8), or a + sign (+8).
A number is negative if it is preceded by a negative sign (-2).
Positive numbers are greater than zero ( > 0 )
Negative numbers are less than zero ( < 0 )
You MUST keep track of the signs of numbers.
This will be VERY important in this course!
Adding a negative number is
the same as subtraction:
3 + (-1) + 4
=3–1+4
=2+4
=6
Subtracting changes the sign
of a negative number (it’s like
adding a positive):
3 – (– 1) – 4
=3+1–4
=4–4
=0
Multiplication: A negative
number times a positive
number equals a negative:
– 3(4) = – 12
(3)( – 4) = – 12
A negative number times a
negative number is ALWAYS (– 3)( – 5) = 15
positive:
Division follows the same
rules:
(– 4) / 2 = -2
(4) / (– 2) = -2
BUT (– 4) / (– 2) = 2
6
Summation

This is the sign called the “summation” sign:
It is the Greek letter “sigma”, and pretty much means to add a series of
scores together – but there are some tricks to watch for!
Let’s work with a set of scores: 2,
X
X
Means add the scores
i
2
i
 X 
2
i
X
i literally
means “the summation
of all the scores”
X
2
i
means “the
sum of the squared
scores:”
 X 
2
means
“the sum of the
scores, squared”
i
4, 6, 8
First square the scores, then add the sum
of the squared scores.
First add the scores, then square the
total.
X
X
2
i
 2  4  6  8  20
i

(2 * 2)  (4 * 4)  (6 * 6)  (8 * 8)
 120
This means you add the numbers first,
then square them:
 X 
2
i
 (2  4  6  8) 2
 (20) 2  400
7
Solving for unknown quantities
In this course, you will need to solve for unknown quantities. Remember that
whatever you do to one side of an equation, you must do to the other.
For example, solve the following equation for X:
5+X=4
We want X on one side, and the numbers (the answer), on the other. We need to
get rid of that 5, so we have to subtract it from the left side of the equation – but we
MUST also subtract it from the right side:
5+X–5=4–5
X = -1
If the equation was
X–5=4
we would ADD 5 to both sides:
5+X–5=4+5
X=9
It is the same for multiplication and division:
3X = 12
divide both sides by 3:
3 X 12

3
3
1X  4 or X = 4
4.2X = 15
divide both sides by 4.2 
4.2 X 15

4.2
4.2
X = 3.57
8
3X + 5 = 26
First, get rid of the 5 by subtracting it from both sides:
3X + 5 – 5 = 26 – 5
3X = 21
then, divide each side by 3:
3 X 21

3
3
1X = 7 or X = 7
3X
 5  20
4
Subtract 5 from each side:
3X
 5  5  20  5
4
3X
 15
then you have:
4
We need to get rid of the 4 in the denominator. So, multiply each side by 4:
3X
3X 4
 4  15  4
  15  4
4 1
4
=
the 4 cancels out of the left side, leaving you with:
3X = 60
X = 20
9
Psych 300A – Math review Practice Questions2
Before you proceed with this course, you must be comfortable with ALL of these operations.
1. Complete each of the following:
a. 17 x 3 =
b. 17(3) =
c. (17)(3) =
d. 17/3 =
e. (42)2 =
f.
113 
2. For the set of scores (X1) of 50, 55, 60, 65, and 70, evaluate (complete) each of the
expressions below:
a.
b.
c.
X 
X 
 X  =
1
2
1
2
1
3. Complete each of the following:
a. 17 + (-3) + (4) + (–2) =
b. 15 – 3 – (–5) + 2 =
c. (–27)(54) =
d. (113)( –2) =
e. (–14)( –100) =
f. –34/–2 =
g. 322/–11 =
h.
2 
i. (–17)2 =
4. Round off each of the following to two places beyond the decimal point:
a. 17.17532
b. 43.119
c. 1076.77337
d. 32.4651152301
e. 32.4751152301
5. Evaluate (complete) each of the following:
2
Most of these questions are from Healey (2005:529-531).
10
a. (3 + 7)/10 =
b. 3 + 7/10 =
c. ((4–3) + (7 + 2) / ((4+5)(10)) =
d.
(7(5  3) 2 ) /((17 / 3)( 4)) 
e. 22 + 44/15 =
6. In a study of Social Science students, 300 were first year, 250 were second year, 100 were
third year, and 50 were fourth year. What percentage of students were in each year?
1 1
7.      =
3  2
1  1
3  2
8.      =
9.  10  2 =
10. –.5 – (–.34) =
11