Master gmat 2010 part 23 pdf

Forms, Relationships,and Sets OVERVIEW • Percents, fractions, and decimals • Simplifying and combining fractions • Decimal place values and operations • Simple percent problems • Percent

Forms, Relationships,

and Sets

OVERVIEW

• Percents, fractions, and decimals

• Simplifying and combining fractions

• Decimal place values and operations

• Simple percent problems

• Percent increase and decrease

• Ratios and proportion

• Altering fractions and ratios

• Ratios with more than two quantities

• Proportion problems with variables

• Arithmetic mean, median, mode, and range

• Standard deviation

• Geometric sequences

• Arithmetic sequences

• Permutations

• Combinations

• Probability

• Summing it up

In this chapter, you’ll focus first on various forms of numbers and

relationships between numbers Specifically, you’ll learn how to:

another

.chapter

• Determine ratios between quantities and determine quantities from ratios

Next, you’ll explore the following topics, all of which involve sets (defined groups) of numbers

or other objects:

whole)

numbers)

occurring)

PERCENTS, FRACTIONS, AND DECIMALS

Any real number can be expressed as a fraction, a percent, or a decimal number For instance, 2

often require you to rewrite one form as another as part of solving the problem at hand You should know how to write any equivalent quickly and confidently

To rewrite a percent as a decimal, move the decimal point two places to the left (and drop the

percent sign) To rewrite a decimal as a percent, move the decimal point two places to the

right (and add the percent sign).

95% 5 0.95 0.004 5 0.4%

To rewrite a percent as a fraction, divide by 100 (and drop the percent sign) To rewrite a fraction as a percent, multiply by 100 (and add the percent sign) Percents greater than 100

are equivalent to numbers greater than 1

81

1 10 3

300

75

1

Beware: Percents greater than 100 or less than 1 (such as 457% and 0.067%) can be confusing, because it’s a bit harder to grasp their magnitude To guard against errors when writing, keep

in mind the general magnitude of the number you’re dealing with For example, think of 0.09% as just less than 0.1%, which is one-tenth of a percent, or a thousandth (a pretty small

ALERT!

Although this is

the most basic of

all the math

review chapters

in this book, don’t

skip it The skills

covered here are

basic building

blocks for other,

more difficult

types of questions

covered in the

following chapters.

To rewrite a fraction as a decimal, simply divide the numerator by the denominator, using

long division A fraction-to-decimal equivalent might result in a precise value, an

approximation with a repeating pattern, or an approximation with no repeating pattern:

5

The equivalent decimal number is precise after three decimal places

5

The equivalent decimal number can only be approximated (the digit

5 repeats indefinitely)

5

The equivalent decimal number can safely be approximated

Certain fraction-decimal-percent equivalents show up on the GMAT more often than others

The numbers in the following tables are the test makers’ favorites because they reward test

takers who recognize quick ways to deal with numbers Memorize these conversions so that

they’re second nature to you on exam day

Percent Decimal Fraction

2

4

4

10

10

10

10

1 3

1 3

2 3

2 3

Percent Decimal Fraction

2 3

1 6

1 3

5 6

5

5

5

5

8

8

8

8

SIMPLIFYING AND COMBINING FRACTIONS

A GMAT question might ask you to combine fractions using one or more of the four basic operations (addition, subtraction, multiplication, and division) The rules for combining fractions by addition and subtraction are very different from the ones for multiplication and division

Addition and Subtraction and the LCD

To combine fractions by addition or subtraction, the fractions must have a common

denominator If they already do, simply add (or subtract) numerators If they don’t, you’ll need

to find one You can always multiply all of the denominators together to find a common denominator, but it might be a big number that’s clumsy to work with So instead, try to find

the least (or lowest) common denominator (LCD) by working your way up in multiples of the

largest of the denominators given For denominators of 6, 3, and 5, for instance, try out successive multiples of 6 (12, 18, 24 ), and you’ll hit the LCD when you get to 30

1. 5

5

5

(A) 15

11

(B) 5

2

(C) 15

6

(D) 10

3

(E) 15

3

The correct answer is (D) To find the LCD, try out successive multiples of 6 until you come

across one that is also a multiple of both 3 and 2 The LCD is 6 Multiply each numerator by the same number by which you would multiply the fraction’s denominator to give you the LCD of 6 Place the three products over this common denominator Then, combine the numbers in the numerator (Pay close attention to the subtraction sign!) Finally, simplify to lowest terms:

5

5

5

10

5

15 6

6

Multiplication and Division

To multiply fractions, multiply the numerators and multiply the denominators The

denominators need not be the same To divide one fraction by another, multiply by the

reciprocal of the divisor (the number after the division sign)

Multiplication:

1

5

1

~1!~5!~1!

5 42

Division:

2 5 3 4

4

~2!~4!

8 15

To simplify the multiplication or division, cancel factors common to a numerator and a

denominator before combining fractions It’s okay to cancel across fractions Take, for instance

4

3

1

3

1

1

1

Apply the same rules in the same way to variables (letters) as to numbers

2. 2

a3

b

a

8

c5 ?

(A) ab

4c

(B) 10b

9c

(C) 8

5

(D) 16b

5ac

(E) 4b

5c

The correct answer is (E) Since you’re dealing only with multiplication, look for factors

and variables (letters) in any numerator that are the same as those in any denominator

Canceling common factors leaves

2

b

1

2

c

5c

Mixed Numbers and Multiple Operations

A mixed number consists of a whole number along with a simple fraction—for example, the

fraction To do so, follow these three steps:

1 Multiply the denominator of the fraction by the whole number

2 Add the product to the numerator of the fraction

3 Place the sum over the denominator of the fraction

~3!~4! 1 2

14 3

To perform multiple operations, always perform multiplication and division before you perform addition and subtraction

3.

2

8

(A) 1

3

(B) 3

8

(C) 11

6

(D) 17

6

(E) 11

2

The correct answer is (A) First, rewrite all mixed numbers as fractions Then, eliminate

the complex fraction by multiplying the numerator fraction by the reciprocal of the denominator fraction (cancel across fractions before multiplying):

9 2 9 8

4

11 3

Then, express each fraction using the common denominator 3, then subtract:

DECIMAL PLACE VALUES AND OPERATIONS

Place value refers to the specific value of a digit in a decimal For example, in the decimal

682.793:

The digit 6 is in the “hundreds” place

The digit 8 is in the “tens” place

The digit 2 is in the “ones” place

The digit 7 is in the “tenths” place

The digit 9 is in the “hundredths” place

The digit 3 is in the “thousandths” place

So you can express 682.793 as follows:

9

3 1,000

To approximate, or round off, a decimal, round any digit less than 5 down to 0, and round any

digit greater than 5 up to 0 (adding one digit to the place value to the left)

The value of 682.793, to the nearest hundredth, is 682.79

The value of 682.793, to the nearest tenth, is 682.8

The value of 682.793, to the nearest whole number, is 683

The value of 682.793, to the nearest ten, is 680

The value of 682.793, to the nearest hundred, is 700

Multiplying Decimals

The number of decimal places (digits to the right of the decimal point) in a product should be

the same as the total number of decimal places in the numbers you multiply So to multiply

decimals quickly, follow these three steps:

Multiply, but ignore the decimal points

Count the total number of decimal places among the numbers you multiplied

Include that number of decimal places in your product

Here are two simple examples:

Example 1

Example 2

Dividing Decimals

When you divide (or compute a fraction), you can move the decimal point in both numbers by the same number of places either to the left or right without altering the quotient (value of the fraction) Here are three related examples:

114

114

114,000

GMAT questions involving place value and decimals usually require a bit more from you than just identifying a place value or moving a decimal point around Typically, they require you to combine decimals with fractions or percents

4 Which of the following is nearest in value to1

1

10,000

(B) 33

100,000

(C) 99

100,000

(D) 33

10,000

(E) 99

10,000

The correct answer is (A) There are several ways to convert and combine the four numbers

1

1

1

9

9

1

TIP

Eliminate decimal

points from

fractions, as well

as from percents,

to help you see

more clearly the

magnitude of the

quantity you’re

dealing with.

SIMPLE PERCENT PROBLEMS

On the GMAT, a simple problem involving percent might ask you to perform any one of these

four tasks:

Find a percent of a percent

Find a percent of a number

Find a number when a percent is given

Find what percent one number is of another

The following examples show you how to handle these four tasks (task 4 is a bit trickier than

the others):

Rewrite 2% as 0.02, then multiply:

0.02 3 0.02 5 0.0004, or 0.04%

Rewrite 35% as 0.35, then multiply:

0.35 3 65 5 22.75 Finding a number when a percent

is given

7 is 14% of what number?

Translate the question into an algebraic equation, writing the percent as either a fraction or decimal:

7 5 14% of x

7 5 0.14x

x 5 7

1

100

Finding what percent one number

is of another

90 is what % of 1500?

Set up an equation to solve for the percent:

90

x

100

1500x 5 9000 15x 5 90

x 590

PERCENT INCREASE AND DECREASE

In example 4, you set up a proportion (90 is to 1500 as x is to 100.) You’ll need to set up a

proportion for other types of GMAT questions as well, including questions about ratios, which you’ll look at a bit later in this chapter

The concept of percent change is one of the test makers’ favorites Here’s the key to answering

questions involving this concept: Percent change always relates to the value before the

change Here are two simple illustrations:

10 increased by what percent is 12? 1 The amount of the increase is 2

2 Compare the change (2) to the original number (10)

or 20%

12 decreased by what percent is 10? 1 The amount of the decrease is 2

2 Compare the change (2) to the original number (12)

2

approxi-mately 16.7%

Notice that the percent increase from 10 to 12 (20%) is not the same as the percent decrease

the two questions

A typical GMAT percent-change problem will involve a story—about a type of quantity such

as tax, profit or discount, or weight—in which you need to calculate successive changes in percent For example:

Whatever the variation, just take the problem one step at a time and you’ll have no trouble handling it

5 A stereo system originally priced at $500 is discounted by 10%, then by another

10% If a 20% tax is added to the purchase price, how much would a customer buying the system at its lowest price pay for it, including tax, to the nearest dollar?

(A) $413

(B) $480

(C) $486

(D) $500 (E) $512