Data Analysis, Statistics, and Probability

T
HIS SECTION WILL
help you become familiar with the word problems on the GED and analyze data
using specific techniques.

Translating Words into Numbers
The most important skill needed for word problems is the ability to translate words into mathematical opera-
tions. This list will assist you in this by giving you some common examples of English phrases and their mathe-
matical equivalents.
■
Increase means add.
A number increased by five = x + 5.
■
Less than means subtract.
10 less than a number = x − 10.
■
Times or product means multiply.
Three times a number = 3x.
CHAPTER
Data Analysis,
Statistics, and
Probability
MANY STUDENTS struggle with word problems. In this chapter,
you will learn how to solve word problems with confidence by trans-
lating the words into a mathematical equation. Since the GED math
section focuses on “real-life” situations, it’s especially important for you
to know how to make the transition from sentences to a math problem.
44
417
■
Times the sum means to multiply a number by a

quantity.
Five times the sum of a number and three =
5(x + 3).
■
Two variables are sometimes used together.
A number y exceeds five times a number x by
ten.
y = 5x + 10
■
Inequality signs are used for at least and at most,
as well as less than and more than.
The product of x and 6 is greater than 2.
x × 6 > 2
When 14 is added to a number x, the sum is
less than 21.
x + 14 < 21
The sum of a number x and four is at least
nine.
x + 4 ≥ 9
When seven is subtracted from a number x,
the difference is at most four.
x − 7 ≤ 4

Assigning Variables in
Word Problems
It may be necessary to create and assign variables in a
word problem. To do this, first identify an unknown and
a known. You may not actually know the exact value of
the “known,” but you will know at least something about
its value.

Examples
Max is three years older than Ricky.
Unknown = Ricky’s age = x.
Known = Max’s age is three years older.
Therefore,
Ricky’s age = x and Max’s age = x + 3.
Lisa made twice as many cookies as Rebecca.
Unknown = number of cookies Rebecca made
= x.
Known = number of cookies Lisa made = 2x.
Cordelia has five more than three times the
number of books that Becky has.
Unknown = the number of books Becky has
= x.
Known = the number of books Cordelia has
= 3x + 5.

Ratio
A ratio is a comparison of a two quantities measured in
the same units. It can be symbolized by the use of a
colon—x:y or
ᎏ
x
y
ᎏ
or x to y. Ratio problems can be solved
using the concept of multiples.
Example
A bag containing some red and some green can-
dies has a total of 60 candies in it. The ratio of

the number of green to red candies is 7:8. How
many of each color are there in the bag?
From the problem, it is known that 7 and 8
share a multiple and that the sum of their prod-
uct is 60. Therefore, you can write and solve the
following equation:
7x + 8x = 60
15x = 60
ᎏ
1
1
5
5
x
ᎏ
=
ᎏ
6
1
0
5
ᎏ
x = 4 Therefore, there are 7x = (7)(4) = 28
green candies and 8x = (8)(4) = 32 red
candies.

Mean, Median, and Mode
To find the average or mean of a set of numbers, add all
of the numbers together and divide by the quantity of
numbers in the set.

Average =
Example
Find the average of 9, 4, 7, 6, and 4.
ᎏ
9+4+7
5
+6+4
ᎏ
=
ᎏ
3
5
0
ᎏ
= 6 The average is 6.
(Divide by 5 because there are 5 numbers in the
set.)
sum of the number set
ᎏᎏᎏ
quantity of set
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DATA ANALYSIS, STATISTICS, AND PROBABILITY
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418
To find the median of a set of numbers, arrange the
numbers in ascending order and find the middle value.
■
If the set contains an odd number of elements,
then simply choose the middle value.
Example

Find the median of the number set: 1, 3, 5, 7, 2.
First, arrange the set in ascending order: 1, 2, 3,
5, 7, and then choose the middle value: 3. The
answer is 3.
■
If the set contains an even number of elements,
simply average the two middle values.
Example
Find the median of the number set: 1, 5, 3, 7, 2, 8.
First, arrange the set in ascending order: 1, 2, 3, 5,
7, 8 and then choose the middle values, 3 and 5.
Find the average of the numbers 3 and 5:
ᎏ
3+
2
5
ᎏ
= 4. The median is 4.
The mode of a set of numbers is the number that occurs
the greatest number of times.
Example
For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the
number 3 is the mode because it occurs the
most often.

Percent
A percent is a measure of a part to a whole, with the
whole being equal to 100.
■
To change a decimal to a percentage, move the

decimal point two units to the right and add a
percentage symbol.
Example
.45 = 45% .07 = 7% .9 = 90% .085 = 8.5%
■
To change a fraction to a percentage, first change
the fraction to a decimal. To do this, divide the
numerator by the denominator. Then change the
decimal to a percentage.
Examples
ᎏ
4
5
ᎏ
= .80 = 80%
ᎏ
2
5
ᎏ
= .4 = 40%
ᎏ
1
8
ᎏ
= .125 = 12.5%
■
To change a decimal to a percentage, move the
decimal point two units to the right and add a
percentage symbol.
■

To change a percentage to a decimal, simply move
the decimal point two places to the left and elimi-
nate the percentage symbol.
Examples
64% = .64 87% = .87 7% = .07
■
To change a percentage to a fraction, put the per-
cent over 100 and reduce.
Examples
64% =
ᎏ
1
6
0
4
0
ᎏ
=
ᎏ
1
2
6
5
ᎏ
75% =
ᎏ
1
7
0
5

0
ᎏ
=
ᎏ
3
4
ᎏ
82% =
ᎏ
1
8
0
2
0
ᎏ
=
ᎏ
4
5
1
0
ᎏ
■
Keep in mind that any percentage that is 100 or
greater will need to reflect a whole number or
mixed number when converted.
Examples
125% = 1.25 or 1
ᎏ
1

4
ᎏ
350% = 3.5 or 3
ᎏ
1
2
ᎏ
Here are some conversions you should be familiar
with. The order is from most common to less common.
Fraction Decimal Percentage
ᎏ
1
2
ᎏ
.5 50%
ᎏ
1
4
ᎏ
.25 25%
ᎏ
1
3
ᎏ
.333 . . . 33.3
៮៮
ᎏ
2
3
ᎏ

.666 . . . 66.6
៮៮
ᎏ
1
1
0
ᎏ
.1 10%
ᎏ
1
8
ᎏ
.125 12.5%
ᎏ
1
6
ᎏ
.1666 . . . 16.6
៮៮
ᎏ
1
5
ᎏ
.2 20%
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
Calculating Interest

Interest is a fee paid for the use of someone else’s money.
If you put money in a savings account, you receive inter-
est from the bank. If you take out a loan, you pay inter-
est to the lender. The amount of money you invest or
borrow is called the principal. The amount you repay is
the amount of the principal plus the interest.
The formula for simple interest is found on the for-
mula sheet in the GED. Simple interest is a percent of the
principal multiplied by the length of the loan:
Interest = principal × rate × time
Sometimes, it may be easier to use the letters of each
as variables:
I = prt
Example
Michelle borrows $2,500 from her uncle for
three years at 6% simple interest. How much
interest will she pay on the loan?
Step 1: Write the interest as a
decimal. 6% = 0.06
Step 2: Substitute the known
values in the formula I = prt
and multiply. = $2,500 × 0.06 × 3
= $450
Michelle will pay $450 in interest.
Some problems will ask you to find the amount that
will be paid back from a loan. This adds an additional
step to problems of interest. In the previous example,
Michelle will owe $450 in interest at the end of three
years. However, it is important to remember that she will
pay back the $450 in interest as well as the principal,

$2,500. Therefore, she will pay her uncle $2,500 + $450
= $2,950.
In a simple interest problem, the rate is an annual, or
yearly, rate. Therefore, the time must also be expressed in
years.
Example
Kai invests $4,000 for nine months. Her invest-
ment will pay 8%. How much money will she
have at the end of nine months?
Step 1: Write the rate as a decimal. 8% = 0.08
Step 2: Express the time as a fraction
by writing the length of time in months
over 12 (the number of months in a year).
9 months =
ᎏ
1
9
2
ᎏ
=
ᎏ
3
4
ᎏ
year
Step 3: Multiply. I = prt
= $4,000 × 0.08 ×
ᎏ
3
4

ᎏ
= $180
Kai will earn $180 in interest.

Probability
Probability is expressed as a fraction and measures the
likelihood that a specific event will occur. To find the
probability of a specific outcome, use this formula:
Probability of an event =
Example
If a bag contains 5 blue marbles, 3 red marbles,
and 6 green marbles, find the probability of
selecting a red marble:
Probability of an event =
=
ᎏ
5+
3
3+6
ᎏ
Therefore, the probability of selecting a red
marble is
ᎏ
1
3
4
ᎏ
.
Helpful Hints about Probability
■

If an event is certain to occur, the probability is 1.
■
If an event is certain not to occur (impossible),
the probability is 0.
■
If you know the probability of all other events
occurring, you can find the probability of the
remaining event by adding the known probabili-
ties together and subtracting their total from 1.
Number of specific outcomes
ᎏᎏᎏᎏ
Total number of possible outcomes
Number of specific outcomes
ᎏᎏᎏᎏ
Total number of possible outcomes
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
Graphs and Tables
The GED exam will test your ability to analyze graphs
and tables. Read each graph or table very carefully before
reading the question. This will help you to process the
information that is presented. It is extremely important
to read all of the information presented, paying special
attention to headings and units of measure. Here is an
overview of the types of graphs you will encounter:
■
Circle graphs or pie charts

This type of graph is representative of a whole
and is usually divided into percentages. Each sec-
tion of the chart represents a portion of the
whole, and all of these sections added together
will equal 100% of the whole.
■
Bar graphs
Bar graphs compare similar things with differ-
ent length bars representing different values. Be
sure to read all labels and legends, looking care-
fully at the base and sides of the graph to see what
the bars are measuring and how much they are
increasing or decreasing.
■
Broken-line graphs
Broken-line graphs illustrate a measurable
change over time. If a line is slanted up, it repre-
sents an increase whereas a line sloping down
represents a decrease. A flat line indicates no
change as time elapses.

Scientific Notation
Scientific notation is a method used by scientists to con-
vert very large or very small numbers to more manage-
able ones. You will have to make a few conversions to
scientific notation on the GED. Expressing answers in
scientific notation involves moving the decimal point
and multiplying by a power of ten.
Example
A space satellite travels 46,000,000 miles from

Earth. What is the number in scientific notation?
Step 1: Starting at the decimal point to the right
of the last zero, move the decimal point until
only one digit remains to its left
46,000,000 becomes 4.6.
Step 2: Count the number of places the decimal
was moved left (in this example, the decimal
point was moved 7 places), and express it as a
power of 10:
10
7
Step 3: Express the full answer in scientific nota-
tion by multiplying the reduced answer from
Step 1 by 10
7
:
4.6 × 10
7
Increase
Decrease
No Change
Increase
Decrease
Change in Time
Unit of Measure
Comparison of Roadwork Funds
of New York and California
2001–2005
New York
California

KEY
0
10
20
30
40
50
60
70
80
90
2001 2002 2003 2004 2005
Money Spent on New Roadwork
in Millions of Dollars
Year
25%
40%
35%
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