# math cram kit

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8/9/2019 Math Cram Kit

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theWorldScholars Cup

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DemiDec, The World Scholars Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation.

Academic Decathlon and USAD are registered trademarks of the United States Academic Decathlon Association.

DemiDec is not affiliated with the United States Academic Decathlon.

MATHCRAM KIT

I. WHAT IS A CRAM KIT?................................................................. 2II. CRAMMING FOR SUCCESS 2III. GENERAL MATH................................. 3IV. ALGEBRA........................................................ 5V. GEOMETRY...........................................................15VI. TRIGONOMETRY...................................................22VII. CALCULUS......................................................................................26VIII. CRUNCH KIT..................................................................... 33IX. ABOUT THE AUTHOR.35

BYSTEVEN ZHUHARVARD UNIVERSITY

FRISCO HIGH SCHOOL

EDITED BY

DEAN SCHAFFERSTANFORD UNIVERSITY

TAFT HIGH SCHOOLSOPHY LEEHARVARD UNIVERSITY

PEARLAND HIGH SCHOOL

DEDICATED TO PYTHAGORAS,

FOR BEING SUCH A HOMIE.

2009 DEMIDEC

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WHAT IS A CRAM KIT?A Word from the Editor

COMPETITION IS NEARING STRUCTURE OF A CRAM KIT

The handful of days before competition can be the mostoverwhelming. You dont have enough time to revieweverything, so a strategic allocation of your resources is

crucial. Cram Kits are designed with one goal in mind-----to provide you with the most testable and most easilyforgotten facts.

Math. The very word strikes fear into the hearts ofmany. But dont be discouraged-----math, like any otherevent, can be mastered through studying, and perhapsmore than any other event, through test-taking.

Sounds simple, right? Unfortunately, Decathlon math isso broad that no guide could possibly hope to cover allof nooks and crannies. This Cram Kit, then, is meant asa quick review tool to cover last-minute formulas and to

correct minor misconceptions that may cost you pointsin competition. I advise you to go through this guidewith textbooks nearby. Doing example problems is thebest way to reinforce the concepts that you learn.

The main body of the Cram Kit is filled with charts anddiagrams for efficient studying. Youll also find helpfulquizzes to reinforce the information as you review.

The Crunch Kit presents the most important formulasthat you need to know for the math test. Realize,however, that knowing when to apply each formula ishalf the battle. Plugging in the numbers is often theeasiest step.

Last, but not least, remember to relax. In the finalmoments before you open your test booklet, confidenceis your most important asset.

Good luck and happy cramming!

Sophy Lee

CRAMMING FOR SUCCESSA Word from the Author

PIECES OF THE MATH PIE

TIME IS TICKING!

If you have one day left, read the whole guide.

*

If you have one hour left, read the Crunch Kit.

*

If you have one minute left, scan the List of Lists

*

If you have one second left, good luck.

10%

30%

30%

20%

10% General Math

Algebra

Geometry

Trigonometry

DifferentialCalculus

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GENERAL MATHThe Deceptively Simple and the Utterly Confusing

INTEGERS, FRACTIONS, DECIMALS,

AND PERCENTSBASIC COUNTING TECHNIQUES

FRACTIONS

Fractions must have a common denominatorbefore we can add or subtract them

When multiplying fractions, try to cancel outcommon factors

When dividing fractions, flip over the secondfraction and multiply it by the first one

Step 1. Turn the second fraction upside-down(the reciprocal):

1 4

4 1

Step 2. Multiply the first fraction by the reciprocal of thesecond:

1 4 = 14 = 4

2 1 21 2Step 3. Simplify the fraction: 2

PERCENTAGES

1% represents one in 100 Divide a percentage by 100 to convert it to a

decimal

Multiply a decimal by 100 to convert it t

25 20$40.00 (1- ) (1- ) $24

100 100x x o a percentage

Formula for sale prices: x(1- )(originalprice)100

Successive discounts do NOT have the sameeffect as a cumulative discount

If more than one discount applies to an item, keepmultiplying the right side of the above formula by

discount(1- )

100

A shirt originally priced $40.00 is markeddown by 25%. Joe uses a 20%-off coupon topurchase the shirt. How much does he haveto pay for the shirt before tax?

MULTIPLICATION PRINCIPLE

Helps us find the total number of possibilities whenwe are choosing one item from each of severalgroups

Multiply the number of choices from each group If Sally can choose an outfit from 4 pairs of jeans,

5 shirts, and 3 pairs of shoes, she has 4 x 5 x 3 =60 outfit choices

FACTORIALS, PERMUTATIONS, AND COMBINATIONS

FACTORIALS

! denotes a factorial (50! 50 49 48... 2 1) x x x x

PERMUTATIONS

Arrangements of a set of objects in which ordermatters

When arranging r objects out of a set of n totalobjects, the number of permutations is n r

n!P

(n-r)!

A club of 12 people wants to elect a president, avice-president, and a treasurer. How manydifferent results can this election have?

The three positions are different, so order matters 12 3 12! 12!P 12 x 11x 10 1320

(n-3)! 9!

COMBINATIONS

Arrangements of a set of objects in which order doesNOT matter

When arranging r objects out of a set of n totalobjects, the number of combinations is

n rn!

C(r!)(n-r)!

A club of 12 people wants to elect three people toa committee. How many different results can thiselection have?

The three seats on the committee are the same,so order does not matter

12 3 12! 12! 12 x 11x 10C 2203!(12-3)! 3!9! 3 x 2 x 1

TRY THIS MNEMONIC!

Permutations = Prizes (order matters) Combinations = Committees (order doesnt matter)

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GENERAL MATHMore Counting; Vegas

BASIC COUNTING TECHNIQUES (PT. 2) PROBABILITY OF EQUALLY LIKELY EVENTS

ARRANGEMENT RULES

ARRANGEMENT PRINCIPLE

When a set has two or more identical objects, weneed to take away the redundant arrangementscaused by the identical objects

To arrange the letters in CALIFORNIA, weneed to find the number of permutations anddivide by the factorials of the identical letters

CALIFORNIA has two As and two Is possible arrangements

ARRANGING OBJECTS IN CIRCLES

When we arrange objects in circles, we need tomake sure that each arrangement represents adistinct ordering of objects, not a mere rotation ofanother arrangement

Number of possible circular arrangements = hk

We have to keep one object in place to mark thebeginning of the arrangements

How many different ways can four people sitaround a circular table?

Keep one person in place and rearrange theother three

53

When arranging keys on a keychain, we mustdivide the result by 2 since we can flip thekeychain over, which makes arrangements thatare mirror images of each other identical

In how many different ways can 4 keys bearranged on a keychain?

53

PROBABILITY

The chance that an event will happen

RULES

The probability that event A happens is P(A) csc 6x 2 8 The probability that independent, unrelated events A

and B will occur is P(A+B) = P(A) x P(B)

If events A and B are not mutually exclusive, theprobability of one or the other occurring isP(A or B) = P(A) + P(B) --- P(A+B)

USEFUL FACTS

A standard poker deck has 52 cards Such a deck has 4 suits (2 red and 2 black) of 13

cards each

A decks face cards are the Jack, Queen, and King ofeach suit (12 face cards total in a standard pokerdeck)

A standard die has 6 facesEXAMPLES

What is the probability of rolling a sum of 9 with twodice?

We have 4 outcomes with a sum of 9 (3-6, 6-3,4-5, 5-4) The total possible number of outcomes is 6 x 6 =

36

The probability of rolling a sum of 9 is Asin(kx h) b or Acos(kx h) b

What is the probability of drawing a red Queen froma standard deck of cards?

A 52-card deck has four Queens, two of whichare red

r2 1

P(Q ) 52 26

What is the probability of a coin landing heads fourtosses in a row?

For each toss, the chance of landing heads is 12

The tosses are independent events, since eachtoss does not affect the result of any other toss

41 1

P(4H) P(H) P(H) P(H) P(H)2 16

CALCULATOR USE

When dealing with permutations and combinations,use the built-in functions on your scientific or

graphing calculator to avoid typing in the formulas.Master these (and other calculator techniques)

beforethe test!

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ALGEBRA

Separate but Equal

SOLVING POLYNOMIAL EQUATIONS

(THE BASICS)

SOLVING POLYNOMIAL EQUATIONS (LINEAR)

EQUATION

A mathematical statement that two expressions areequal

Examples 3 + 7 = 14 --- 4 4x + 5 = 2y

POLYNOMIAL

An expression containing variables 4 25x x 23

9

The variables cannot be contained in fractiondenominators

The variables also cannot be contained inexponents

Polynomials with only one term are calledmonomials

212x y is an example of a monomial Even though the expres