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Fractions, Decimals, & Percents
Clarifies Concepts and Builds Computation Skills

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x

50

X 4 _ X l
X

2 _ J_

25

x ~ 50 x = 100

Then, cross-multiply: x = 100.

2. 4,250:
2x

Ax

y

8,500

2x _

x

~y ” 2,125

First, simplify the ratio on the right-hand side of the equation.

Then, cross-multiply: 4,250x = xy.
Divide both sides of the equation by *:

3. 11: Write a proportion to solve this problem:

Cross-multiply to solve:

red

2

y —4,250.

22

2x —22 x — 11

4. 43: First, establish the starting number of men and women with a proportion, and simplify:
5 men

35 men

7 women

x women

Xm X a n
1

7 women

7 men

x women

Cross-multiply: x —49.
If 6 women leave the room, there are 49 - 6 = 43 women left.

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71

Ratios

Chapter 4

5. 33 hours: Use an equation with the Unknown Multiplier to represent the total hours put in by the three people:

2x + 3x + 5x= 110
10*= 110 x= 11
Therefore, the hardest working person put in 5(11) = 55 hours, and the person who worked the least put in 2(11) = 22 hours. This represents a difference of 55 - 22 = 33 hours.
6. 3 mL: The correct ratio is 1:4, which means that there should be x parts bleach and 4x parts water.

x

However, Alexandra put in half as much bleach as she should have, so she put in — parts bleach. You

x

can represent this with an equation: — + 4x = 27.

x+ 8x= 54
9* = 5 4 x= 6
You were asked to find how much bleach Alexandra used. This equalled x!2, so Alexandra used 6/2 = 3 mL of bleach.

7. 8 cm3:
Real: V= s5

64 = s3 ^ s = 4

Model 0.5 cm
Real

1m

xcm
4m

The length of a side on the real sculpture is 4 m.
Set up a proportion and solve for x, the side of the model 05_x
1 ” 4

x —2 cm

The length of a side on the model is 2 cm.

Model: K = ^ = (2)3 = 8

The volume of the model is 8 cm3.

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C h a p te r^
Fractions, Decimals, & Percents

FDPs

Common FDP Equivalents

,

Converting Among Fractions, Decimals and Percents
When to Use Which Form

FDPs
GMAT problems often do not test fractions, decimals, and percents in isolation. Instead, many prob­ lems that test your understanding of non-integer numbers involve some kind of combination of frac­ tions, decimals, and percents.
For this reason, we refer to these problems as FDPs (an abbreviation for fraction-decimal-percent). In order to achieve success with FDP problems on the GMAT, you must understand the connections be­ tween fractions, decimals, and percents; you should be able to shift amongst the three comfortably and quickly. In a very real sense, fractions, decimals, and percents are three different ways of expressing the exact same thing: a part-whole relationship:
A fraction expresses a part-whole relationship in terms of a numerator (the part) and a denominator (the whole).
A decimal expresses a part-whole relationship in terms of place value (a tenth, a hun­ dredth, a thousandth, etc.).

A percent expresses the special part-whole relationship between a number (the part) and one hundred (the whole).

FP
Ds

Chapter 5

Common FDP Equivalents
You should memorize the following common equivalents:

Fraction

Decimal

Percent

Fraction

Decimal

Percent

Xoo

0.01

1%

3/
/5

0.6

60%

Xo

0.02

2%

5/
/8

0.625

62.5%

X 5

0.04

4%

2/
73

0.6 = 0.667

= 66.7%

Xo

0.05

5%

0.7

70%

Xo

0.10

10%

0.75

75%

X

0.11 = 0.111

= 11.1%

X

0.125

12.5%

X

0.16 = 0.167

= 16.7%

X

0.2

20%

X

0.25

25%

Xo

0.3

30%

X

0.3 = 0.333

= 33.3%

3
/
/8

0.375

37.5%

2/

0.4

40%

1/

0.5

50%

/5

72

7
6

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%
V

/5
5/
/6

0.8

80%

0.83 = 0.833

= 83.3%

7/
/8

0.875

87.5%

0.9

90%

1

100%

1.25

125%

4/
/3

1.3 = 1.33

133%

3/
72

1.5

150%

%

1.75

175%

%.

X
X

FP
Ds

Converting Among Fractions, Decimals, and Percents
The following chart reviews the ways to convert from fractions to decimals, from decimals to fractions, from fractions to percents, from percents to fractions, from decimals to percents, and from percents to decimals. FROM 7 0

I

FRACTIO N 8

DECIMAL 0.375
Divide the numerator by the denominator:
3 + 8 = 0.375

FRACTION
3
8

PERCENT 37.5%
Divide the numerator by the denominator and move the decimal two places to the right:
3 + 8 = 0.375 -► 37.5%

Use long division if necessary. DECIMAL
0.375

PERCENT
37.5%

Move the decimal point two places to the right:
0.375 -► 37.5%

Use the place value of the last digit in the decimal as the denominator, and put the decimals digits in the numerator. Then simplify:
375 3
1,000 8
Use the digits of the percent for the numerator and 100 for the denominator. Then simplify: 37.5 _ 3
100 8

Find the percent’ decimal s point and move it two places to the left:
37.5% -► 0.375

When to Use Which Form____________________
Fractions are good for cancelling factors in multiplications. They are also the best way of exactly expressing proportions that do not have clean decimal equivalents, such as 1/7. Switch to fractions if there is a handy fractional equivalent of the decimal or percent and/or you think you can cancel lots of factors.For example:

What is 37.5% of 240?
If you simply convert the percent to a decimal and multiply, you will have to do a fair bit of arithmetic:
0.375
x 240
0
15000
75000

9 .0 0
00

Alternatively, you can recognize that 0.375 = —.
8

(

So we have (0.375)(240) = -

(24tf 30) = 3(30) = 90.

This is much faster!

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FP
Ds
2
A dress is marked up 1 6 -% to a final price of $140. What is the original price of the dress?
2
From the previous page, you know that 16 —%

3

l i is equivalent to —. Thus, adding — a number to of 6

6

1 7 itself is the same thing as multiplying by 1 + —= —:
6 6

—x = 140
6

x = \ — ]l40 = ' A ' JU4CT20 = 120. The original price is $120.
7
j

J

Decimals, on the other hand, are good for estimating results or for comparing sizes. The reason is that the basis of comparison is equivalent (there is no denominator). The same holds true for percents. The implied denominator is always 100, so you can easily compare percents (of the same whole) to each other. To convert certain fractions to decimals or percents, multiply top and bottom by the same number:

25

17X4 = i 8_ = 0.68 = 68%
2 5 x 4 100

This process is faster than long division, but it only works when the denominator has only 2s and/or 5 s as factors.
In some cases, you might find it easier to compare a bunch of fractions by giving them all a common denominator, rather than by converting them all to decimals or percents. The general rule is this: Prefer fractions for doing multiplication or division, but prefer decimals and percents for doing addition or subtraction, for estimating numbers, or for comparing numbers.

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FP
Ds

C a te 5 hp r

Problem Set
1.

Express the following as fractions and simplify:

2.45

0.008

2.

Express the following as fractions and simplify:

420%

8%

3.

Express the following as decimals and simplify:

9
2

3.000
10.000

4.

Express the following as decimals and simplify:

5.

4

123

Express the following as percents:

1,000
10

25
8

6.

Express the following as percents:

80.4

0.0007

7.

Order from least to greatest:

8
18

0.8

40%

8.

Order from least to greatest:

1.19

120
84

131.44%

9.

What number is 62.5% of 192?

10. 200 is 16% of what number?

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7
9

FP
Ds

C a te 5 hp r

Solutions
1. To convert a decimal to a fraction, write it over the appropriate power of 10 and simplify:
45
9
49
2.45 = 2 ----- = 2 — (mixed) = — (im proper)

100

0.008 =

20

20

^

v

8
1,000

125

2. To convert a percent to a fraction, write it over a denominator of 100 and simplify:
420 21
1
420% = — = "
(im proper) = 4 ~ (mixed)
8
2
8% = -----= —
100 25
3. To convert a fraction to a decimal, divide the numerator by the denominator:
9
- = 9 + 2 = 4.5
2
It often helps to simplify the fraction before you divide:
3’00° . = A = o.3

10,000

10

4. To convert a mixed number to a decimal, simplify the mixed number first, if needed:
27
3
1— = 1 + 6 - = 7.75
4
4
1 2 ^ = 12 + 2 — = 14— = 14.6
3
3
3
Note: You do not have to know the “repeating bar” notation, but you should know that 2/3 = 0.6666.
5. To convert a fraction to a percent, rewrite the fraction with a denominator of 100:

1,000 10,000
----- = — -------= 10,000%
10
100
Or, convert the fraction to a decimal and shift the decimal point two places to the right:
2 5 - 8 = 3 ^ = 3.125 = 312.5%
6. To convert a decimal to a percent, shift the decimal point two places to the right:
80.4 = 8,040%
0.0007 = 0.07%

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FDPs

Chapter 5

7. 40% < — < 0.8: To order from least to greatest, express all the terms in the same form:
8
4
— = - = 0 .4 4 4 4 ... = 0.4
18 9
0.8 = 0.8
40% = 0.4
0.4 < 0.4 < 0.8
8. 1.19 < 131.44% <

120

: To order from least to greatest, express all the terms in the same form:

1.19= 1.19

120
84

= 1.4286

131.44% = 1.3144
1.19 < 1.3144 < 1.4286
9. 120: This is a percent vs. decimal conversion problem. If you simply recognize that 62.5% = 0.625
= ~ , this problem will be a lot easier: —Xl92 = —x 2 4 = 120. Multiplying 0.625 x 240 will take much
8
1
8
longer to complete.
10. 1,250: This is a percent vs. decimal conversion problem. If you simply recognize that 16% = 0.16 =
16
4
4
25
-----= — , this problem will be a lot easier: — x = 200, so x = 200 x — = 50 x 25 = 1,250. Dividing
100
25
25
4 out 200 -r 0.16 will probably take longer to complete.

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C h a p te r^
Fractions, Decimals, & Percents

FDP Strategies

Data Sufficiency Basics
What Does "Sufficient" Mean?
The DS Process
Putting It All Together
Putting It All Together (Again)
Concrete Values vs. Relative Values
Smart Numbers: Multiples o f the Denominators
When Not to Use Smart Numbers
Smart Numbers: Pick 100
Benchmark Values

FDP Strategies
The following five sections appear in all 5 quant strategy guides. If you are familiar with this informa­ tion, skip ahead to page 92 for new content.

Data Sufficiency Basics______________________
Every Data Sufficiency problem has the same basic form:
The Question Stem is (sometimes) made up of two parts:
(1) The Question: “
What day of the week is the party on?”
(2) Possible Additional Info: "Jons birthday party is this week”
This might simply be background OR could provide additional constraints or equations needed to solve the problem.
Jon's birthday party is this week. What day of the week is the party on?
(1) The party is not on Monday or Tuesday.
(2) The party is not on Wednesday, Thursday, or Friday.
(A) Statement (1) ALONE is sufficient, but statement (2) is NOT sufficient
(B) Statement (2) ALONE is sufficient, but statement (1) is NOT sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement
ALONE is sufficient
(D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient
-------------------------......................................

Following the question are two
Statements labeled (1) and (2).
To answer Data Sufficiency prob­ lems correctly, you need to decide

whether the statements provide enough information to answer the question. In other words, do you have sufficient data?

1 —

Lastly, you are given the Answer Choices.
These are the same for every Data Sufficiency problem so memorize them as soon as possible.

FDP Strategies

What Does "Sufficient" Mean?
The key to Data Sufficiency is to remember that it does not require you to answer the question asked in the question stem. Instead, you need to decide whether the statements provide enough information to answer the question.
Notice that in answer choices (A), (B), and (D), you are asked to evaluate each of the statements sepa­ rately. You must then decide if the information given in each is sufficient (on its own) to answer the question in the stem.
The correct answer choice will be:
(A) when Statement (1) provides enough information by itself, but Statement (2) does not,

(B) when Statement (2) provides enough information by itself, but Statement (1) does not,
OR

(D) when BO TH statements, independently, provide enough information.
But what happens when you cannot answer the question with either statement individually? Now you must put them together and decide if all of the information given is sufficient to answer the question in the stem.
If you must use the statements together, the correct answer choice will be:
(C) if together they provide enough information (but neither alone is sufficient),
OR
(E) if the statements, even together, do not provide enough information.
We will revisit the answer choices when we discuss a basic process for Data Sufficiency.

The PS Process____________________________
Data Sufficiency tests logical reasoning as much as it tests mathematical concepts. In order to master
Data Sufficiency, develop a consistent process that will help you stay on task. It is very easy to forget what you are actually trying to accomplish as you answer these questions.
To give yourself the best chance of consistently answering DS questions correctly, you need to be me­ thodical. The following steps can help reduce errors on every DS problem.

Step 1: Separate additional info from the actual question.
If the additional information contains constraints or equations, make a note on your scrap paper.

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FDP Strategies
Step 2: Determine whether the question is Value or Yes/No.
Value: The question asks for the value of an unknown (e.g., W hat is x?).
A statement is Sufficient when it provides 1 possible value.
A statement is Not Sufficient when it provides more than 1 possible value.

Yes/No: The question that is asked has two possible answers: Yes or No (e.g., Is x even?).
A statement is Sufficient when it provides a definite Yes or definite No.
A statement is Not Sufficient when the answer could be Yes or No.

Sufficient

Not Sufficient

Value

1 Value

More than 1 Value

Yes/No

1 Answer (Yes or No)

More than 1 Answer (Yes AND No)

Step 3: Decide exactly what the question is asking.
To properly evaluate the statements, you must have a very precise understanding of the question asked in the question stem. Ask yourself two questions:
1.

W hat, precisely, would be sufficient?

2.

W hat, precisely, would not be sufficient!

For instance, suppose the question is, “W hat is x?”
1.

W hat, precisely, would be sufficient? One value for x (e.g., x = 5).

2.

W hat, precisely, would not be sufficient? More than one value for x (e.g., x is prime).

Step 4: Use the Grid to evaluate the statements.
The answer choices need to be evaluated in the proper order. The Grid is a simple but effective tool to help you keep track of your progress. Write the following on your page:

AD
BCE

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Chapter 6

FDP Strategies
The two columns below will tell you how to work through the Grid:
First, evaluate Statement (1).

—

If Statement (1) is Sufficient, only choices (A) and (D) are possible.
Cross off the bottom row.

If Statement (1) is Not Sufficient, only choices (B), (C), and (E) are possible. Cross off the top row.

AD

AB

B €E

BCE
Next, evaluate Statement (2).

®D

A®

A©

A©

B €E

B €E

®CE

BCE

Notice that the first two steps are always the same: evaluate Statement (1) then evaluate Statement (2).
If neither Statement by itself is sufficient, then the only two possible answers are (C) and (E). The next step is to look at the Statements TOGETH ER:

A©

B€(D

8
8

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A©
B©E

FDP Strategies

Chapter 6

Putting It All Together_______________________
Now that you know the process, it’s time to work through the practice problem start to finish.

Jon's birthday party is this week. What day of the week is the party on?
(1)The party is not on Monday or Tuesday.
(2)The party is not on Wednesday, Thursday, or Friday.
(A) Statement (1) ALONE is sufficient, but statement (2) is NOT sufficient
(B) Statement (2) ALONE is sufficient, but statement (1) is NOT sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement
ALONE is sufficient
(D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient

Step 1: Separate additional info from the actual question.
Question

What day of the week is the party on?

Additional Info

Jon's birthday party is this week.

Step 2: Determine whether the question is Value or Yes/No.
You need to know the exact day of the week that the party is on.
This is a Value question.

Step 3: Decide exactly what the question is asking.
W hat, precisely, would be sufficient? One possible day of the week.
W hat, precisely, would not be sufficient? More than one possible day of the week.

Step 4; Use the Grid to evaluate the statements.
Evaluate Statement (1): Statement (1) tells you that the party is not on Monday or Tuesday. The party could still be on Wednesday, Thursday, Friday, Saturday, or Sunday. Statement (1) is N ot Sufficient.
A©
BCE

Evaluate Statement (2): Statement (2) tells you that the party is not on Wednesday, Thursday, or Friday.
The party could still be on Saturday, Sunday, Monday, or Tuesday. Statement (2) is N ot Sufficient.
A©

BCE

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FDP Strategies
Now that you’ve verified neither statement is sufficient on its own, it’s time to evaluate the statements taken together.
Evaluate (1) A N D (2): Taking both statements together, you know the party is not on Monday, Tues­ day, Wednesday, Thursday, or Friday. The party could still be on Saturday or Sunday. Statements (1) and (2) together are N ot Sufficient.
A©
fi€(E)
The correct answer is (E).

Putting It All Together (Again)
Now try a different, but related, question:

It rains all day every Saturday and Sunday in Seattle, and never on any other day.
Is it raining in Seattle right now?
(1) Today is not Monday or Tuesday.
(2)Today is not Wednesday, Thursday, or Friday.
(A) Statement (1) ALONE is sufficient, but statement (2) is NOT sufficient
(B) Statement (2) ALONE is sufficient, but statement (1) is NOT sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient (D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient
The statements are exactly the same as in the previous example, but the question has changed. The process is still the same.

Step 1; Separate additional info from the actual question.
Question

Additional Info

Is it raining in Seattle right now?

It rains all day every Saturday and Sunday in
Seattle, and never on any other day.

Step 2: Determine whether the question is Value or Yes/No.
There are two possible answers to this question:
1.

Yes, it is raining in Seattle right now.

2.

No, it is not raining in Seattle right now.

This is a Yes/No question.

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GMAT

FDP Strategies

Chapter 6

Step 3: Decide exactly what the question is asking.
Be careful. This part of the process is usually more complicated when the question is Yes/No. Sufficient is defined as providing a definite answer to the Yes/No question. Since the statements often allow for multiple possible values, you have to ask the Yes/No question for all the possible values.
Before you look at the statements, keep in mind there are only 7 days of the week. You know the answer to the question on each of those days as well. If today is Saturday or Sunday, the answer is yes, it is

raining in Seattle right now. If today is Monday, Tuesday, Wednesday, Thursday, or Friday, the an­ swer is no, it is not raining in Seattle right now.
W hat, precisely, would be sufficient? It is definitely raining (Saturday or Sunday) OR it is definitely

NOT raining (Monday through Friday).
W hat, precisely, would not be sufficient? It may be raining (e.g., Today is either Friday or Saturday).

Step 4: Use the Grid to evaluate the statements.
Evaluate Statement (1): Statement (1) tells you that today is not Monday or Tuesday. Today could still be Wednesday, Thursday, Friday, Saturday, or Sunday. It might be raining in Seattle right now. You cannot know for sure. Statement (1) is N ot Sufficient.
AD

taSiBM.

BCE

Evaluate Statement (2): Statement (2) tells you that today is not Wednesday, Thursday, or Friday. To­ day could still be Saturday, Sunday, Monday, or Tuesday. It might be raining in Seattle right now. You cannot know for sure. Statement (2) is N ot Sufficient.
AB
BCE
Now that you’ve verified neither statement is sufficient on its own, its time to evaluate the statement taken together.

Evaluate (1) AND (2): Taking both statements together, you know that today is not Monday, Tuesday,
Wednesday, Thursday, or Friday. Today could still be on Saturday or Sunday. If today is Saturday, you know that it is raining in Seattle. If today is Sunday, you know that it is raining in Seattle. Either way, you can say definitely that yes, it is raining in Seattle right now. Taken together, Statements (1) and
(2) are Sufficient.
A©
f i© E
The correct answer is (C).

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„

FDP Strategies

Concrete Values vs. Relative Values
The GMAT likes using Fractions, Decimals, Percents, and Ratios (FDPRs) in Data Sufficiency ques­ tions because a little information can go a long way.
This section will talk about FDPR word problems. There are two types of information presented in these problems: concrete values and relative values.
1. Concrete values are actual amounts (# of tickets sold, liters of water, etc.), and
2. relative values relate two quantities using fractions, decimals, percents, or ratios
(twice as many, 60% less, ratio of 2:3, etc.).

A company sells only two kinds of pie: apple pie and cherry pie. What fraction of the total pies sold last month were apple pies? A company sells only two kinds of pie: apple pie and cherry pie. How many apple pies did the company sell last month?
(1) The company sold 460 pies last month.

(1) The company sold 460 pies last month.
(2) The company sold 30% more cherry pies than apple pies last month.

(2) The company sold 30% more cherry pies than apple pies last month.

The question itself will also ask for either a concrete value or a relative value.
The question on the right asks for a concrete value (How many apple pies) while the question on the left asks for a relative value (W hat fraction of the pies).
The difference is important because you need more information to find a concrete value.
Take a look at the question on the left first. You can label the number of apple pies sold a and the num­ ber of cherry pies sold c.
You need to know what fraction of the total pies sold were apple pies: apple pies _ total pies
The total number of apple pies sold is a, and the total number of pies sold is a + c.

C -\- C
L
Look at Statement 1. If the total number of pies sold was 460, then a + c = 460. You can plug this value into your question:

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FDP Strategies

460

?

You don’t have enough information to answer the question. Eliminate answer choices (A) and (D).
Now look at Statement 2. If the company sold 30% ,more cherry pies than apple pies, then the number of cherry pies sold was 130% of the number of apple pies sold:
1.3^ = c
O n the surface this may not seem like enough information. But watch what happens if you replace c with 13a in the rephrased question.

a+ c a _

a + l.3a a _ 1
2.3 a ~ 2.3
Statement 2 gave you enough information to find the value of our fraction. The correct answer is (B).
Now the question becomes, is there a way you could have recognized sooner that Statement 2 was suf­ ficient? Yes, there is!
Remember that this Data Sufficiency question was asking for a relative value (W hat fraction of the pies). Relative values are intimately related to ratios. The ratio in this question is: apple pies sold : cherry pies sold : total pies sold.
One way to rephrase the question is to say that it asked for the ratio of apple pies sold to total pies sold.
Statement 2 was sufficient because it provides the ratio of apple pies sold to cherry pies sold.

If a Data Sufficiency question asks for the relative value of two pieces of a ratio, ANY statement that gives the relative value of ANY two pieces of the ratio will be sufficient.
Try contrasting this to the question that asked for a concrete value:

A company sells only two kinds of pie: apple pie and cherry pie. How many apple pies did the company sell last month?
(1)
(2)

The company sold 460 pies last month.
The company sold 30% more cherry pies than apple pies last month.

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FDP Strategies

Chapter 6

This question is asking for the value of a .
Take a look at Statement 1:

a + c = 460
W ithout the value of c this will not allow you to solve for a. You can eliminate answer choices (A) and
>
(D).
Now look at Statement 2:

1.3a = c
Once again, without the value of c, you cant solve for the value of a . Cross off answer choice (B).
Now look at the two statements together:

a + c = 460
1.3a = c
W ith two variables and two equations, you’ll be able to solve for the value of a. Together, the statements are sufficient. The right answer is C.
Once again, there is a general principle here. Statement 1 provided a concrete value for the total number of pies sold. Statement 2 provided the ratio of apple pies sold to cherry pies sold (a relative value).

If a Data Sufficiency question asks for the concrete value of one element of a ratio, you will need
BOTH the concrete value of another element of the ratio AND the relative value of two elements of the ratio.

Smart Numbers: Multiples of the Denominators
Sometimes, fraction problems on the GMAT include unspecified numerical amounts; often these unspecified amounts are described by variables. In these cases, pick real numbers to stand in for the variables. To make the computation easier, choose Smart Numbers equal to common multiples of the denominators of the fractions in the problem.
For example, consider this problem:

The Crandalls' hot tub is half filled. Their swimming pool, which has a capacity four times that of the tub, is filled to four-fifths of its capacity. If the hot tub is drained into the swimming pool, to what fraction of its capacity will the pool be filled? 9
4

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GMAT

FDP Strategies
The denominators in this problem are 2 and 5. The Smart Number is the least common denominator, which is 10. Therefore, assign the hot tub a capacity of 10 units. Since the swimming pool has a capac­ ity 4 times that of the pool, the swimming pool has a capacity of 40 units. You know that the hot tub is only half filled; therefore, it has 5 units of water in it. The swimming pool is four-fifths of the way filled, so it has 32 units of water in it.
Add the 5 units of water from the hot tub to the 32 units of water that are already in the swimming pool: 32 + 5 = 37.
37
W ith 37 units of water and a total capacity of 40, the pool will be filled to — of its total capacity.

swimming pool capacity: 40 units
4/5 filled: 32 units

hot tub capacity: 10 units
1/2 filled: 5 units

When Not to Use Smart Numbers
In some problems, even though an amount might be unknown to you, it is actually specified in the problem in another way. In these cases, you cannot use Smart Numbers to assign real numbers to the variables. For example, consider this problem:

Mark's comic book collection contains 1/3 Killer Fish comics and 3/8 Shazaam
Woman comics. The remainder of his collection consists of Boom! comics. If Mark has 70 Boom! comics, how many comics does he have in his entire collection?
Even though you do not know the number of comics in Mark s collection, you can see that the total is not completely unspecified. You know a piece of the total: 70 Boom! comics. You can use this informa­ tion to find the total. Do not use Smart Numbers here. Instead, solve similar problems by figuring out how big the known piece is; then, use that knowledge to find the size of the whole. You will need to set up an equation and solve:
1
3
17
— Killer Fish H
— Shazaam Woman = — comics that are not Boom!
3
8
24
7
Therefore, — of the comics are Boom! comics:
24

MANHATTAN
GMAT

Chapter 6

FDP Strategies
— x = 70
24
™ 24 x = 70x —
7
*

= 240

Mark has 240 comics.
In summary, do pick smart numbers when no amounts are given in the problem, but do not pick smart numbers when any amount or total is given!

Smart Numbers: Pick 100
More often than not, percent problems on the GMAT include unspecified numerical amounts; often these unspecified amounts are described by variables. For example:

A shirt that initially cost d dollars was on sale for 20% off. If s represents the sale price of the shirt, d is what percentage of s?
This is an easy problem that might look confusing. To solve percent problems such as this one, simply pick 100 for the unspecified amount (just as you did when solving successive percents).
If the shirt initially cost $100, then d = 100. If the shirt was on sale for 20% off, then the new price of the shirt is $80. Thus, s = 80.
The question asks: d is what percentage of s, or 100 is what percentage of 80? Using a percent table, fill in 80 as the whole and 100 as the part (even though the part happens to be larger than the whole in this case). You are looking for the percent, so set up a proportion, cross-multiply, and solve:

PART
WHOLE

100
80

X

100

x

100

80

100

80*= 10,000 * = 125
Therefore, d is 125% of s.

The important point here is that, like successive percent problems and other percent problems that include unspecified amounts, this example is most easily solved by plugging in a real value. For percent problems, the easiest value to plug in is generally 100. The fastest way to success with GMAT percent

problems with unspecified amounts is to pick 100.

M ANHATTAN
GMAT

FOP Strategies

Ch,

Benchmark Values__________________________
You will use a variety of estimating strategies on the GMAT. One important strategy for estimating with fractions is to use Benchmark Values. These are simple fractions with which you are already familiar:

1 1 1 1 1 2

,3

— and —
10 5 4 3 2 3
4
You can use Benchmark Values to compare fractions:

.......
+ 127
162 7
Which is greater:----- o r ------ ?
255
320
If you recognize that 127 is less than half of 255, and 162 is more than half of 320, you will save your­ self a lot of cumbersome computation.
You can also use Benchmark Values to estimate computations involving fractions:

10
5
What is — of — of 2,000?
22
18
If you recognize that these fractions are very close to the Benchmark Values ^ and ~ , you can estimate:

— of - of 2,000 = 250. Therefore, — of — of 2,000 * 250.
2
4
22
18
Notice that the rounding errors compensated for each other:
— «— =—

You decreased the denominator, so you rounded up:

— «— =—
18 20 4

You increased the denominator, so yourounded down: — > —.
18 4

22

20

2

— < —.

22

2

If you had rounded — to — = — instead, then you would have rounded both fractions up. This would

18

18

3

•

lead to a slight but systematic overestimation:

— x -x 2 ,0 0 0 = 333
2 3
Try to make your rounding errors cancel by rounding some numbers up and others down.

MANHATTAN
GMAT

;r 6

FDP Strategies

Benchmark Values: 10%
To find 10% of any number, just move the decimal point to the left one place:
10% of 500 is 50

10% of 34.99 = 3.499

10% of 0.978 is 0.0978

You can use the Benchmark Value of 10% to estimate percents. For example:

Karen bought a new television, originally priced at $690. However, she had a cou­ pon that saved her $67. For what percent discount was Karen's coupon?
You know that 10% of 690 would be 69. Therefore, 67 is slightly less than 10% of 690.

The Heavy Division Shortcut
Some division problems involving decimals can look rather complex. But sometimes, you only need to find an approximate solution. In these cases, you often can save yourself time by using the Heavy Divi­ sion Shortcut: Move the decimals in the same direction and round to whole numbers. For example:

What is 1,530,794 -s- (31.49 x 104) to the nearest whole number?
Step 1: Set up the division problem in fraction form:

Step 2: Rewrite the problem, eliminating powers of 10:

1,530,794
31.49X104
1,530,794
314,900

Step 3: Your goal is to get a single digit to the left of the decimal in the denominator. In this problem, you need to move the decimal point backward 5 spaces. You can do this to the denominator as long as you do the same thing to the numerator. (Technically, what you are doing is dividing top and bottom by the same power of 10: 100,000.) Thus:
1,530,794 = 15.30794
314,900

3.14900

Now you have the single digit 3 to the left of the decimal in the denominator.
Step 4: Focus only on the whole number parts of the

15.30794 _ 15 _ ^

numerator and denominator and solve:

3.14900

3

An approximate answer to this complex division problem is 5. If this answer is not precise enough, keep one more decimal place and do long division (e.g., 153 -s- 31 « 4.9).

M AN H ATTAN
GMAT

FDP Strategies

Ch.

Problem Set_______________________________
1.
2.
3.

2
Put these fractions in order fromleast to greatest: 3
4 549 982 344
Estimate to the nearest 10,000: —----- ------ —
5.342x104

3
—
13

5
7

2
—
9

3
5
Lisa spends — of her monthly paycheck on rent and — on food. Her roommate,
Carrie, who earns twice as much as Lisa, spends — of her monthly paycheck on rent
1
4 and — on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate? 11

6

4.

Estimate to theclosest integer: What is — of — of 120? a 30
20

5.

Rob spends - of hismonthly paycheck, after taxes,on rent. He spends - on food
1
2
3
and — on entertainment. If he donates the entire remainder, $500, to charity, what

1

1

8

is Rob's monthly income, after taxes?
6.

Bradley owns b video game cartridges. If Bradley's total is one-third the total owned by Andrew and four times the total owned by Charlie, how many video game cartridges do the three of them own altogether, in terms of bl
(A) ^ b
3

7.

(B) — b
4

(C) ^ b
4

12

(D) ^ - b (E)± - b
12

Company X has exactly two product lines and no other sources of revenue. If the consumer products line experiences a k% increase in revenue (where k is a posi­ tive integer) in 2010 from 2009 levels and the machine parts line experiences a k% decrease in revenue in 2010 from 2009 levels, did Company X's overall revenue increase or decrease in 2010?
(1)
(2)

In 2009, the consumer products line generated more revenue than the ma­ chine parts line. k-8 8.

Company Z only sells chairs and tables. What percent of its revenue in 2008 did
Company Z derive from its sales of chairs?

9.

What is the ratio x : y : z !
(1)
(2)

x+y=2z
2x + 3y = z

M ANHATTAN

GMAT

FDP Strategies

Ch<

Solutions_________________________________
2
3 2 5
3
2
1
1. —< — < —< —: Using Benchmark Values, you should notice that — and — are both less than —,
9 13 3 7
7
13
9
2
2
5
1
and that — and — are both more than —. Use cross-multiplication to compare each pair of fractions:

3X9

=

27

2 x 7 = 14

— V '' 13 ^
9

2 x 1 3 = 26

-

5 x 3 = 15

V
3 ^ 7

2 3
This makes it easy to order the fractions: — < —
9 13

— > 13
9

- < 3 7

2
< —
<
3

5
—.
7

2. 90,000: Use the Heavy Division Shortcut to estimate:
4,549,982,344

4,500,000,000

53,420

450,000

50,000

= 90,000

17
3. — : Use Smart Numbers to solve this problem. Since the denominators in the problem are 8, 12,
4, and 2, assign Lisa a monthly paycheck of $24. Assign her roommate, who earns twice as much, a monthly paycheck of $48. The two womens monthly expenses break down as follows:
Rent

Food

Left over

Lisa

^ of 24 = 9

A

2 4 - ( 9 + 10) =

Carrie

-of48=12
4

- o f 48 = 24
2

8

12

o f 24 = 10

5

48 - (12 + 24) = 12

The women will donate a total of $17 out of their combined monthly income of $72.
4. A pproxim ately 13: Use Benchmark Values to estimate: — is slightly more than — and — is
30
3
20
slightly less than — . Therefore, — of — of 120 should be approximately — of — of 120, or
3
30
20
3
3

,
9

which is slightly more than 13.
Another technique to solve this problem would be to write the product and cancel common factors:

11.. 6
30 20

(11 )0 3 (120) _ (ll)qaS6)
(M)(20)
(5)t*0

66
5

M ANHATTAN
GMAT

FDP Strategies
Note that for estimation problems, there is not necessarily one best way to estimate. The key is to arrive at an estimate that is close to the exact answer— and to do so quickly!
5. $12,000: You cannot use Smart Numbers in this problem, because the total amount is specified.
Even though the exact figure is not given in the problem, a portion of the total is specified. This means that the total is a certain number, although you do not know what it is. In fact, the total is exactly what you are being asked to find. Clearly, if you assign a number to represent the total, you will not be able to accurately find the total.
First, use addition to find the fraction of Rob’s money that he spends on rent, food, and entertain1 1 1 12
8
3
23 T1 r .
. . .
. .
1
ment: —H
-----b —= -----1
------ 1----= — . 1 hererore, the $500 that he donates to charity represents —
2 3 8 24 24 24 24
7 v
24
of his total monthly paycheck. We can set up a proportion:

x

= — . Thus, Robs monthly income is

24

$500 x 24, or $12,000.
6. (C): This problem can be answered very effectively by picking numbers to represent how many video game cartridges everyone owns. Bradley owns 4 times as many cartridges as Charlie, so you should pick a value for b that is a multiple of 4.
If b = 4, then Charlie owns 1 cartridge and Andrew owns 12 cartridges. Together they own
4 + 1 + 12 = 17 cartridges. Plug b —4 into the answer choices and look for the one that yields 17:
(A) — (4) = — = 21—
3
3
3
(B) ^ ( 4 ) = 17
4
(C) ^ ( 4 ) = 13
4
(D) — (4) = — = too small to be 17
12
3
(E) — (4) = —= too small to be 17

12 3

The correct answer is (C).
7. (A): This question requires you to employ logic about percents. No calculation is required, or even possible. Heres what you know so far (use new variables c and m to keep track of your information):
2009:
consumer products makes c dollars machine parts makes m dollars total revenue = c + m

M AN HATTAN
GMAT

FDP Strategies
2010:
consumer products makes “ dollars increased by k%” c machine parts makes “m dollars decreased by k%” total revenue: ?
W hat would you need to answer the question, “Did Company X’s overall revenue increase or decrease in 2010?” Certainly, if you knew the values of c, m , and k>you could achieve sufficiency, but the
GMAT would never write such an easy problem. W hat is the minimum you would need to know to answer definitively?
Since you already know that k is a positive integer, you know that c increases by the same percent by which m decreases. So, you don’t actually need to know k . You already know that k percent of which­ ever number is bigger (c or m) will constitute a bigger change to the overall revenue.
All this question is asking is whether the overall revenue went up or down. If c started off bigger, then a k% increase in c means more new dollars coming in than you would lose due to a k% decrease in the smaller number, m . If c is smaller, then the k% increase would be smaller than what you would lose due to a k% decrease in the larger number, m .
The question can be rephrased, “W hich is bigger, c or mT
(1) SUFFICIENT: This statement tells you that c is bigger than m. Thus, a k% increase in c is larger than a k% decrease in m , and the overall revenue went up.
(2) INSUFFICIENT: Knowing the percent change doesn’t help, since you don’t know whether c or m is bigger. Note that you could plug in real numbers if you wanted to, although the problem is faster with logic.
Using Statement (1) only:
2009:
consumer products makes $200 machine parts makes $100 total revenue = $300
2010: i f k = 50

consumer products makes $300 machine parts makes $50 total revenue = $350
You can change k to any positive integer (you don’t need to know what k is). As long as c> m, you will get the same result. The increase to the larger c will be larger than the decrease to the smaller m.
The correct answer is (A).

M ANHATTAN
GMAT

FDP Strategies
8. (C): First of all, notice that the question is only asking for the percent of the revenue the company derived from chairs. The question is asking for a relative value.
The revenue for the company can be expressed by the following equation:
Revenue^o m p a n y Z = Revenue,,,. s + Revenue,,, a ir s
7
.
C
T a b le
Ch
If you can find the relative value of any two of these Revenues, you will have enough information to answer the question.
Also, note that the GMAT will expect you to know that Revenue = Price x Quantity Sold. This rela­ tionship is discussed in more detail in Chapter 1 of the Word Problems Strategy Guide.
The revenue derived from tables is the price per table multiplied by the number of tables sold. The rev­ enue derived from chairs is the price per chair multiplied by the number of chairs sold. You can create some variables to represent these unknown values:
R e v e n u e ^ , = PT x QT
Revenuecha>,s = Pc x Qc
(1) INSUFFICIENT: This statement gives you the relative value of the price of tables to the price of chairs. If the price of tables was 10% higher than the price of chairs, then the price of tables was 110% of the price of chairs: p t = i .i p c

However, this information by itself does not give you the relative value of their revenues.
(2) INSUFFICIENT: If the company sold 20% fewer tables than chairs, then the number of tables sold is 80% of the number of chairs sold:
Q T = 0.80,,
This information by itself, however, does not give you the relative value of their revenues.
(1) A N D (2) SUFFICIENT: Look again at the equation for Revenue derived from the sale of tables:
R evenue,.^ = pT x Q T
Replace PT with 1.1 Pc and replace Q T with 0.80^.:
P t x Q t =(1.1P c) x (0.8Qc)
P t x Q t = (0.88)(PC x
R e v e n u e ,^ = (0.88)Revenuechairs

M ANHATTAN
GMAT

FD Strategies
P

Chapter 6

Taken together, the two statements provide the relative value of the revenues. No further calculation is required to know that you can find the percent of revenue generated from the sale of chairs. The state­ ments together are sufficient.
If you do want to perform the calculation, it would look something like this:
RevenueC
hairs _
Total Revenue

Revenuechairs

_________ Revenuechairs_________ =

RevenueT + RevenueC ablcs hairs

(0.88)Revenuechairs + Revenuechaini

_
(1 -88) Rejceaue^Tf

i

_ 53%

1.88

Save time on Data Sufficiency questions by avoiding unnecessary computation. Once you know you can find the percent, you should stop and move on to the next problem.
The correct answer is (C).
9. (C): For this problem, you do not necessarily need to know the value of x, y, or z . You simply need to know the ratio x : y : z ( in other words, the value of x!y AND the value of ylz). You need to manipulate the information given to see whether you can determine this ratio.
(1) INSUFFICIENT: There is no way to manipulate this equation to solve for a ratio. If you simply solve for x!y> for example, you get a variable expression on the other side of the equation:

x + y =2 z x = 2z —y x _ 2 z —y _ 2 z

y

y

j

y

(2) INSUFFICIENT: As in the previous example, there is no way to manipulate this equation to solve for a ratio. If you simply solve for x!y, for example, you get a variable expression on the other side of the equation: 2x + 3y = z
2x = z - 3y x _ z —3y _ z

3

y

2

2y

2y

MANHATTAN

GMAT

,0
5

Chapter 6

FDP Strategies
(1) A N D (2) SUFFICIENT: Here, you can use substitution to combine the equations in helpful ways:

x + y = 2z
2x+ 3y —z
Since z = 2x + 3jy, you can substitute:

x + y = 2(2*+ 3y) x + y = 4x + 6y
Therefore, you can arrive at a value for the ratio x/y:

-3 x = 5y

Divide by y.

Divide by - 3
*

5

y
You can also substitute for * to get a value for the ratio ytz:

x + y = 2z x = 2z —y
2x + 3y = z
2(2z - y ) + 3y = z
4z —2y + 3y —z y = -3z z This tells you that x \ y ——
5/3, a n d y ^ = -3/1. Both ratios contain a 3 for the y variable and both also contain a negative sign, so assign the value - 3 to y. This means that * must be 5 and z must be 1.
Therefore, the ratio * : j / : £ = 5 : - 3 : 1 .
You can test the result by choosing * = 5, y - -3 , and z = 1, or * = 10, y = - 6 , and z - 2. In either case, the original equations hold up.
The correct answer is (C).

16 M AN H ATTAN
0
GMAT

Fractions, Decimals, & Percents

Extra FDPs

Repeating Decimals
Terminating Decimals
Using Place Value on the GMAT
The Last Digit Shortcut
Unknown Digits Problems
Formulas That Act on Decimals
Chemical Mixtures
Percents and Weighted Averages
Percent Change and Weighted Averages
Estimating Decimal Equivalents

Extra FDPs
This chapter outlines miscellaneous extra topics within the area of Fractions, Decimals, Percents, & Ratios,

Repeating Decimals_________________________
Dividing an integer by another integer yields a decimal that either terminates or that never ends and repeats itself:

2+9=?

2 + 9 = 0.2222... = 0.2
The bar above the 2 indicates that the digit 2repeats forever.
You will not have to use the bar on the GMAT; it is simply a convenient shorthand.

Generally, you should just do long division to determine the repeating cycle. However, it is worth not­ ing the following patterns, which have appeared in published GMAT questions:
4 + 9 = 0.4444... = 0.4

—
11

99

= — = 0.0909... = 0.09

23 - 99 = 0.2323... = 0.23

— = — = 0.2727... = 0.27
11

99

If the denominator is 9, 99, 999 or another number equal to a power of 10 minus 1, then the numera­ tor gives you the repeating digits (perhaps with leading zeroes). Again, you can always find the decimal pattern by simple long division.

Chapter 7

Extra FDPs

Terminating Decimals_______________________
Some numbers, like \[ l and tt, have decimals that never end and never repeat themselves. The GMAT will only ask you for approximations for these decimals (e.g., yjl ~ 1.4). Occasionally, though, the
GMAT asks you about properties of “terminating” decimals; that is, decimals that end. You can tack on zeroes, of course, but they do not matter. Some examples of terminating decimals are 0.2, 0.47, and
0.375.
Terminating decimals can all be written as a ratio of integers (which might be reducible):
Some integer
Some power of 10
2
1
0.2 = - = -

10

5

47
0.47 = —

0.375

100

375 _ 3
1,000

8

Positive powers of 10are composed of only 2s and 5s as prime factors. This means that when you reduce this fraction, you only have prime factors of 2s and/or 5s in the denominator. Every terminating decimal shares this characteristic. If, after being fully reduced, the denominator has any prime factors besides 2 or 5, then its decimal will not terminate. If the denominator only has factors of 2 and/or 5, then the decimal will terminate.

Using Place Value on the GMAT________________
Some difficult GMAT problems require the use of place value with unknown digits. For example”

A and B are both two-digit numbers, with A > 6. If A and B contain the same digits, but in reverse order, what integer must be a factor of (A - B)1
(A) 4

(B) 5

(C)6

(D)8

(E)9

To solve this problem, assign two variables to be the digits in A and B : x and y. Let A =[x][3/] (not the product of * and y: x is in the tens place, and y is in the units place). The boxes remind you that x and y stand for digits. A is therefore the sum of * tens and y ones. Using algebra, write A = 10x+y.
Since 5 s digits are reversed, B =jj/][x]. Algebraically, B can be expressed as 10j/ + x. The difference of A and B can be expressed as follows:

A —B — lOx+j y - (10jy + x) = 9 x - 9 y = 9 ( x - y )
Clearly, 9 must be a factor o f A - B . The correct answer is (E).
You can also make up digits for * and y and plug them in to create A and B. This will not necessarily yield the unique right answer, but it should help you eliminate wrong choices.

10 M AN HATTAN
1
GMAT

Extra FDPs

Chapter 7

In general, for unknown digits problems, be ready to create variables (such as x, y , and z) to represent the unknown digits. Recognize that each unknown is restricted to at most 10 possible values (0 through
9). Then apply any given constraints, which may involve number properties such as divisibility or odds
& evens.

The Last Digit Shortcut_____________________
Sometimes the GMAT asks you to find a units digit, or a remainder after division by 10:

What is the units digit of (7)2(9)2(3)3
?
In this problem, you can use the Last Digit Shortcut:
To find the units digit of a product or a sum of integers, only pay attention to the units digits of the numbers you are working with. Drop any other digits.
This shortcut works because only units digits contribute to the units digit of the product:
STEP 1: 7 x 7 = 4 9

Drop the tens digit and keep only the last digit: 9.

STEP 2: 9 x 9 = 81

Drop the tens digit and keep only the last digit: 1.

STEP 3: 3 x 3 x 3 = 27

Drop the tens digit and keep only the last digit: 7.

STEP 4: 9 x 1 x 7 = 63

Multiply the last digits of each of the products.

The units digit of the final product is 3.

Unknown Digits Problems__________________
Occasionally, the GMAT asks tough problems involving unknown digits. These problems look like
“brainteasers”; it seems it could take all day to test the possible digits.
However, like all other GMAT problems, these digit “brainteasers” must be solvable under time con­ straints. As a result, you always have ways of reducing the number of possibilities:
Principles:
(1) Look at the answer choices first, to limit your search.
(2) Use other given constraints to rule out additional possibilities.
(3) Focus on the units digit in the product or sum. This units digit is affected by the fewest other digits.
(4) Test the remaining answer choices.

M ANHATTAN
GMAT

Extra FDPs
Example:

AB x CA
DEBC
In the multiplication above, each letter stands for a different non-zero digit, with
A x B < 10. What is the two-digit number AB?
(A) 23

(B) 24

(C) 25

(D) 32

(E) 42

It is often helpful to look at the answer choices. Here, you see that the possible digits for A and B are 2,
3, 4, and 5.
Next, apply the given constraint that A x B < 10. This rules out answer choice (C), 25, since 2 x 5 = 10.
Now, test the remaining answer choices. Notice that A x B = C, the units digit of the product. There­ fore, you can find all the needed digits and complete each multiplication.
Compare each result to the template. The two positions of the B digit must match:
23

x 62

24 x 82

1,426

1,968

The B’s do not match

The B’s do not match

32

x 63

42 x 84

2,016

3,528

The B’s do not match

The B’s match

Answer is (E).
Note that you could have used the constraints to derive the possible digits (2, 3, and 4) without using the answer choices. However, for these problems, you should take advantage of the answer choices to restrict your search quickly.

Formulas That Act on Decimals
Occasionally, you might encounter a formula or special symbol that acts on decimals. Follow the for­ mula’s instructions precisely.
Let us define symbol [x] to represent the largest integer less than or equal to x:

What is [5.1]?

M AN HATTAN
GMAT

Extra FDPs
According to the definition you are given, [5.1] is the largest integer less than or equal to 5.1. That integer is 5. So [5.1] = 5.

What is [0.8]?
According to the definition again, [0.8] is the largest integer less than or equal to 0.8. That integer is
0. So [0.8] = 0. Notice that the result is not 1. This particular definition does not round the number.
Rather, the operation seems to be truncation— simply cutting off the decimal. However, you must be careful with negatives.

What is [-2.3]?
Once again, [-2.3] is the largest integer less than or equal to -2.3. Remember that “less than” on a number line means “to the left of.” A “smaller” negative number is further away from zero than a “big­ ger” negative number. So the largest integer less than -2 .3 is -3 , and [-2.3] = -3 . Notice that the result is n o t— this bracket operation is not truncation.
2;
Be sure to follow the instructions exactly whenever you are given a special symbol or formula involv­ ing decimals. It is easy to jump to conclusions about how an operation works; for instance, finding the largest integer less than x is not the same as rounding x or truncating x in all cases. Also, do not confuse this particular set of brackets [x] with parentheses (x) or absolute value signs \x\.

Chemical Mixtures__________________________
Another type of GMAT percent problem bears mention: the chemical mixture problem:

A 500 mL solution is 20% alcohol by volume. If 100 mL of water is added, what is the new concentration of alcohol, as a percent of volume?
Chemical mixture problems can be solved systematically by using a mixture chart:

Volume (mL)

Original

Change

New

Alcohol
Water
Total Solution
Note that Original + Change = New. Moreover, the rows contain the parts of the mixture and sum to a total. Only insert actual amounts; compute percents off on the side.

M ANHATTAN
GMAT

Chapter 7

Extra FDPs
First, fill in the amounts that you know. Put 500 mL of solution in the Original column. You can also put +100 mL of water in the Change column. Since no alcohol was added or removed, put 0 mL of alcohol in the Change column. This tells you that our total Change is 100 mL as well. You do not need to input the units (mL):

Volume (mL)

Original

Alcohol

Change

New

0

Water

+100

Total Solution

500

+100

Since the Original solution is 20% alcohol, you can compute the mL of alcohol in the Original solution by asking: How many mL of alcohol is 20% of 500 mL? Solve this using the decimal equivalent: x= (0.20)(500 mL) = 100 mL.
Now, fill in all the remaining numbers:

Volume (mL)

Original

Change

New

Alcohol

100

0

100

Water

400

+100

500

Total Solution

500

+100

600

Alcohol 100
1
Finally, you can find the new alcohol percentage: ---------- = ------= - « 0.167 = 16.7%.
Total
600 6
Note that with this chart, you can handle proportions of many kinds. For instance, you might have been asked the concentration of water in the final solution. Simply take the quantities from the proper rows and columns and calculate a proportion.

Percents and Weighted Averages_____________
A mixture chart can be used to solve weighted average problems that involve percents:

Kris-P cereal is 10% sugar by weight, whereas healthier but less delicious Bran-0 cereal is 2% sugar by weight. To make a delicious and healthy mixture that is 4% sugar, what should be the ratio of Kris-P cereal to Bran-0 cereal, by weight?
First, set up a mixture chart. This time, instead of Original/Change/New, put the cereal brands and
Total across the top. Also, put the parts of each cereal in the rows:

14 M AN HATTAN
1
GMAT

Chapter 7

Extra FDPs
Pounds (lbs)

Bran-0

Kris-P

Total

Sugar
Other stuff
Total Cereal
You are not given any actual weights in this problem, nor are you asked for any such weights. As a re­ sult, you can pick one Smart Number. Pick the total amount of Kris-P: 100 pounds (lbs). Now you can compute how much sugar is in that Kris-P: (0.10)(100) = 10 lbs. Do not bother computing the weight of the “other stuff”; it rarely matters.

Pounds (lbs)
Sugar

Kris-P

Bran-0

Total

10

Other stuff
Total Cereal

100

Now set the total amount of Bran-O as * lb (you cannot pick another Smart Number). Since Bran-O is only 2% sugar, the mass of sugar in the Bran-O will be (0.02)* lb. You can now add up the bottom row: the total amount of all cereals is 100 + * lb. Since the total mixture is 4% sugar, the weight of sugar in the mixture is (0.04)(100 + *) lb:

Pounds (lbs)
Sugar

Kris-P

Bran-0

Total

10

(0.02)x

(0.04)(10 0 +x)

100

X

100+x

Other stuff
Total Cereal

Finally, you can write an equation summing the top row (the amounts of sugar):
10 + (0.02)* = (0.04)(100 + *)
10 + (0.02)* = 4 + (0.04)*

6 = (0.02)*
300 = *

The ratio of Kris-P to Bran-O is 100 : 300 or 1 : 3.
This result should make sense: to make a 4% mixture out of 10% and 2% cereals, you need much more of the 2%. In fact, 4% is the average of 10% and 2%, weighted 1 to 3.

MANHATTAN
GMAT

15
1

Extra FDPs

Percent Change and Weighted Averages
Weighted averages can also show up in “percent change” problems:

A company sells only pens and pencils. The revenue from pen sales in 2007 was up 5% from 2006, but the revenue from pencil sales declined 13% over the same period. If overall revenue was down 1% from 2006 to 2007, what was the ratio of pencil revenue to pen revenue in 2006?
First, set up a chart. You can use the Original/Change/New framework, but write 2006 and 2007 in the column headers. Write Pen and Pencil Revenue in the row headers:

Dollars ($)

2006

Change

2007

Pen Revenue
Pencil Revenue
Total Revenue
As in the previous problem, you are not given any actual amounts (in this case, dollar revenue), nor are you asked for any such revenue in dollar terms. Rather, you are asked for a ratio of revenue. As a result, you can pick one Smart Number. Pick $100 for the 2006 Pen Revenue. Since that revenue went up 5%, the change is +$5, and the 2007 Pen Revenue is $105. Remember, all amounts are in some monetary unit (say, dollars):

Dollars ($)
Pen Revenue

2006

Change

2007

100

+5

105

Pencil Revenue
Total Revenue
Now set the 2006 Pencil Revenue equal to $x. Remember, you cannot pick another Smart Number, since you do not know what the ratio of 2006 revenue will be. Since the Pencil Revenue went down
13%, the change in dollar terms is -0.13*, and the 2007 Pencil Revenue is 0.87* dollars.
You can also write the 2006 Total Revenue as the sum of that column. Since the Total Revenue went down 1%, the change (again, in dollar terms) is -0.01(100 + x), and the 2007 Total Revenue is 0.99(100
+ x):

Dollars ($)
Pen Revenue
Pencil Revenue
Total Revenue

M AN HATTAN
GMAT

2006

Change

2007

100

+5

105

X

-0.13x

0.87x

100 + x

-0.01(100+ x)

0.99(100+ x)

Extra FDPs
Finally, you can write an equation summing the 2007 column:
105 + 0.87* = 0.99(100 + *)
105+ 0.87* = 9 9 + 0.99*
6 = 0 . 12*

600 = 12*
50 = *
Since the 2006 Pen Revenue is $100, the ratio of Pencil Revenue to Pen Revenue in 2006 is 50 : 100, or

1 : 2.
Be sure to answer the question exactly as given! The problem could easily ask for the ratio of Pen Rev­ enue to Pencil Revenue in 2007, or for the ratio of either part to the total.
Again, this result should make sense. A 5% increase in Pen Revenue and a 13% decline in Pencil Rev­ enue only average to a 1% decline overall if there is proportionally more Pen Revenue to start with. If the 2006 revenue of pens and pencils were equal, then the average change would just be a straight aver­ age (arithmetic mean):
+ 5% + (-13% ) -8 %
/n/
------------------- = ------- = —
4%
2
2
As it stands, however, the overall percent change is a weighted average of the two percent changes. The weights are the 2006 (original) revenues:
(+5%) (100) + (—
13%)(50) +5 - 6.5 -1 .5
10/
---------------------------------- = ------------= ------ = — i% 100 + 50
150
150
You can use a similar formula to solve for the $50 and thus the revenue ratio. The algebraic steps are the same as they are with the chart:
(+5%)(100) + (-1 3 % )(* )_

1%

100 + *
In fact, to solve this equation for *, you can simply leave the percents as percents, rather than change them to decimals.
Last, do not forget that on any real GMAT problem, you can plug in answer choices. You will always be given the correct ratio in one of the answer choices. Simply pick an answer choice (say, a ratio of 1 : 3) and invent revenues based on that ratio (say, $100 : $300). Then work forward from there, finding the changes in revenue per product and overall revenue. Compare your results to the overall change given. Repeat as necessary. This method can be computationally intensive, but it will produce the correct answer eventually in many situations.
For more on Weighted Averages, see the Word Problems Strategy Guide.

M ANHATTAN
GMAT

Chapter 7

Extra FDPs

Estimating Decimal Equivalents
W hen you are estimating the decimal equivalent of a fraction, you often have a few choices:

9
9
Estimate a decimal equivalent for — . (By long division, — ~ 0.173077...)
Choice (1): Make the denominator the nearest factor of 100 or another power of 10:
9
9 18
— * — = -----= 0.18 > real value
52 50 100

High estimate: Lower the denominator.
&

Choice (2): Change the numerator or denominator to make the fraction simplify easily:
9
—

9

1
= —= 0.16< real value

Low estimate: Raise the denominator.

Try not to change both the numerator and denominator, especially in opposite directions. But in a pinch, you can adjust both numbers— especially if your estimation does not have to be that precise (e.g., in order to eliminate answers of a drastically different size):
— = 0.2 »

real value

Raise the top and lower the bottom.

If you need a more precise estimate, you can average a couple of methods, or you can think about small

percent adjustments'.
* 100,000 m

,

100,000

-

Estimate----------- . (By the calculator,----------- = 1,041.6)
96
96
First adjust the denominator to 100 and perform the division:
100,000 100,000 , ^
, ,
„ . , ,
-----------« ----------- = 1,000 < real value Raise the denominator.
96
100
Now, you can make the following approximation , as long as you realize it is never exact, and that you can only use it for small adjustments. Use with caution!
You increased the denominator from 96 to 100. That is approximately a 4% increase:
Change _ 4
Original

96

4
100

This means that you can increase the result by 4%, to make your estimate more accurate:
1,000(1.04) = 1,040
Notice how close this estimate is to the real value (1,040 is 99.84% of 1,041.6).

„o MANHATTAN
"8 GMAT

Extra FDPs

Ch,

Problem Set
1.

What is the units digit of
V

6s

/

2.

What is the units digit of (2)s(3)3(4)2?

3.

What is the sum of all the possible three-digit numbers that can be constructed us­ ing the digits 3,4, and 5 if each digit can be used only once in each number?

4.
5.

3
What is the length of the sequence of different digits in the decimal equivalent of - ?
Which of the following fractions will terminate when expressed as a decimal?
(Choose all that apply.)
1

27

100

(A26 (B 5