Data Analysis, Statistics, and Probability

■ If the set contains an odd number of elements, then simply choose the middle value.. ■ If the set contains an even number of elements, simply average the two middle values.. ■ To chang

TH I S S E C T I O N W I L L help you become familiar with the word problems on the GED and analyze data

using specific techniques

 Tr a n s l a t i n g Wo r d s i n t o N u m b e r s

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.

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

■ 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

 A s s i g n i n g Va r i a b l e s i n

Wo r d P r o b l e m s

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.

 R a t i o

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 orx y 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 cancan-dies 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

1155x= 6105

x = 4 Therefore, there are 7x = (7)(4) = 28

green candies and 8x = (8)(4) = 32 red

candies

 M e a n , M e d i a n , a n d M o d e

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

9 + 4 + 75+ 6 + 4= 350= 6 The average is 6

(Divide by 5 because there are 5 numbers in the set.)

sum of the number set

quantity of set

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:

3 +25 = 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

 P e r c e n t

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

45= 80 = 80% 25= 4 = 40% 18= 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% = 16040 = 1265 75% = 17050= 34 82% = 18020= 4510

■ 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 114

350% = 3.5 or 312

Here are some conversions you should be familiar with The order is from most common to less common

1 3 .333 33.3

2 3 .666 66.6

1 6 .1666 16.6

 C a l c u l a t i n g I n t e r e s t

Interest is a fee paid for the use of someone else’s money.

If you put money in a savings account, you receive

est from the bank If you take out a loan, you pay

inter-est to the lender The amount of money you invinter-est 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

Step 2: Substitute the known

= $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 = 192= 34year

= $4,000 × 0.08 ×34

= $180 Kai will earn $180 in interest

 P r o b a b i l i t y

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 =

= 5 +33 + 6

Therefore, the probability of selecting a red marble is 134

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

Total number of possible outcomes

Number of specific outcomes

Total number of possible outcomes

 G r a p h s a n d Ta b l e s

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 porsec-tion 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 represdiffer-enting differdiffer-ent 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

 S c i e n t i f i c N o t a t i o n

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:

107

Step 3: Express the full answer in scientific nota-tion by multiplying the reduced answer from Step 1 by 107:

4.6 × 107

Increase

Decrease

No Change

Increase Decrease

Change in Time

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

Year

25%

40%

35%

An amoeba is 000056 inch long What is its

length in scientific notation?

Step 1: Move the decimal point to the right until

there is one digit other than zero to the left of

the decimal

.000056 becomes 5.6

Step 2: Count the number of places moved to

the right—5 However, because the value of the

number is being increased as it is expressed in

scientific notation, it is written as a negative

exponent

10−5

Step 3: Express the full answer in scientific

notation:

.0000056 becomes 5.6 × 10−5

 G e n e r a l S t r a t e g i e s f o r M a t h

Q u e s t i o n s

■ Skipping and returning.

If you are unsure of what you are being asked

to find, if you don’t know how to solve a problem,

or if you will take a long time to find the correct

answer, skip the question and come back to it

later Do the easy problems first The GED is not arranged with increasingly difficult questions The difficult questions appear alongside the easier questions Therefore, it is important to skip difficult problems and come back to them

■ Plugging in.

There will be times when you should use the answer choices to find the correct answer This can be done when you have a problem that gives you a formula or equation Plug in answers when you feel it will be quicker than solving the prob-lem another way, and when you have enough information to do so

■ Eliminating.

Eliminate choices you know are wrong so that you can spend more time considering choices that might be right It may sound like a simple strategy, but it can make a big difference

■ Making educated guesses.

It’s important to remember you are not penal-ized for a wrong answer If you don’t know the answer to a question and you are approaching the time limit, simply use the last few minutes to make an educated guess to the remaining ques-tions If you can eliminate some of the answer choices, you will improve your odds of getting

it right