The gre anlytycal writting section 4 pptx

Here are some helpful rules for how even and odd numbers behave when added or multiplied: F ACTORS AND M ULTIPLES Factors are numbers that can be divided into a larger number without a r

( + ) ( – ) = – (+) ( – ) ( – ) = + (–)

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

A DDING

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

S UBTRACTING

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

R EMAINDERS

Dividing one integer by another results in a remainder of either zero or a positive integer For example:

1 R1

–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

O DD AND E VEN N UMBERS

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 numnum-ber 2, for example, 1, 3, 5, 7, 9, 11, 13, and so on The even and odd numnum-bers 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:

F ACTORS AND M ULTIPLES

Factors are numbers that can be divided into a larger number without a remainder.

Example:

12 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

P RIME AND C OMPOSITE N UMBERS

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 x represents 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 a b, where a and b are integers and b 0 In a

b, the a is called the numerator and

the b is called the denominator Since the fraction a bmeans a

ing with fractions, it is necessary to understand some basic concepts The following are math rules for fractions with variables:



b

a 

d

b

a+

b c=

a

b 

d c= a

b d

b

a+

d c=

Dividing by Zero

Dividing by zero is not possible This is important when solving for a variable in the denominator of a fraction

Example: 

a –

6 3



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

bd

a d

b c

a + c

b

a c

b d

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:

4

5  6

7 = 2

3

4 5



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 reciprorecipro-cals of the following numbers are:

1

3 ⇒3

1 = 3 x⇒1

5 ⇒5

4  5 ⇒1

5  –

2 1

⇒– 1 2

 = –2

When dividing fractions, simply multiply the dividend by the divisor’s reciprocal to get the answer For example:

1

2

2

1

 3

4 = 1

2

2 1

 4

3 = 4 6

8 3

= 1 2

6 1



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:

1

7 + 5

7 = 6

7  and 5

8 – 2

8 = 3

8 

■ 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 3 = 24, and

12 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:

1

3 + 2

5 = 5

5

(

(

1 3

) )

+ 3 3

( (

2 5

) )

=  1

5 5

+  1

6 5

= 1 1

1 5



Mixed Numbers and Improper Fractions

A mixed number is a fraction that contains both a whole number and a fraction For example, 412is 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 412can be expressed as the improper fraction 92 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

For example, convert 513to an improper fraction.

1 First, multiply the denominator by the whole number: 5 3 = 15

2 Now add the numerator to the product: 15 + 1 = 16.

3 Write the sum over the denominator (which stays the same):136

Therefore, 513can be converted to the improper fraction 136

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 1,000) + (2 100) + (6 10) + (8 1) + (3 .1) + (4 .01) + (5 .001) + (7 .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 T H O U S A N D S

2 H 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

Operations with Decimals

To add and subtract decimals, you must always remember to line up the decimal points:

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 .916, 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