The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction.. Remember to reverse the direction of the
Trang 1T h e F O I L M e t h o d
The FOIL method can be used when multiplying
bino-mials FOIL stands for the order used to multiply the
terms: First, Outer, Inner, and Last To multiply
binomi-als, you multiply according to the FOIL order and then
add the like terms of the products
Example
(3x + 1)(7x + 10)
3x and 7x are the first pair of terms,
3x and 10 are the outermost pair of terms,
1 and 7x are the innermost pair of terms, and
1 and 10 are the last pair of terms
Therefore, (3x)(7x) + (3x)(10) + (1)(7x) +
(1)(10) = 21x2+ 30x + 7x + 10.
After we combine like terms, we are left with the
answer: 21x2+ 37x + 10.
F a c t o r i n g
Factoring is the reverse of multiplication:
2(x + y) = 2x + 2y Multiplication
2x + 2y = 2(x + y) Factoring
Three Basic Types of Factoring
1 Factoring out a common monomial.
10x2− 5x = 5x(2x − 1) and
xy − zy = y(x − z)
2 Factoring a quadratic trinomial using the reverse
of FOIL:
y2− y − 12 = (y − 4) (y + 3) and
z2− 2z + 1 = (z − 1)(z − 1) = (z − 1)2
3 Factoring the difference between two perfect
squares using the rule:
a2− b2= (a + b)(a − b) and
x2− 25 = (x + 5)(x − 5)
Removing a Common Factor
If a polynomial contains terms that have common
fac-tors, the polynomial can be factored by dividing by the
greatest common factor
Example
In the binomial 49x3+ 21x, 7x is the greatest
common factor of both terms
Therefore, you can divide 49x3+ 21x by 7x to
get the other factor
49x37+x21x = 479xx3+ 271xx = 7x2+ 3
Thus, factoring 49x3+ 21x results in 7x(7x2+ 3)
Q u a d r a t i c E q u a t i o n s
A quadratic equation is an equation in which the great-est exponent of the variable is 2, as in x2+ 2x− 15 = 0 A quadratic equation has two roots, which can be found by breaking down the quadratic equation into two simple equations
Example
Solve x2+ 5x + 2x + 10 = 0.
x2+ 7x + 10 = 0 Combine like terms.
(x + 5)(x + 2) = 0 Factor.
x + 5 = 0 or x + 2 = 0
x−=5−−55 x−=2−−22
Now check the answers
−5 + 5 = 0 and −2 + 2 = 0
Therefore, x is equal to both −5 and −2.
I n e q u a l i t i e s
Linear inequalities are solved in much the same way as
simple equations The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction
Example
10 > 5 but if you multiply by −3,
(10) − 3 < (5)−3
−30 < −15
Solving Linear Inequalities
To solve a linear inequality, isolate the variable and solve the same as you would in a first-degree equation Remember to reverse the direction of the inequality sign
if you divide or multiply both sides of the equation by a negative number
– A L G E B R A , F U N C T I O N S , A N D PAT T E R N S –
Trang 2If 7 − 2x > 21, find x
Isolate the variable
7 − 2x > 21
−−27x > −147
Because you are dividing by a negative number,
the direction of the inequality symbol changes
direction
−−22x> −14 2
x < −7
The answer consists of all real numbers less than −7
E x p o n e n t s
An exponent tells you how many times the number,
called the base, is a factor in the product.
Example
25 exponent= 2 × 2 × 2 × 2 × 2 = 32
base
– A L G E B R A , F U N C T I O N S , A N D PAT T E R N S –
Trang 3TH 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.
C H A P T E R
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
Trang 4■ 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
– D ATA A N A LY S I S , S TAT I S T I C S , A N D P R O B A B I L I T Y –
Trang 5To 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 8 .125 12.5%
1 6 .1666 16.6
– D ATA A N A LY S I S , S TAT I S T I C S , A N D P R O B A B I L I T Y –
Trang 6C 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
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 = 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
– D ATA A N A LY S I S , S TAT I S T I C S , A N D P R O B A B I L I T Y –