Master gmat 2010 part 24 ppsx

Finding a Ratio A GMAT question might ask you to determine a ratio based on given quantities.. We’ll place special emphasis on how the test makers incorporate algebraic features into GMA

The correct answer is (C) After the first 10% discount, the price was $450 ($500 minus

10% of $500) After the second discount, which is calculated based on the $450 price, the price

of the stereo is $405 ($450 minus 10% of $450) A 20% tax on $405 is $81 Thus, the customer

has paid $405 1 $81 5 $486

A percent-change problem might also involve an accompanying chart or graph, which provides

the numbers needed for the calculation

6.

200

160

180

140

120

100

80

60

40

20

Year

Holden Software

Stock Price

0

Based on the graph above, the average low price of Holden Software stock for the

two-year period 2000–2001 was approximately what percent lower than its average

high price for the two-year period 2003–2004?

(A) 25

(B) 37

(C) 45

(D) 52

(E) 75

The correct answer is (D) Annual low prices (represented by black bars) for 2000 and 2001

were $60 and $80, respectively, which yield an average of $70 for the two-year period Annual

high prices (represented by gray bars) for 2003 and 2004 were approximately $190 and $100,

respectively, which yield an average of $145 The percent decrease from $145 to $70 ' 52%

The only possible answer choice is (D)

Something to keep in mind: If a question based on a bar graph, line graph, or pie chart asks

for an approximation, the test makers are telling you that it’s okay to round off numbers you

glean from the chart or graph For example, in the preceding question, a rough estimate of

$190 for the high 2003 stock price was close enough to determine the correct answer choice

RATIOS AND PROPORTION

A ratio expresses proportion or comparative size—the size of one quantity relative to the size

of another As with fractions, you can simplify ratios by dividing common factors For example, given a class of 28 students—12 freshmen and 16 sophomores:

• The ratio of freshmen to sophomores is 12:16, or 3:4

• The ratio of freshmen to the total number of students is 12:28, or 3:7

• The ratio of sophomores to the total number of students is 16:28, or 4:7

Finding a Ratio

A GMAT question might ask you to determine a ratio based on given quantities This is the easiest type of GMAT ratio question

7 A class of 56 students contains only freshmen and sophomores If 21 of the students

are sophomores, what is the ratio of the number of freshmen to the number of sophomores in the class?

(A) 3:5

(B) 5:7

(C) 5:3

(D) 7:4 (E) 2:1

The correct answer is (C) Since 21 of 56 students are sophomores, 35 must be freshmen.

The ratio of freshmen to sophomores is 35:21 To simplify the ratio to simplest terms, divide both numbers by 7, giving you a ratio of 5:3

Determining Quantities from a Ratio (Part-to-Whole Analysis)

You can think of any ratio as parts adding up to a whole For example, in the ratio 5:6,

5 parts 1 6 parts 5 11 parts (the whole) If the actual total quantity were 22, you’d multiply each element by 2: 10 parts 1 12 parts 5 22 parts (the whole) Notice that the ratios are the same: 5:6 is the same ratio as 10:12

You might be able to solve a GMAT ratio question using this part-to-whole approach

8 A class of students contains only freshmen and sophomores If 18 of the students are

sophomores, and if the ratio of the number of freshmen to the number of sophomores

in the class is 5:3, how many students are in the class?

(A) 30

(B) 36

(C) 40

(D) 48 (E) 56

The correct answer is (D) Using a part-to-whole analysis, look first at the ratio and the

sum of its parts: 5 (freshmen) 1 3 (sophomores) 5 8 (total students) These aren’t the actual quantities, but they’re proportionate to those quantities Given 18 sophomores altogether, sophomores account for 3 parts—each part containing 6 students Accordingly, the total number of students must be 6 3 8 5 48

Determining Quantities from a Ratio (Setting Up a Proportion)

Since you can express any ratio as a fraction, you can set two equivalent, or proportionate,

ratios equal to each other, as fractions So the ratio 16:28 is proportionate to the ratio 4:7

because16

285

4

7 If one of the four terms is missing from the equation (the proportion), you can

solve for the missing term using algebra So if the ratio 3:4 is proportionate to 4:x, you can

solve for x in the equation 3

45

4

x Using the cross-product method, equate products of

numerator and denominator across the equation:

~3!~x! 5 ~4!~4!

3x 5 16

x 516

3, or 5

1 3

Or, since the numbers are simple, shortcut the algebra by asking yourself what number you

multiply the first numerator (3) by for a result that equals the other numerator (4):

3 34

35 4 (a no-brainer calculation).

So you maintain proportion (equal ratios) by also multiplying the first denominator (4) by 4

3:

4 34

35

16

3 (another no-brainer calculation)

Even if the quantities in a question strike you as decidedly “unround,” it’s a good bet that

doing the math will be easier than you might first think

9 If 3 miles are equivalent to 4.83 kilometers, then 11.27 kilometers are equivalent to

how many miles?

(A) 1.76

(B) 5.9

(C) 7.0

(D) 8.4

(E) 16.1

The correct answer is (C) The question essentially asks, “3 is to 4.83 as what is to 11.27?”

Set up a proportion, then solve for x by the cross-product method:

3

4.835

x

11.27

~4.83!~x! 5 ~3!~11.27!

x 5~3!~11.27!

4.83

x 533.81

4.83, or 7 Notice that, despite all the intimidating decimal numbers, the solution turns out to be a tidy

number: 7 That’s typical of the GMAT

Now let’s focus on more advanced applications of fractions, percents, decimals, ratios, and proportion We’ll place special emphasis on how the test makers incorporate algebraic features into GMAT questions covering these concepts:

• Altering fractions and ratios

• Ratios involving more than two quantities

• Proportion problems with variables We’ll take a look at how test makers design tougher-than-average GMAT questions involving:

• Arithmetic mean (simple average) and median (two ways that a set of numbers can

be measured as a whole)

• Standard deviation (a quantitative expression of the dispersion of a set of measurements)

• Geometric sequences (the pattern from one number to the next in an exponential list

of numbers)

• Permutations (the possibilities for arranging a set of objects)

• Combinations (the possibilities for selecting groups of objects from a set)

• Probability (the statistical chances of a certain event, permutation, or combination occurring)

ALTERING FRACTIONS AND RATIOS

An average test taker might assume that adding the same positive quantity to a fraction’s numerator (p) and to its denominator (q) leaves the fraction’s valueSp

qDunchanged But this

is true if and only if the original numerator and denominator were equal to each other.

Otherwise, the fraction’s value will change Remember the following three rules, which apply

to any positive numbers x, p, and q (the first one is the no-brainer you just read):

If p 5 q, then p

q5

p 1 x

q 1 x (The fraction’s value remains unchanged and is always 1.)

If p q, then p

q.

p 1 x

q 1 x (The fraction’s value will decrease.)

If p , q, then p

q,

p 1 x

q 1 x (The fraction’s value will increase.)

A GMAT question might ask you to alter a ratio by adding or subtracting from one (or both) terms in the ratio The rules for altering ratios are the same as for altering fractions In either case, set up a proportion and solve algebraically for the unknown term

10 A drawer contains exactly half as many white shirts as blue shirts If four more

shirts of each color were to be added to the drawer, the ratio of white to blue shirts

would be 5:8 How many blue shirts does the drawer contain?

(A) 14

(B) 12

(C) 11

(D) 10

(E) 9

The correct answer is (B) Represent the original ratio of white to blue shirts by the

fraction x

2x , where x is the number of white shirts, then add 4 to both the numerator and

denominator Set this fraction equal to 5

8 (the ratio after adding shirts) Cross-multiply to

solve for x:

x 1 4

2x 1 45

5 8

8x 1 32 5 10x 1 20

12 5 2x

x 5 6

The original denominator is 2x, or 12.

RATIOS WITH MORE THAN TWO QUANTITIES

You approach ratio problems involving three or more quantities the same way as those

involving only two quantities The only difference is that there are more “parts” that make up

the “whole.”

11 Three lottery winners—X, Y, and Z—are sharing a lottery jackpot X’s share is1

5of

Y’s share and1

7 of Z’s share If the total jackpot is $195,000, what is the dollar

amount of Z’s share?

(A) $15,000

(B) $35,000

(C) $75,000

(D) $105,000

(E) $115,000

The correct answer is (D) At first glance, this problem doesn’t appear to involve ratios.

(Where’s the colon?) But it does The ratio of X’s share to Y’s share is 1:5, and the ratio of X’s

share to Z’s share is 1:7 So you can set up the following triple ratio:

X:Y:Z 5 1:5:7

TIP

Remember: when you add (or subtract) the same number from both the numerator and denominator of a fraction—or from each term in a ratio—you alter the fraction or ratio, unless the original ratio was 1:1 (in which case the ratio is unchanged).

X’s winnings account for 1 of 13 equal parts (1 1 5 1 7) of the total jackpot. 1

13of $195,000 is

$15,000 Accordingly, Y’s share is 5 times that amount, or $75,000, and Z’s share is 7 times

that amount, or $105,000

In handling word problems involving ratios, think of a whole as the sum of its fractional parts,

as in the method used to solve the preceding problem: 1

13 (X’s share) 1

5

13 (Y’s share) 1 7

13(Z’s share) 5 1 (the whole jackpot).

PROPORTION PROBLEMS WITH VARIABLES

A GMAT proportion question might use letters instead of numbers—to focus on the process

rather than the result You can solve these problems algebraically or by using the plug-in strategy

12 A candy store sells candy only in half-pound boxes At c cents per box, which of the

following is the cost of a ounces of candy? [1 pound 5 16 ounces]

(A) c

a

(B) a

16c

(C) ac

(D) ac

8

(E) 8c

a

The correct answer is (D) This question is asking: “c cents is to one box as how many cents

are to a ounces?” Set up a proportion, letting x equal the cost of a ounces Because the question asks for the cost of ounces, convert 1 box to 8 ounces (a half pound) Use the

cross-product method to solve quickly:

c

85

x a 8x 5 ac

x 5 ac

8 You can also use the plug-in strategy for this question, either instead of algebra or, better yet,

to check the answer you chose using algebra Pick easy numbers to work with, such as 100 for

c and 16 for a At 100 cents per 8-ounce box, 16 ounces of candy cost 200 cents Plug your numbers for a and c into each answer choice Only choice (D) gives you the number 200 you’re

looking for

ARITHMETIC MEAN, MEDIAN, MODE, AND RANGE

Arithmetic mean (simple average), median, mode, and range are four different ways to

describe a set of terms quantitatively Here’s the definition of each one:

arithmetic mean (average): In a set of n measurements, the sum of the

measure-ments divided by n.

median: The middle measurement after the measurements are ordered by size (or

the average of the two middle measurements if the number of measurements is

even)

mode: The measurement that appears most frequently in a set.

range: The difference between the greatest measurement and the least

measurement

For example, given a set of six measurements, {8,24,8,3,2,7}:

mean 5 4 (8 2 4 1 8 1 3 1 2 1 7) 4 6 5 24 4 6 5 4

median 5 5 The average of 3 and 7—the two middle measurements in the set

ordered in this way: {24,2,3,7,8,8}

mode 5 8 8 appears twice (more frequently than any other measurement)

range 5 12 The difference between 8 and 24

For the same set of values, the mean (simple average) and the median can be, but are not

necessarily, the same For example: {3,4,5,6,7} has both a mean and median of 5 However, the

set {22,0,5,8,9} has a mean of 4 but a median of 5

The GMAT covers arithmetic mean far more frequently than median, mode, or range, so let’s

focus on problems involving mean First of all, in finding a simple average, be sure the

numbers being added are all of the same form or in terms of the same units

13 What is the average of1

5, 25%, and 0.09?

(A) 0.18

(B) 20%

(C) 1

4

(D) 0.32

(E) 1

3

The correct answer is (A) Since the answer choices are not all expressed in the same form,

first rewrite numbers as whichever form you think would be easiest to work with when you

add the numbers together In this case, the easiest form to work with is probably the decimal

form So rewrite the first two numbers as decimals, and then find the sum of the three

numbers: 0.20 1 0.25 1 0.09 5 0.54 Finally, divide by 3 to find the average: 0.54 4 3 5 0.18

To find a missing number when the average of all the numbers in a set is given, plug into the

arithmetic-mean formula all the numbers you know—which include the average, the sum of

the other numbers, and the number of terms Then, use algebra to find the missing number.

Or, you can try out each answer choice, in turn, as the missing number until you find one that results in the average given

14 The average of five numbers is 26 Four of the numbers are 212, 90, 226, and 10.

What is the fifth number?

(A) 16

(B) 42

(C) 44

(D) 68 (E) 84

The correct answer is (D) To solve the problem algebraically, let x 5 the missing number.

Set up the arithmetic-mean formula, then solve for x:

26 5~90 1 10 2 12 2 26! 1 x

5

26 562 1 x

5

130 5 62 1 x

68 5 x

Or, you can try out each answer choice in turn Start with the middle value, 44 (choice (C)) The sum of 44 and the other four numbers is 106 Dividing this sum by 5 gives you 21.2—a number less than the average of 26 that you’re aiming for So you know the fifth number is greater than 44—and that leaves choices (D) and (E) Try out the number 68 (choice (D)), and you’ll obtain the average of 26

If the numbers are easy to work with, you might be able to determine a missing term, given the simple average of a set of numbers, without resorting to algebra Simply apply a dose

of logic

15 If the average of six consecutive multiples of 4 is 22, what is the greatest of these

integers?

(A) 22

(B) 24

(C) 26

(D) 28 (E) 32

The correct answer is (E) You can answer this question with common sense—no algebra

required Consecutive multiples of 4 are 4, 8, 12, 16, etc Given that the average of six such numbers is 22, the two middle terms (the third and fourth terms) must be 20 and 24 (Their average is 22.) Accordingly, the fifth term is 28, and the sixth and greatest term is 32

TIP

Numerical answer

choices are listed

in ascending

order of value, so

if you’re working

backward from

the choices, start

with (C), the

median value If

(C) is either too

great or too little,

you’ve narrowed

down the options

either to (A) and

(B) or to (D)

and (E).

On the GMAT, easier questions involving simple average might ask you to add numbers

together and divide a sum A tougher question might ask you to perform the following task

(which involves algebra) such as:

Find the value of a number that changes an average from one number to another

When an additional number is added to a set, and the average of the numbers in the set

changes as a result, you can determine the value of the number that’s added by applying the

arithmetic-mean formula twice

16 The average of three numbers is 24 If a fourth number is added, the arithmetic

mean of all four numbers is 21 What is the fourth number?

(A) 210

(B) 2

(C) 8

(D) 10

(E) 16

The correct answer is (C) To solve the problem algebraically, first determine the sum of

the three original numbers by the arithmetic-mean formula:

24 5a 1 b 1 c

3 Then, apply the formula again accounting for the additional (fourth) number The new

average is 21, the sum of the other three numbers is 212, and the number of terms is 4 Solve

for the missing number (x):

21 5212 1 x

4

24 5 212 1 x

8 5 x

You approach arithmetic-mean problems that involve variables instead of (or in addition to)

numbers in the same way as those involving only numbers Just plug the information you’re

given into the arithmetic-mean formula, and then solve the problem algebraically

17 If A is the average of P, Q, and another number, which of the following represents

the missing number?

(A) 3A 2 P 2 Q

(B) A 1 P 1 Q

(C) A 1 P 2 Q

(D) A 2 P 1 Q

(E) 3A 2 P 1 Q

The correct answer is (A) Let x 5 the missing number Solve for x by the arithmetic-mean

formula:

A 5 P 1 Q 1 x

3

3A 5 P 1 Q 1 x 3A 2 P 2 Q 5 x

STANDARD DEVIATION

Standard deviation is a measure of dispersion among members of a set Computing standard

deviation involves these five steps:

Compute the arithmetic mean (simple average) of all terms in the set

Compute the difference between the mean and each term

Square each difference you computed in step (2)

Compute the mean of the squares you computed in step (3)

Compute the non-negative square root of the mean you computed in step (4) For example, here’s how you’d determine the standard deviation of Distribution A: {21, 2, 3, 4}:

• Arithmetic mean 521 1 2 1 3 1 4

8

45 2

• The difference between the mean (2) and each term:

2 2 (21) 5 3; 2 2 2 5 0; 3 2 2 5 1; 4 2 2 5 2

• The square of each difference:

{32, 02, 12, 22} 5 {9, 0, 1, 4}

• The mean of the squares:

9 1 0 1 1 1 4

14

4 5

7 2

• The standard deviation of Distribution A 5Î7

2

A GMAT question might ask you to calculate standard deviation (as in the preceding

example) Or, a question might ask you to compare standard deviations You might be able to

ALERT!

Don’t try the

plug-in strategy to

solve the problem

on this page; it’s

too complex Be

flexible and use

shortcuts

wherever you

can—but

recognize their

limitations.