GMAT exam success Episode 2 Part 8 pps

If an even number of negatives is used, multiply or divide as usual, and the answer is positive.. If an odd number of negatives is used, multiply or divide as usual, and the answer is ne

3 If you are subtracting, change the subtraction sign to addition, and change the sign of the number

fol-lowing to its opposite Then follow the rules for addition:

a –5 + –6 = –11 b –12 + (+7) = –5

Remember: When you subtract, you add the opposite

M ULTIPLYING AND D IVIDING I NTEGERS

1 If an even number of negatives is used, multiply or divide as usual, and the answer is positive.

a –3 × –4 = 12 b (–12 –6) × 3 = 6

2 If an odd number of negatives is used, multiply or divide as usual, and the answer is negative.

a –15 5 = –3 b (–2 × –4) × –5 = –40

This is helpful to remember when working with powers of a negative number If the power is even, the answer is positive If the power is odd, the answer is negative

Fractions

A fraction is a ratio of two numbers, where the top number is the numerator and the bottom number is the denominator.

R EDUCING F RACTIONS

To reduce fractions to their lowest terms, or simplest form, find the GCF of both numerator and denominator Divide each part of the fraction by this common factor and the result is a reduced fraction When a fraction

is in reduced form, the two remaining numbers in the fraction are relatively prime.

When performing operations with fractions, the important thing to remember is when you need a com-mon denominator and when one is not necessary

A DDING AND S UBTRACTING F RACTIONS

It is very important to remember to find the least common denominator (LCD) when adding or subtract-ing fractions After this is done, you will be only addsubtract-ing or subtractsubtract-ing the numerators and keepsubtract-ing the com-mon denominator as the bottom number in your answer

6

15 10

15 16

15

3 x

y x  4

xy3x 4

xy

2 3

5 32 5

3 5

LCD  xy LCD 15

3

y 4

xy

2

52

3

32x 4xy8

y

6

92

3

M ULTIPLYING F RACTIONS

It is not necessary to get a common denominator when multiplying fractions To perform this operation, you can simply multiply across the numerators and then the denominators If possible, you can also

cross-can-cel common factors if they are present, as in example b.

D IVIDING F RACTIONS

A common denominator is also not needed when dividing fractions, and the procedure is similar to multi-plying Since dividing by a fraction is the same as multiplying by its reciprocal, leave the first fraction alone, change the division to multiplication, and change the number being divided by to its reciprocal

Decimals

The following chart reviews the place value names used with decimals Here are the decimal place names for the number 6384.2957

It is also helpful to know of the fractional equivalents to some commonly used decimals and percents, especially because you will not be able to use a calculator

0.4 40% 2

5

0.3 331

3% 1

3

0.1 10%  1

10

T

H

O

U

S

A

N

D

S

H

U

N

D

R

E

D

S

T

E

N

S

O

N

E

S

D E C I M A L P O I N T

T E N T H S

H U N D R E D T H S

T H O U S A N D T H S

T E N

T H O U S A N D T H S

6 3 8 4 2 9 5 7

3x

y 12x 5xy3 1x1

y1 5xy1

12 4x1 5x

4 4

5 4

3 41

5 3

4 1 3 5

12

25 5

3124

25 5 51

3 4 5 1

3 2

32

9

– A R I T H M E T I C –

3 3 0

A DDING AND S UBTRACTING D ECIMALS

The important thing to remember about adding and subtracting decimals is that the decimal places must be lined up

M ULTIPLYING D ECIMALS

Multiply as usual, and count the total number of decimal places in the original numbers That total will be the amount of decimal places to count over from the right in the final answer

34.5

× 5.4

1,380

+ 17,250

18,630

Since the original numbers have two decimal places, the final answer is 186.30 or 186.3 by counting over two places from the right in the answer

D IVIDING D ECIMALS

Start by moving any decimal in the number being divided by to change the number into a whole number Then move the decimal in the number being divided into the same number of places Divide as usual and keep track of the decimal place

.3 5.11.53  ⇒ 5115.3 

0 1.53 5.1

0.75 75% 3

4

0.6 66 2

3% 2

3

0.5 50% 1

2

A ratio is a comparison of two or more numbers with the same unit label A ratio can be written in three ways:

a: b

a to b

or

A rate is similar to a ratio except that the unit labels are different For example, the expression 50 miles

per hour is a rate—50 miles/1 hour

Proportion

Two ratios set equal to each other is called a proportion To solve a proportion, cross-multiply.

Cross multiply to get:

Percent

A ratio that compares a number to 100 is called a percent.

To change a decimal to a percent, move the decimal two places to the right

.25 = 25%

.105 = 10.5%

.3 = 30%

To change a percent to a decimal, move the decimal two places to the left

36% = 36

125% = 1.25

8% = 08

Some word problems that use percents are commission and rate-of-change problems, which include sales and interest problems The general proportion that can be set up to solve this type of word problem is

, although more specific proportions will also be shown

Part

Whole %

100

x 121

2

4x

4 50

4

4x  50

4

510

x

a

b

– A R I T H M E T I C –

3 3 2

C OMMISSION

John earns 4.5% commission on all of his sales What is his commission if his sales total $235.12?

To find the part of the sales John earns, set up a proportion:

Cross multiply

R ATE OF C HANGE

If a pair of shoes is marked down from $24 to $18, what is the percent of decrease?

To solve the percent, set up the following proportion:

Cross multiply

Note that the number 6 in the proportion setup represents the discount, not the sale price

S IMPLE I NTEREST

Pat deposited $650 into her bank account If the interest rate is 3% annually, how much money will she have

in the bank after 10 years?

x 25% decrease in price

24x

24 600

24

24x 600

6

24  x

100

24

100

part

whole  change

original cost  %

100

x 10.5804  $10.58

100x

100 1058.04

100

100x 1058.04

x

235.12 4.5

100

part

whole  change

original cost  %

100

Interest = Principal (amount invested) × Interest rate (as a decimal) × Time (years) or I = PRT.

Substitute the values from the problem into the formula I = (650)(.03)(10).

Multiply I = 195

Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her account

Exponents

The exponent of a number tells how many times to use that number as a factor For example, in the expres-sion 43, 4 is the base number and 3 is the exponent, or power Four should be used as a factor three times: 43

= 4× 4 × 4 = 64

Any number raised to a negative exponent is the reciprocal of that number raised to the positive expo-nent:

Any number to a fractional exponent is the root of the number:

Any nonzero number with zero as the exponent is equal to one: 140° = 1

Square Roots and Perfect Squares

Any number that is the product of two of the same factors is a perfect square

1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25,

Knowing the first 20 perfect squares by heart may be helpful You probably already know at least the first ten

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

2561  2 2564  4

271  2 273  3

25

1

 2 25  5

3  11

3221

9

– A R I T H M E T I C –

3 3 4

A square root symbol is also known as a radical sign The number inside the radical is the radicand

To simplify a radical, find the largest perfect square factor of the radicand

32 =16 × 2

Take the square root of that number and leave any remaining numbers under the radical

32 =42

To add or subtract square roots, you must have like terms In other words, the radicand must be the same If you have like terms, simply add or subtract the coefficients and keep the radicand the same

Examples

1. 32 +22 =52

2. 42–2= 32

3. 62 + 35 cannot be combined because they are not like terms

Here are some rules to remember when multiplying and dividing radicals:

Multiplying: x× y=  xy

2× 3 = 6

Dividing:

Counting Problems and Probability

The probability of an event is the number of ways the event can occur, divided by the total possible outcomes

The probability that an event will NOT occur is equal to 1 – P(E).

P1E2Number of ways the event can occur

Total possible outcomes

B

25

16  2 25

2 16  5

4

B

x

y  2 x

2 y

The counting principle says that the product of the number of choices equals the total number of

pos-sibilities For example, if you have two choices for an appetizer, four choices for a main course, and five choices for dessert, you can choose from a total of 2 × 4 × 5 = 40 possible meals

The symbol n! represents n factorial and is often used in probability and counting problems.

n! = (n) × (n – 1) × (n – 2) × × 1 For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Permutations and Combinations

Permutations are the total number of arrangements or orders of objects when the order matters The formula

is , where n is the total number of things to choose from and r is the number of things to

arrange at a time Some examples where permutations are used would be calculating the total number of dif-ferent arrangements of letters and numbers on a license plate or the total number of ways three difdif-ferent peo-ple can finish first, second, and third in a race

Combinations are the total number of arrangements or orders of objects when the order does not mat-ter The formula is , where n is the total number of objects to choose from and r is the size

of the group to choose An example where a combination is used would be selecting people for a commit-tee

Statistics

Mean is the average of a set of numbers To calculate the mean, add all the numbers in the set and divide by

the number of numbers in the set Find the mean of 2, 3, 5, 10, and 15

The mean is 7

Median is the middle number in a set To find the median, first arrange the numbers in order and then

find the middle number If two numbers share the middle, find the average of those two numbers

Find the median of 12, 10, 2, 3, 15, and 12

First put the numbers in order: 2, 3, 10, 12, 12, and 15

Since an even number of numbers is given, two numbers share the middle (10 and 12) Find the aver-age of 10 and 12 to find the median

The median is 11

10  12

2 22

2

2  3  5  10  15

5

n C r n!

r! 1n

n P r n!

– A R I T H M E T I C –

3 3 6

Mode is the number that appears the most in a set of numbers and is usually the easiest to find.

Find the mode of 33, 32, 34, 99, 66, 34, 12, 33, and 34

Since 34 appears the most (three times), it is the mode of the set

NOTE: It is possible for there to be no mode or several modes in a set.

Range is the difference between the largest and the smallest numbers in the set.

Find the range of the set 14, –12, 13, 10, 22, 23, –3, 10

Since –12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them

23 – (–12) = 23 + (+12) = 35 The range is 35

 Tr a n s l a t i n g E x p r e s s i o n s a n d E q u a t i o n s

Translating sentences and word problems into mathematical expressions and equations is similar to trans-lating two different languages The key words are the vocabulary that tells what operations should be done and in what order Use the following chart to help you with some of the key words used on the GMAT® quan-titative section

SUM MORE THAN

ADDED TO

PLUS INCREASED BY

PRODUCT TIMES MULTIPLIED BY

QUOTIENT DIVIDED BY

EQUAL TO TOTAL

DIFFERENCE LESS THAN SUBTRACTED FROM MINUS DECREASED BY FEWER THAN

Algebra

21

The following is an example of a problem where knowing the key words is necessary:

Fifteen less than five times a number is equal to the product of ten and the number What is the number?

Translate the sentence piece by piece:

Fifteen less than five times the number equals the product of 10 and x.

Subtract 5x from both sides: 5x – 5x – 15 = 10x – 5x

Divide both sides by 5:

–3 = x

It is important to realize that the key words less than tell you to subtract from the number and the key word product reminds you to multiply.

 C o m b i n i n g L i k e Te r m s a n d P o l y n o m i a l s

In algebra, you use a letter to represent an unknown quantity This letter is called the variable The number preceding the variable is called the coefficient If a number is not written in front of the variable, the coeffi-cient is understood to be one If any coefficoeffi-cient or variable is raised to a power, this number is the exponent.

3x Three is the coefficient and x is the variable.

xy One is the coefficient, and both x and y are the variables.

–2x3y Negative two is the coefficient, x and y are the variables, and three is the exponent of x.

Another important concept to recognize is like terms In algebra, like terms are expressions that have

exactly the same variable(s) to the same power and can be combined easily by adding or subtracting the coef-ficients

Examples

3x + 5x These terms are like terms, and the sum is 8x.

4x2y + –10x2y These terms are also like terms, and the sum is –6x2y.

2xy2+ 9x2y These terms are not like terms because the variables, taken with their powers,

are not exactly the same They cannot be combined

–15

5 5x

5

– A L G E B R A –

3 4 0

A polynomial is the sum or difference of many terms and some have specific names:

8x2 This is a monomial because there is one term.

3x + 2y This is a binomial because there are two terms.

4x2+ 2x – 6 This is a trinomial because there are three terms.

 L a w s o f E x p o n e n t s

■ When multiplying like bases, add the exponents: x2× x3= x2 + 3 = x5

■ When dividing like bases, subtract the exponents:

■ When raising a power to another power, multiply the exponents:

■ Remember that a fractional exponent means the root:x = x12and 3

x

 = x13

The following is an example of a question involving exponents:

Solve for x: 2 x + 2= 83

a 1

b 3

c 5

d 7

e 9

The correct answer is d To solve this type of equation, each side must have the same base Since 8 can

be expressed as 23, then 83= (23)3= 29 Both sides of the equation have a common base of 2, so set the

expo-nents equal to each other to solve for x x + 2 = 9 So, x = 7.

 S o l v i n g L i n e a r E q u a t i o n s o f O n e Va r i a b l e

When solving this type of equation, it is important to remember two basic properties:

■ If a number is added to or subtracted from one side of an equation, it must be added to or subtracted from the other side

■ If a number is multiplied or divided on one side of an equation, it must also be multiplied or divided

on the other side

1x223 x23 x6

x5

x2= x5 – 2 = x3

Linear equations can be solved in four basic steps:

1 Remove parentheses by using distributive property.

2 Combine like terms on the same side of the equal sign.

3 Move the variables to one side of the equation.

4 Solve the one- or two-step equation that remains, remembering the two previous properties.

Examples

Solve for x in each of the following equations:

a 3x – 5 = 10

Add 5 to both sides of the equation: 3x – 5 + 5 = 10 + 5

Divide both sides by 3:

x = 5

b 3 (x – 1) + x = 1

Use distributive property to remove parentheses:

3x – 3 + x = 1

Combine like terms: 4x – 3 = 1

Add 3 to both sides of the equation: 4x – 3 + 3 = 1 + 3

Divide both sides by 4:

x = 1

c 8x – 2 = 8 + 3x

Subtract 3x from both sides of the equation to move the variables to one side:

8x – 3x – 2 = 8 + 3x – 3x

Add 2 to both sides of the equation: 5x – 2 + 2 = 8 + 2

Divide both sides by 5:

x = 2

 S o l v i n g L i t e r a l E q u a t i o n s

A literal equation is an equation that contains two or more variables It may be in the form of a formula You may be asked to solve a literal equation for one variable in terms of the other variables Use the same steps that you used to solve linear equations

5x

5 10 5

4x

4 4 4

3x

3 15 3

– A L G E B R A –

3 4 2

25 61  25 64 

27 1  27 3 

25

1

 25 

3 ... keep the radicand the same

Examples

1. 3 2 +2 2 =5 2

2. 4 2? ?? 2= 3 2

3. 6 2 + 35 cannot be combined because they are... the set 14, – 12, 13, 10, 22 , 23 , –3, 10

Since – 12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them

23 – (– 12) = 23 + (+ 12) = 35 The