Loyola University MBA Program

Loyola University MBA Program
GMAT/GRE Boot Camp Loyola University MBA Program

Instructor Nate Straight, Loyola M.B.A. ‘08
College of Business’ Director of Assessment; Business Statistics & Quantitative Methods Instructor GMAT Score of 760, 99th percentile

Materials

Basic Information Structure and Scoring

GMAT Test Structure 4 sections, taken in this order:
Analytical Writing – 30 minutes Integrated Reasoning – 30 minutes Math / Quantitative – 75 minutes Verbal Reasoning – 75 minutes

GMAT Test Structure Analytical Writing (30 mins.):
1 essay requiring analysis of the reasoning behind a presented argument and communication of a critique of the argument

GMAT Test Structure Integrated Reasoning (30 mins.):
12 multiple-choice questions Interpretation of graphics / tables Reasoning from multiple sources Multiple-step/two-part analysis

GMAT Test Structure Math / Quantitative (75 mins.):
37 multiple-choice questions 50% problem-solving 50% “data sufficiency”

GMAT Test Structure Verbal Reasoning (75 mins.):
41 multiple-choice questions 40% sentence correction 30% reading comprehension 30% critical reasoning

GMAT Scoring Scores use the following ranges:
Analytical Writing – 0 to 6 Integrated Reasoning – 1 to 8 Math / Quantitative – 0 to 60 Verbal Reasoning – 0 to 60 Total Score – 200 to 800

GMAT Scoring Your GMAT Score Report: Loyola’s average GMAT score is a 520 Math % Verb Total AWA IR 36 42 30 56 550 48 4.5 38 6 79 Your total score ↑ depends only on your math and your verbal reasoning scores.

Basic Strategies Improving your Score

More Test Structure The GMAT is taken on a computer, one question at a time, and you can not skip or come back to questions. The math / quantitative and verbal sections are “computer-adaptive”.

Computer-Adaptive The math and verbal sections start with medium-difficulty questions and ramp up or down the difficulty based on your prior performance. Your score increases with difficulty.

Computer-Adaptive The goal of adaptive scoring is to gradually hone in on your level of ability, which determines your score. Earlier questions identify your general ability and have a large impact on your score. Later questions are ‘fine-tuning’.

Computer-Adaptive

Computer-Adaptive Strategies for adaptive testing:
Early questions are very important; spend your time getting them right. Especially early, difficult questions can dramatically improve your score. Plan to never miss an easy question.

Unanswered questions at the end will decrease your score; even if not much individually, they will add up. If you are running out of time, use process of elimination, then guess.

Early, easy-sounding questions will have easy answers; later, as difficulty rises these “easy” answers are a trap. If an answer seems obviously right, rethink it carefully before choosing it.

Process of Elimination
“18 percent of American corn is grown in Iowa. If the total amount of corn grown in America is 12 billion bushels, how many of those bushels are grown outside of Iowa?” 2.2 billion 11.8 billion 6 billion 13.2 billion 9.8 billion

“A company’s sales increased 10% from 2012 to 2013, and then 20% from 2013 to By approximately what percent did sales increase overall from 2012 to 2014?” 20% 32% 24% 40% 30%

“An MBA student took 2 classes in Fall with an average grade of 80, then 3 classes in the Spring with an average grade of 90. What is the student’s average grade for the year?” 83.3 86 85 88 85.5

General Strategies There is no substitute for content knowledge. Process of elimination and other strategies are a last resort. It will always be quicker to solve the problem directly than to work through 5 answers trying to find the right one.

General Strategies There is also no substitute for knowing your own ability level and limitations. You should spend as little test time as possible trying to jog your faded high-school algebra or grammar memory bank in search of long-lost knowledge.

General Strategies The first seconds after reading a problem should be spent considering a) whether you know how to solve the problem; b) if not, whether you can figure out how to identify the answer. Be fully honest in this self-reflection.

General Strategies If you know that you do not know how to solve a problem, and cannot even come up with a way to find an answer, just take your best guess and move on. Staring at the problem will not make you understand it; it only wastes time.

General Strategies Practice exams are more helpful than random study. There are 2 full-length tests available at: the-gmat-exam/prepare-for-the-gmat-exam.aspx Take the first test, review in detail all the content you missed, then re-take.

Math / Quantitative Overview and Strategies

Math Overview 2 types of questions: 50% problem-solving
50% “data sufficiency” A calculator is neither provided nor allowed for the math section. Do not use one while you study for the exam.

Math Overview Problem-solving content: Arithmetic, order of operations
Factors, multiples, prime numbers Fractions, ratios, and proportions Operations using fractions/ratios Percentages and percent change

Math Overview Problem-solving content: Probability and permutations
Mean, median, and mode Range and standard deviation Exponents and square roots Operations using exponents

Math Overview Problem-solving content: Manipulating algebraic terms
Algebra for “word problems” Solving systems of equations Using algebraic functions Algebraic factorization

Math Overview Problem-solving content: Distance/rate/time problems
Work/rate/time problems Geometry: Angles and area Solids: Volume and surface area Lines: Slope, intercept, and areas

Math Overview Data sufficiency content:
All of the above concepts, focused on the “how” and “why” of a process rather than the “what [is the answer]” Evaluate sufficiency of 2 statements

Math Overview Data sufficiency example:
“What is the value of x? 1) x2 = ) x is negative” Statement 1 alone is sufficient; Statement 2 alone is not Statement 2 alone is sufficient; Statement 1 alone is not Both statements 1 and 2 together are sufficient; neither statement alone is sufficient to answer the question Each statement alone is sufficient to answer the question Statements 1 and 2 together are not sufficient

Math Strategies The GMAT contains relatively few “one-step” problems. Nearly every question will require more than one calculation or test multiple concepts. Remember, the easiest, most obvious answers are often set up as traps.

Math Strategies There are two good ways to tackle a question you don’t know how to solve: If the answers are algebraic, plug in a number to simplify the question If the answers are numbers, plug each value into the question itself.

Math Strategies t / 100 10 / t t / 10 100 / t t
“Jeremy can run 5 miles in t minutes. If he can run x miles in 20 minutes, which of the following is equal to the value of x?” Let’s make t = 50, for 5 “10-minute” miles.

Math Strategies 50 / 100 10 / 50 50 / 10 100 / 50 50 “Jeremy can run 5 miles in 50 minutes. If he can run x miles in 20 minutes, which of the following is equal to the value of x?” Let’s make t = 50, for 5 “10-minute” miles.

Math Strategies 50 / 100 = .5 10 / 50 = .2 50 / 10 = 5 100 / 50 = 2
10 / 50 = .2 50 / 10 = 5 100 / 50 = 2 = 50 “Jeremy can run 5 miles in 50 minutes. If he can run x miles in 20 minutes, which of the following is equal to the value of x?”

Math Strategies “Mr. Hooper’s Ice Cream Shop sells only vanilla and chocolate. Today the ratio of vanilla to chocolate cones sold was 2 to 3, but if 5 more vanilla cones had been sold the ratio would have been 3 to 4. How many vanilla cones did he sell today?” 20 25 30 35 40

Math Strategies “… the ratio of vanilla to chocolate cones sold was 2 to 3. If 5 more vanilla cones had been sold the ratio would have been 3 to 4. How many vanilla cones did he sell today?” 20 25 30 35 40 If you don’t know how to solve it with algebra, plug in the answers. Start with C.

Math Strategies “… the ratio of vanilla to chocolate cones sold was 2 to 3. If 5 more vanilla cones had been sold the ratio would have been 3 to 4. How many vanilla cones did he sell today?” 30 / x = 2 / 3, so x = 45 chocolate cones 5 more than 30 is 35, so the ratio would be 35 vanilla to 45 chocolate… which isn’t 3/4.

Math Strategies “… the ratio of vanilla to chocolate cones sold was 2 to 3. If 5 more vanilla cones had been sold the ratio would have been 3 to 4. How many vanilla cones did he sell today?” 20 25 30 35 40 C is out. The numbers in the ratio were too close together, because C was small. Try E.

Math Strategies “… the ratio of vanilla to chocolate cones sold was 2 to 3. If 5 more vanilla cones had been sold the ratio would have been 3 to 4. How many vanilla cones did he sell today?” 40 / x = 2 / 3, so x = 60 chocolate cones 5 more than 40 is 45, so the ratio would be 45 vanilla to 65 chocolate… so E is correct.

Math / Quantitative Specific Content Review

Numbers & Operations You should be familiar with primes and factors/multiples of numbers. You should be aware of the “inner workings” of arithmetic procedures You should practice working “long” division and multiplication by hand.

Numbers & Operations If p is a negative number and 0<𝑠< 𝑝 then which of the following must also be negative? (p + s)2 p2 – s2 (p – s)2 s2 – p2 (s – p)2

Numbers & Operations If x and y are the tens and units digit, respectively, of the product of 725,278 and 67,066, what is the value of x + y? 48 12 16 10 14

Numbers & Operations What is the smallest integer n such that the following inequality is true? 1 2 𝑛 < 0.001 5 500 10 501 11

Order of Operations Perform arithmetic in this order:
Powers and square-roots Multiplication and division Addition and subtraction But, anything in parentheses should be solved first, from the inside out.

Order of Operations Solve: −5 2 -20 200 5 400 128

Powers, Exponents, Roots
Exponents follow a few basic rules: 4 3 ∗ 4 2 = = 4 5 4∗4∗4 ∗(4∗4) = 6 4−2 = 6 2 6∗6∗6∗6 (6∗6) =(6∗6)

Exponents follow a few basic rules: 5 2 ∗ 6 2 = 5∗6 2 5∗5 ∗ 6∗6 = 5∗6 ∗(5∗6) 2 3 ∗ 3 5 = 2 3 ∗ = 2∗3 3 ∗ 3 2 2∗2∗2 ∗ 3∗3∗3∗3∗3 = 2∗3 ∗ 2∗3 ∗ 2∗3 ∗3∗3

Exponents follow a few basic rules: = 7 2∗3 = ∗7 ∗ 7∗7 ∗ 7∗7 2 −5 = 8 1/3 = 3 8

General facts about exponents: A negative number to an even power will be positive : −3 2 =−3∗−3=9 A negative number to an odd power will be negative: −2 3 =−2∗−2∗−2

/ / /2 6 9 8 ∗ / 3 6 ∗ 1/2 9 2 ∗ 3 −3 = ∗ 3 −3 3 4 ∗ 3 −3 = 3 4−3 = 3 1

Fractions, Ratios, Proportions
A fraction is a part divided by a whole: 7 𝑚𝑎𝑙𝑒𝑠 15 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑠 or 7 divided by 15 A ratio is one part divided by another: 7 𝑚𝑎𝑙𝑒𝑠 8 𝑓𝑒𝑚𝑎𝑙𝑒𝑠 or a ratio of 7 : 8

Fractions may be written in many different equivalent ways. To find an equivalent fraction, multiply or divide the top and bottom of the original fraction by the same number: 3 5 → 3 5 ∗ 2 2 → 3 ∗ 2 5 ∗ 2 → 6 10

A proportion is a collective term for a set of equivalent fractions or ratios: 3 5 = = = = =𝑒𝑡𝑐 1 2 = 2 4 = 5 10 = 6 12 = =𝑒𝑡𝑐

You will often need to be able to “solve a proportion” on GMAT problems, as in the prior ice cream shop example: “If the ratio of vanilla to chocolate cones sold is 2 to 3 and 30 vanilla cones were sold today, how many chocolate cones were sold today?”

“If the ratio of vanilla to chocolate is 2 to 3 and 30 vanilla cones were sold, how many chocolate cones were sold?” Solve this step by setting up a set of two equivalent ratios, a proportion: 2 3 = 30 𝑥

2 3 ∗ = 30 𝟒𝟓 2 3 = 30 𝑥 You could solve this proportion using algebra, or you can just think of what number we multiplied the top and bottom of the original fraction by:

Operations using Fractions
Simplifying fractions: Simplifying means to divide the top and bottom by the same number; this process may be repeated: → 1/2 1/2 → 220/10 200/10 = 22 20 → 220/10 200/10 = → 22/2 20/2 = 11 10

Reciprocals of fractions: If asked to divide 1/x where x is any fraction, the answer is to “flip” x: = 5 3

Reciprocal relationships: If two fractions are equal, you can flip both and they will still be equal: = → = =

Addition or subtraction: Make each fraction have the same bottom number using a proportion, then add/subtract the top numbers: → 3 ∗ 4 5 ∗ ∗ 5 4 ∗ 5

Addition or subtraction: Make each fraction have the same bottom number using a proportion, then add/subtract the top numbers: 3 ∗ 4 5 ∗ ∗ 5 4 ∗ 5 → → 22 20 3 ∗ 4 5 ∗ ∗ 5 4 ∗ 5 3 ∗ 4 5 ∗ ∗ 5 4 ∗ 5 → 3 ∗ 4 5 ∗ ∗ 5 4 ∗ 5 → → → 11 10

Multiplication or division: For multiplication, multiply the 2 top and 2 bottom numbers; for division, flip the 2nd fraction, then multiply: 2 3 ∗ 4 5 → 2 ∗ 4 3 ∗ 5 → 8 15 2 3 ∗ 4 5

Multiplication or division: For multiplication, multiply the 2 top and 2 bottom numbers; for division, flip the 2nd fraction, then multiply: 2 3 / 4 5 → 2 3 ∗ 5 4 → 2 ∗ 5 3 ∗ 4 → → 5 6 2 3 / 4 5 → 2 3 ∗ 5 4 2 3 / 4 5

Combining numbers and fractions: Any number x may be written as x/1, or as any fraction that simplifies to it: 3∗ → 3 1 ∗ → = 27 6 3∗ 3∗ → 3 1 ∗

The quantities S and T are positive and related by the equation 𝑆= 𝑘 𝑇 where k is a constant. If the value of S were to increase by 50%, then the value of T would decrease by what percent? 25% 67% 33% 75% 50%

If 1 𝑥 = 3 2 solve: 1 𝑥+2 2 9/64 64/9 16/9 9/16 4/9

“At a football game, 4/5 of seats in the lower deck were sold. If 1/4 of all the seats are in the lower deck, and 2/3 of total seats sold, what fraction of the unsold seats were in the lower deck?” 3 20 1 6 1 5 1 3 7 15

Percent & Percent Change
A percentage is a fraction over 100: 57%= ‘x percent of y’ is x written as a fraction over 100 multiplied by the value of Y: 35% 𝑜𝑓 400 𝑖𝑠…?→ ∗400=120 A percentage is a fraction over 100: 57%= ‘x percent of y’ is x written as a fraction over 100 multiplied by the value of Y: 35% 𝑜𝑓 400 𝑖𝑠…?→ ∗400 A percentage is a fraction over 100: 57%= ‘x percent of y’ is x written as a fraction over 100 multiplied by the value of Y: 35% 𝑜𝑓 400 𝑖𝑠…?→ ∗400

Use proportions to solve percentages: Of the 400 trucks for sale on a used car lot, 120 have air-conditioning; what percent of trucks have a/c? =𝑥%?→ = 𝑥 100 Use proportions to solve percentages: Of the 400 trucks for sale on a used car lot, 120 have air-conditioning; what percent of trucks have a/c? =𝑥%? Use proportions to solve percentages: Of the 400 trucks for sale on a used car lot, 120 have air-conditioning; what percent of trucks have a/c? =𝑥%?→ = 𝑥 100 → ∗100

An “x percent change” is calculated as the proportion of the amount of the change over the original amount: “If a home was purchased for $200k and sold for $230k, by what percent did the home appreciate in value?”

“If a home was purchased for $200k and sold for $230k, by what percent did the home appreciate in value?” 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = = 𝑥 100 → =𝑥 𝑥 1 “If a home was purchased for $200k and sold for $230k, by what percent did the home appreciate in value?” 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = = 𝑥 100 → ∗100=𝑥 𝑥 1 “If a home was purchased for $200k and sold for $230k, by what percent did the home appreciate in value?” 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = = 𝑥 100 “If a home was purchased for $200k and sold for $230k, by what percent did the home appreciate in value?” 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = = 𝑥 100

The GMAT contains many problems that make use of multiple percentages. Although it is possible to solve these using proportions and/or algebra, it is better to plug in a starting value of 100 and calculate other values as you go.

“60 percent of Loyola seniors are female. If 40 percent of females and 30 percent of males have studied abroad in this year’s senior class, what is the total percent of Loyola seniors that have studied abroad?” 24 30 36 42 70

“60 percent female… 40 percent of females and 30 percent of males studied abroad…” 24 30 36 42 70 Imagine there are 100 seniors, 60 females 40 percent of 60 females is 40/100*60 = 24 30 percent of 40 males is 30/100*40 = 12 24 females + 12 males = 36/100 = 36%

Combinations & Permutations
A permutation (or combination) is the number of ways to order (or select) a number of choices from a given set. It is best to treat these problems as a multistep series of consecutive choices and adjust if necessary if not ordered.

For an ordered set (a permutation or “arrangement”), imagine “steps” I, II, III, etc & multiply together the number of choices still remaining at each step. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑦𝑜𝑢 𝑠𝑒𝑎𝑡 4 𝑝𝑒𝑜𝑝𝑙𝑒 𝑖𝑛 𝑎 𝑟𝑜𝑤 𝑜𝑓 5 𝑐ℎ𝑎𝑖𝑟𝑠?

The “steps” I, II, III, etc are the points at which a selection is to be made. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑦𝑜𝑢 𝑠𝑒𝑎𝑡 4 𝑝𝑒𝑜𝑝𝑙𝑒 𝑖𝑛 𝑎 𝑟𝑜𝑤 𝑜𝑓 5 𝑐ℎ𝑎𝑖𝑟𝑠? I II III IV = 5 ∗ 4 ∗ 3 ∗ 2 =120 𝑠𝑒𝑎𝑡𝑖𝑛𝑔𝑠 = 5 ∗ 4 ∗ 3 ∗ 2

For an unordered group (combination or “selection”), first imagine the boxes for steps I, II, III, etc, then adjust for possible orderings of choices A, B, etc. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑦𝑜𝑢 𝑐ℎ𝑜𝑜𝑠𝑒 3 𝑑𝑒𝑙𝑒𝑔𝑎𝑡𝑒𝑠 𝑓𝑟𝑜𝑚 7 𝑛𝑜𝑚𝑖𝑛𝑒𝑒𝑠?

𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑦𝑜𝑢 𝑐ℎ𝑜𝑜𝑠𝑒 3 𝑑𝑒𝑙𝑒𝑔𝑎𝑡𝑒𝑠 𝑓𝑟𝑜𝑚 7 𝑛𝑜𝑚𝑖𝑛𝑒𝑒𝑠? I II III =7 ∗ 6 ∗ 5 =210 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 For example, delegates B, D, & G chosen: B D G

For example, delegates B, D, & G chosen: B D G But B, D, & G could have been chosen in many different equivalent “orderings”. D B G B G D D G B

Consider how many orderings B, D, & G could have been chosen in as a group: i ii iii =3 ∗ 2 ∗ 1 =6 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑛𝑔𝑙𝑒 ′ 𝑠 𝑒𝑙𝑒𝑐𝑡𝑖𝑜 𝑛 ′ 𝑜𝑓 𝐵, 𝐷, & 𝐺

3 𝑑𝑒𝑙𝑒𝑔𝑎𝑡𝑒𝑠 𝑓𝑟𝑜𝑚 7 𝑛𝑜𝑚𝑖𝑛𝑒𝑒𝑠: I II III =7 ∗ 6 ∗ 5 =210 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 But every delegate set is repeated 6x: i ii iii =3 ∗ 2 ∗ 1 =6 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔𝑠 𝑔𝑟𝑜𝑢𝑝

3 𝑑𝑒𝑙𝑒𝑔𝑎𝑡𝑒𝑠 𝑓𝑟𝑜𝑚 7 𝑛𝑜𝑚𝑖𝑛𝑒𝑒𝑠: I II III i ii iii =210 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 / 6 𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔𝑠 There are 210 total selections, but there are 6 equivalent orderings for each set, or 210 / 6 = 35 combinations

In the state of New Texas, license plates have 2 letters from the 26 letters of the alphabet followed by 3 1-digit numbers. How many different license plates can New Texas have if repetition of letters and numbers is allowed? 23,400 60,840 67,600 608,400 676,000

“To fill a number of vacant positions, an IT firm needs to hire 2 database administrators from 6 applicants, and 3 developers from 5 applicants. How many selections could be made?” 25 60 150 180 1800

Averages & Variation “Mean, median, and mode”:
Mean (or average) is the sum of all values divided by the count of values. Median is the “middle” number in a set of values when put in rank-order. Mode is the most frequent value.

Averages & Variation {3, 4, 6, 7, 7, 9} Mean = 3+4+6+7+7+9 6 = 36 6 =6
Median = ½-way between 6 and 7 = 6.5 Mode = 7

Averages & Variation The mean (or average) has 3 parts: 𝑚𝑒𝑎𝑛 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 GMAT questions commonly test the ability to apply this formula in an unexpected “direction” or context

Averages & Variation “A company has two sales teams, A and B, with 4 and 6 salespeople respectively. If the average number of monthly sales generated by salespeople in team A is 20 sales and the average in team B is 30 sales, what is the overall average number of sales generated?”

Averages & Variation “If average sales in team A (4 people) is 20 and average sales in team B (6 people) is 30, what is the overall average # of sales?” 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 To solve for the overall average, take this formula and start filling in what you know.

Averages & Variation 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 4 𝑓𝑟𝑜𝑚 𝐴+6 𝑓𝑟𝑜𝑚 𝐵 The number of sales figures being averaged is 4 from team A and 6 from team B, or 10.

Averages & Variation 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝐴+𝑠𝑢𝑚 𝑜𝑓 𝐵 10
𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝐴+𝑠𝑢𝑚 𝑜𝑓 𝐵 10 To find the sum of values from each team, you have to use the average of each team.

Averages & Variation 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝐴+𝑠𝑢𝑚 𝑜𝑓 𝐵 10
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝐴 (20)= 𝑠𝑢𝑚 𝑜𝑓 𝐴 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝐴 (4) The average of A, 20, is equal to the sum of A (unknown) divided by the 4 salespeople. 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 20 ∗𝑛𝑢𝑚𝑏𝑒𝑟 4 =𝑠𝑢𝑚 𝑜𝑓 𝐴

Averages & Variation 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 80+𝑠𝑢𝑚 𝑜𝑓 𝐵 10
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 30 ∗𝑛𝑢𝑚𝑏𝑒𝑟 6 =𝑠𝑢𝑚 𝑜𝑓 𝐵 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒= 𝑓 10 𝑓 = =26 So the overall average number of sales is 26 which is different from the obvious answer.

Averages & Variation “The average of a set of 9 numbers is 8. Two of the numbers are 11 and 12. What is the average of the remaining values?” 4.5 5 5.4 6 7

Averages & Variation Range and Standard Deviation:
The range of a set of values is the highest value – the lowest value. The standard deviation is a number that represents an average “distance from the mean” in the set of values.

Averages & Variation {3, 4, 6, 7, 7, 9} Range = 9 – 3 = 6
St. Dev. = approx. 2 Mean ↓ Data → Dist. = 2 Dist.=1 Dist. = 3 Dist. = 3

Averages & Variation Standard Deviation concepts:
Two sets of data with different means do not necessarily have different standard deviations. Distance from the mean, not other data points, determines the std. dev.

Averages & Variation {2, 10} {10, 18} {7, 13} {9, 11} {16, 16}
Set S has a mean of 10 and a std. dev. of 3. We are going to add two numbers to Set S. Which pair of numbers would decrease the std. dev. the most? {2, 10} {10, 18} {7, 13} {9, 11} {16, 16}

Math / Quantitative Algebra Content Review

Fundamentals of Algebra
Algebra is a process that has 2 parts: Being able to translate a problem into mathematical expressions, functions, or algebraic equations Knowing how to use or modify the resulting expressions or equations

Unknown, to-be-solved-for values are represented by letters in algebra: “The price of a pair of shoes is equal to 3 times the price of a pair of jeans.” Unknown, to-be-solved-for values are represented by letters in algebra: “The price of a pair of shoes is equal to 3 times the price of a pair of jeans.” 𝑠=3∗𝑗

You can perform the same operation, any operation, to both sides of an equation and it will remain equal. 𝑠=3∗𝑗 (𝑠) 40 +2= (3∗𝑗) 40 +2

Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side: 𝑥−5 = 2+3𝑥 −3𝑥 −3𝑥 −2𝑥 = 7

Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side: −2𝑥 = 7 −2𝑥 − = 7 −2

Usually, you do this in order to simplify, not add to, an equation with the goal of a single letter (unknown value) on one side: 𝑥 =− =−3.5

“If x is equal to 1 more than the product of 3 and z, and y is equal to 1 less than the product of 2 and z, then how much greater is 2x than 3y when z has a value of 4?” 1 2 3 5 6 𝑥=3𝑧+1→3∗4+1=13→2𝑥=26 𝑥=3𝑧+1→3∗4+1=13→2𝑥=26 𝑥=3𝑧+1→3∗4+1=13→2𝑥=26 𝑦=2𝑧−1→2∗4−1=7→3𝑦=21 𝑦=2𝑧−1→2∗4−1=7→3𝑦=21 𝑦=2𝑧−1→2∗4−1=7→3𝑦=21

Systems of Equations In general, to solve for a given number of unknown values (i.e. letters), you need as many equations as letters. 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10

Systems of Equations 2 ways to solve a “system of equations”:
Solve one equation for a letter (y), then plug into the other equation. Change one equation so that if you add it to the other equation, one letter (y) will be cancelled out.

Systems of Equations 1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10 −5𝑥 −5𝑥 −𝑦=10−5𝑥 𝑦=−10+5𝑥 Solve for y →

Systems of Equations 1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10 Solve for y →

Systems of Equations 1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 𝑦=−10+5𝑥 Plug in y →

Systems of Equations 1. Solve for y, plug in, then solve for x: 3𝑥+2(−10+5𝑥)=6 3𝑥+2(−10)+2(5𝑥)=6 3𝑥−20 +10𝑥 =6 13𝑥=26→𝑥= =2 Plug in y →

Systems of Equations 1. Solve for y, plug in, then solve for x: 3𝑥+2𝑦=6 𝑦=−10+5𝑥 𝑦=−10+10 Plug in y → → 𝑥=2 𝑦=−10+5(2) → 𝑦=0

Systems of Equations 2. Change equations so y cancels out: 3𝑥+2𝑦=6 5𝑥 − 𝑦 =10 2∗ 5𝑥−𝑦 =10∗2 2 5𝑥 +2 −𝑦 =20 10𝑥−2𝑦=20 Change →

Systems of Equations 2. Change equations so y cancels out: 3𝑥+2𝑦=6
10𝑥−2𝑦=20 13𝑥 =26 𝑥 =2 Change → / /13

Systems of Equations “A symphony sells 3 kinds of tickets: box seats for $40, general admission for $20, and student admission for $10. On a recent night they sold 200 tickets, 40 to students, and made $4,000 in all. How many general admission tickets did the symphony sell?” 96 120 140 160 180

Systems of Equations “15 years ago, Adam was 3 times as old as Bob. Today, Adam is twice as old as Bob. How old will Adam be 5 years from now?” 35 45 50 60 65 𝐴−15 =3∗(𝐵−15) 𝐴=2𝐵

Systems of Equations Cautions about systems of equations:
If the 2 equations are equivalent to each other, you cannot solve them. 3𝑥+2𝑦=6 6𝑥+4𝑦=12

Systems of Equations On the other hand, sometimes you don’t need 2 equations; GMAT will put 2 variables in 1 equation, but do so such that 1 variable cancels out. 15 𝐴+𝐵 −6 𝐵 2 =30+3𝐵(5−2𝐵) 15𝐴+15𝐵−6 𝐵 2 =30+15𝐵−6 𝐵 2

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