The gre anlytycal writting section 4 pptx

( + ) ϫ ( – ) = – (+) Ϭ (–) = –
( – ) ϫ ( – ) = + (–) Ϭ (–) = +
A simple rule for remembering these patterns is that if the signs are the same when multiplying or divid-
ing, the answer will be positive. If the signs are different, the answer will be negative.
ADDING
Adding two numbers with the same sign results in a sum of the same sign:
( + ) + ( + ) = + and ( – ) + (– ) = –
When adding numbers of different signs, follow this two-step process:
1. Subtract the absolute values of the numbers.
2. Keep the sign of the number with the larger absolute value.
Examples:
–2 + 3 =
Subtract the absolute values of the numbers: 3 – 2 = 1.
The sign of the number with the larger absolute value (3) was originally positive, so the answer is positive.
8 + –11 =
Subtract the absolute values of the numbers: 11 – 8 = 3
The sign of the number with the larger absolute value (11) was originally negative, so the answer is –3.
SUBTRACTING
When subtracting integers, change the subtraction sign to addition and change the sign of the number being
subtracted to its opposite. Then follow the rules for addition.
Examples:
(+10) – (+12) = (+10) + (–12) = –2
(–5) – (–7) = (–5) + (+7) = +2
REMAINDERS
Dividing one integer by another results in a remainder of either zero or a positive integer. For example:
1 R1
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155
4ͤ5
ෆ
–4

1
If there is no remainder, the integer is said to be “divided evenly,” or divisible by the number.
When it is said that an integer n is divided evenly by an integer x, it is meant that n divided by x results
in an answer with a remainder of zero. In other words, there is nothing left over.
ODD AND
EVEN NUMBERS
An even number is a number divisible by the number 2, for example, 2,4, 6, 8, 10, 12,14, and so on.An odd num-
ber is not divisible by the number 2, for example, 1, 3, 5, 7, 9, 11, 13, and so on. The even and odd numbers are
also examples of consecutive even numbers and consecutive odd numbers because they differ by two.
Here are some helpful rules for how even and odd numbers behave when added or multiplied:
even + even = even and even ϫ even = even
odd + odd = even and odd ϫ odd = odd
odd + even = odd and even ϫ odd = even
F
ACTORS AND MULTIPLES
Factors are numbers that can be divided into a larger number without a remainder.
Example:
12 ϫ 3 = 4
The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12.
The common factors of two numbers are the factors that are the same for both numbers.
Example:
The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 18 = 1, 2, 3, 6, 9, 18.
From the previous example, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. This list
also shows that we can determine that the greatest common factor of 24 and 18 is 6. Determining the greatest com-
mon factor is useful for reducing fractions.
Any number that can be obtained by multiplying a number x by a positive integer is called a multiple of x.
Example:
Some multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40 . . .
Some multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56 . . .

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156
PRIME AND COMPOSITE NUMBERS
A positive integer that is greater than the number 1 is either prime or composite, but not both.
■
A prime number has exactly two factors: 1 and itself.
Example:
2,3,5,7,11,13,17,19,23,
■
A composite number is a number that has more than two factors.
Example:
4,6,9,10,12,14,15,16,
The number 1 is neither prime nor composite.
Variables
In a mathematical sentence,a variable is a letter that represents a number. Consider this sentence:x+ 4 = 10.It is easy
to determine that xrepresents 6.However, problems with variables on the GRE will become much more complex than
that, and there are many rules and procedures that you need to learn. Before you learn to solve equations with vari-
ables,you must learn how they operate in formulas.The next section on fractions will give you some examples.
Fractions
A fraction is a number of the form
ᎏ
a
b
ᎏ
,where a and b are integers and b  0.In
ᎏ
a
b
ᎏ
, the a is called the numerator and

the b is called the denominator. Since the fraction
ᎏ
a
b
ᎏ
means a Ϭ b, b cannot be equal to zero. To do well when work-
ing with fractions, it is necessary to understand some basic concepts. The following are math rules for fractions
with variables:
ᎏ
b
a
ᎏ
ϫ
ᎏ
d
c
ᎏ
=
ᎏ
b
a
ᎏ
+
ᎏ
b
c
ᎏ
=
ᎏ
a

b
ᎏ
Ϭ
ᎏ
d
c
ᎏ
=
ᎏ
a
b
ᎏ
ϫ
ᎏ
d
c
ᎏ
=
ᎏ
b
a
ᎏ
+
ᎏ
d
c
ᎏ
=
Dividing by Zero
Dividing by zero is not possible. This is important when solving for a variable in the denominator of a fraction.

Example:
ᎏ
a –
6
3
ᎏ
a – 3  0
a  3
In this problem, we know that a cannot be equal to 3 because that would yield a zero in the denominator.
ab + bc
ᎏ
bd
a ϫ d
ᎏ
b ϫ c
a + c
ᎏ
b
a ϫ c
ᎏ
b ϫ d
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157
Multiplication of Fractions
Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the
numerators and the denominators, writing each in the respective place over or under the fraction bar.
Example:
ᎏ
4
5

ᎏ
ϫ
ᎏ
6
7
ᎏ
=
ᎏ
2
3
4
5
ᎏ
Division of Fractions
Dividing by a fraction is the same thing as multiplying by the reciprocal of the fraction. To find the recipro-
cal of any number, switch its numerator and denominator. For example, the reciprocals of the following
numbers are:
ᎏ
1
3
ᎏ
⇒
ᎏ
3
1
ᎏ
= 3 x ⇒
ᎏ
1
x

ᎏᎏ
4
5
ᎏ
⇒
ᎏ
5
4
ᎏ
5 ⇒
ᎏ
1
5
ᎏ
ᎏ
–
2
1
ᎏ
⇒
ᎏ
–
1
2
ᎏ
= –2
When dividing fractions, simply multiply the dividend by the divisor’s reciprocal to get the answer.
For example:
ᎏ
1

2
2
1
ᎏ
Ϭ
ᎏ
3
4
ᎏ
=
ᎏ
1
2
2
1
ᎏ
ϫ
ᎏ
4
3
ᎏ
=
ᎏ
4
6
8
3
ᎏ
=
ᎏ

1
2
6
1
ᎏ
Adding and Subtracting Fractions
■
To add or subtract fractions with like denominators, just add or subtract the numerators and leave the
denominator as it is. For example:
ᎏ
1
7
ᎏ
+
ᎏ
5
7
ᎏ
=
ᎏ
6
7
ᎏ
and
ᎏ
5
8
ᎏ
–
ᎏ

2
8
ᎏ
=
ᎏ
3
8
ᎏ
■
To add or subtract fractions with unlike denominators, you must find the least common denominator,or
LCD.In other words, if the given denominators are 8 and 12, 24 would be the LCD because 8 ϫ 3 = 24, and
12 ϫ 2 = 24. So, the LCD is the smallest number divisible by each of the original denominators. Once you
know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator
by the necessary number to get the LCD, and then add or subtract the new numerators. For example:
ᎏ
1
3
ᎏ
+
ᎏ
2
5
ᎏ
=
ᎏ
5
5
(
(
1

3
)
)
ᎏ
+
ᎏ
3
3
(
(
2
5
)
)
ᎏ
=
ᎏ
1
5
5
ᎏ
+
ᎏ
1
6
5
ᎏ
=
ᎏ
1

1
1
5
ᎏ
Mixed Numbers and Improper Fractions
A mixed number is a fraction that contains both a whole number and a fraction. For example, 4
ᎏ
1
2
ᎏ
is a mixed
number. To multiply or divide a mixed number, simply convert it to an improper fraction. An improper frac-
tion has a numerator greater than or equal to its denominator. The mixed number 4
ᎏ
1
2
ᎏ
can be expressed as the
improper fraction
ᎏ
9
2
ᎏ
. This is done by multiplying the denominator by the whole number and then adding the
numerator. The denominator remains the same in the improper fraction.
– THE GRE QUANTITATIVE SECTION–
158
For example, convert 5
ᎏ
1

3
ᎏ
to an improper fraction.
1. First, multiply the denominator by the whole number: 5 ϫ 3 = 15.
2. Now add the numerator to the product: 15 + 1 = 16.
3. Write the sum over the denominator (which stays the same):
ᎏ
1
3
6
ᎏ
.
Therefore, 5
ᎏ
1
3
ᎏ
can be converted to the improper fraction
ᎏ
1
3
6
ᎏ
.
Decimals
The most important thing to remember about decimals is that the first place value to the right is tenths. The
place values are as follows:
In expanded form, this number can also be expressed as:
1268.3457 = (1 ϫ 1,000) + (2 ϫ 100) + (6 ϫ 10) + (8 ϫ 1) + (3 ϫ .1) + (4 ϫ .01) + (5 ϫ .001) + (7 ϫ .0001)
Comparing Decimals

Comparing decimals is actually quite simple. Just line up the decimal points and fill in any zeroes needed to
have an equal number of digits.
Example: Compare .5 and .005
Line up decimal points and add zeroes: .500
.005
Then ignore the decimal point and ask, which is bigger: 500 or 5?
500 is definitely bigger than 5, so .5 is larger than .005
1
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A
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D
S
2
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U
N
D
R
E
D
S
6
T
E
N

S
8
O
N
E
S
3
T
E
N
T
H
S
4
H
U
N
D
R
E
D
T
H
S
5
T
H
O
U
S

A
N
D
T
H
S
7
T
E
N

T
H
O
U
S
A
N
D
T
H
S
.
D
E
C
I
M
A
L

P
O
I
N
T
– THE GRE QUANTITATIVE SECTION–
159
Operations with Decimals
To add and subtract decimals, you must always remember to line up the decimal points:
356.7 3.456 8.9347
+ 34.9854 + .333 – 0.24
391.6854 3.789 8.6947
To multiply decimals, it is not necessary to align decimal points. Simply perform the multiplication as if there
were no decimal point. Then, to determine the placement of the decimal point in the answer, count the numbers
located to the right of the decimal point in the decimals being multiplied. The total numbers to the right of the
decimal point in the original problem is the number of places the decimal point is moved in the product.
For example:
To divide a decimal by another, such as 13.916 Ϭ 2.45 or 2.45ͤ13
ෆ
.9
ෆ
16
ෆ
, move the decimal point in the
divisor to the right until the divisor becomes a whole number. Next, move the decimal point in the dividend
the same number of places:
This process results in the correct position of the decimal point in the quotient. The problem can now be
solved by performing simple long division:
Percents

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 percent-
age symbol.
245
1391.6
5.68
–1225
166 6
–1470
1960
1391.6
245
1 2.3 4
2
2
x .5 6
1
2
3
4
7 4 0 4
6 1 7 0 0
6.9 1 0 4
1
2
3
4
= TOTAL #'s TO THE RIGHT OF
THE DECIMAL POINT = 4

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160