math for 800 04 integers, fractions and percents

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CONTENTS

INTEGERS

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

EVEN-ODD NUMBERS

INTEGERS

CONSECUTIVE NUMBERS

Consecutive Positive Integers

are integers that follow each other in

order:1, 2, 3, 4, 5, …

CONSECUTIVE INTEGERS

are even integers that follow each other in

order:2, 4, 6, 8, 10, …

EVEN NUMBERS

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

CONSECUTIVE EVEN INTEGERS

Consecutive Odd Integers

are odd integers that follow each other in

order:1, 3, 5, 7, …

ODD NUMBERS

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

CONSECUTIVE ODD INTEGERS

even eveneven odd oddodd

odd oddodd even eveneven

…, n – 2, n – 1, n , n + 1, n + 2, n + 3, …

EVEN/ODD NUMBERS

even even even

odd odd even

even odd odd

odd even odd

EVEN / ODD NUMBERS

EVEN / ODD NUMBERS

even even even

odd even even

even odd even

odd odd odd

EVEN / ODD NUMBERS

EVEN / ODD NUMBERS

Consecutive Prime Numbers

are prime numbers that follow each other

in order:2, 3, 5, 7, 11, …

CONSECUTIVE

NUMBERS

COUNTING INTEGERS

COUNTING CONSEC. INTEGERS

Counting Consecutive Integers

12, 13, 14, 15, 16, 17, 18, 19, 20

20 – 12 + 1 = 9

COUNTING CONSEC. EVEN/ODD INTEGERS

If the result is an integer number,

that is the answer.

Counting Consecutive

Even/Odd Integers

12, 13, 14, 15, 16, 17, 18, 19

19 – 12 + 1 = 8

8 / 2 = 4

Counting Consecutive

Even/Odd Integers

11, 12, 13, 14, 15, 16, 17, 18

18 – 11 + 1 = 8

8 / 2 = 4

Subtract the smallest number

from the largest number and

add 1, divide by 2.

COUNTING CONSEC. EVEN/ODD INTEGERS

If the result is not an integer number,

see how the series starts and ends.

Counting Consecutive

Even/Odd Integers

11, 12, 13, 14, 15, 16, 17, 18, 19

19 – 11 + 1 = 9

9 / 2 = 4.5

Counting Consecutive

Even/Odd Integers

10, 11, 12, 13, 14, 15, 16, 17, 18

18 – 10 + 1 = 9

9 / 2 = 4.5

CONSECUTIVE

NUMBERS

DIVISIBILITY

FACTOR / DIVISOR

FACTOR / DIVISIOR

a number that can be divided by another number without a

remainder.

MULTIPLE / DIVISIBLE

MULTIPLE / DIVISIBLE

Multiples of 2

Multiples of 3

DIVISIBILITY FACTS

1 is a factor/divisor of every integer.

0 is a multiple of every integer.

The factors of an integer include

positive and negative integers.

Factors of 4

Factors of 12

Prime Numbersare natural numbers

that has no positive divisors other than 1

and itself.

PRIM

E N

UM

BERS

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 …

If is prime number then, it won’t have any

factor such that .

2 1, 2

3 1, 3

5 1, 5

7 1, 7

11 1, 11

13 1, 13

17 1, 17

19 1, 19

Current Largest Prime

257,885,161 – 117,425,170 digits longJan 25, 2013, University of Central Missouri

Composite Number

a natural number greater than 1 that is not a prime number.

CO

MPO

SIT

E

NU

MBERS

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 …

If k is composite number then, it

will have at least one factor p such

that 1 < p < k.

4 1, 2, 4

6 1, 2, 3,6

8 1, 2, 4, 8

9 1, 3, 9

12 1, 2, 3, 4, 6, 12

14 1, 2, 7, 14

15 1, 3, 5, 15

16 1, 2, 4, 8, 16

1 1 11 1, 11 21 1, 3, 7, 21

2 1, 2 12 1, 2, 3, 4, 6, 12 22 1, 2, 11, 22

3 1, 3 13 1, 13 23 1, 23

4 1, 2, 4 14 1, 2, 7, 14 24 1, 2, 3, 4, 6, 8, 12, 24

5 1, 5 15 1, 3, 5, 15 25 1, 5, 25

6 1, 2, 3, 6 16 1, 2, 4, 8, 16 26 1, 2, 13, 26

7 1, 7 17 1, 17 27 1, 3, 9, 27

8 1, 2, 4, 8 18 1, 2, 3, 6, 9, 18 28 1, 2, 4, 7, 14, 18

9 1, 3, 9 19 1, 19 29 1, 29

10 1, 2, 5, 10 20 1, 2, 4, 5, 10, 20 30 1, 2, 3, 5, 6, 10, 15, 30

Prime Factorization

is the decomposition of a composite number

into prime factors,

which when multiplied together equal the

original integer.

PRIME FACTORIZATION

PRIME FACTORIZATION

Eighter way, the result is

2 2 3 5 = 60 or 22 3 5 = 60

60

6 10

2 3 2 5

60

2 30

3

2 15

5

NUMBER OF DIVISORS

60 = 22 3

1 51

= 60

1 x 60

2 x 30

3 x 20

4 x 15

5 x 12

6 x 103 x 2 x 2 = 12

• Take all the exponents from the

prime factorization and add 1 to

each of them.

• Multiply the modified exponents

together.

of two integers is the largest positive integer that

divides the numbers without a remainder.

GCF

GCF

Prime factors of :18 = 2 × 3 × 3

Prime factors of :24 = 2 × 2 × 2 × 3

There is one 2 and one 3 in common.

The GCF of 18 and 24 is 2 × 3 = 6

GCF (GCD)

36

4 9

2 2 3 3

54

6 9

2 3 3 3

Shared Factors: 2, 3, 3

Multiply (GCF): 2 3 3 = 18

Find the GCF of 36 and 54:

The Least Common Multiple (LCM)

of two integers or more integers, is the smallest positive integer that is

divisible by all the numbers.

LCM

LCM

LCM

Factors Multiples1 2 3 4 6 12 12 24 36 48 60 72 84 96 108 …

1 2 3 6 9 18 18 36 54 72 90 108 126 …

GCF = 6 LCM = 36

GCF and LCM

The remainder

is the amount "left over" after performing

the division of two integers which do not

divide evenly.

REMAINDER

1- 6

3

2 7divisor

remainder

dividend

quotient

7 = 2∙3 + 1

dividend = divisor∙quotient + remainder

The remainder r when n is divided by a

nonzero integer d is zero if and only if n is

a multiple of d.

Dividing by 4

Divisible by

means that when you divide one number by another the result is a

whole number.

DIVISIBILITY BY 2

2, 40, 258, 1020

Last digit is even

DIVISIBILITY BY 3

69 6+9 = 15

504 5+0+4 = 9

1938 1+9+3+8 = 21

Sum of digits is a multiple of 3

DIVISIBILITY BY 4

512, 720, 1424, 1620

Last two digits are multiple of 4

DIVISIBILITY BY 5

25, 50, 560, 1005

Last digit is 5 or 0

DIVISIBILITY BY 6

72 7+2 = 9

1200 1+2+0+0 = 3

1860 1+8+6+0 = 15

Sum of the digits is multiple

of 3 and the last digit is even

DIVISIBILITY BY 7

3101 310 – 2 = 308

308 30 – 16 = 14

Take the last digit off the

number, double it and

subtract the doubled number

from the remaining number

DIVISIBILITY BY 9

729 7+2+9 = 18

810 8+1+0 = 9

9918 9+9+1+8 = 27

Sum of digits is a multiple of 9

DIVISIBILITY BY 10

30, 70, 100, 250, 560

Last digit is 0

Divisibility Rules

A number is divisible by … DivisibleNot

Divisible

2 If the last digit is even 3,728 357

3 If the sum of the digits is a multiple of 3 120 155

4 If the last two digits form a number divisible by 4 144 142

5 If the last digit is 0 or 5 150 123

6 If the number is divisible by both 2 and 3 48 20

9 If the sum of the digits is divisible by 9 729 811

10 If the last digit is 0 50 53

DIVISIBILITY

FRACTIONS

EQUIVALENT FRACTIONS

NAMING FRACTIONS

FRACTIONS

It is useful to think of a fraction

bar as a symbol for division.

The denominator of a fraction

can’t be equal to zero.

SIGNS IN A FRACTION

numeratorfraction

denominator

Any two of the

three signs of a

fraction may be

changed without

altering the value

of the fraction.

SIGNS IN A FRACTION

2 2 2 2

5 5 5 5

2 2 2 2

5 5 5 5

COMPARING FRACTIONS

Same Denominator

COMPARING FRACTIONS

Same Numerator

1

3

1

4

1

5

1

6

5 4?

8 7Cross- multiplication

COMPARING FRACTIONS

5 7?4 8

35 32

5 4

8 7

2 7

15 15

Make sure the denominators are the same.

Add the numerators, put the answer over

the denominator.

Simplify the fraction.

ADDING FRACTIONS

9

15

3

5

Make sure the denominators are the same.

Subtract the numerators. Put the

answer over the same denominator.

Simplify the fraction.

SUBTRACTING FRACTIONS

2 9 1

15 10 5

25

30

4 27 6

30

4 27 6

30 30 30

5

6

5 6

7 4

3 3 7 217

5 5 1 5

Multiply the numerators.

Multiply the denominators.

Simplify the

fraction.

MULTIPLYING FRACTIONS

30

28

15

14

1 3

2 5

Turn the second fraction upside-down

(this is now a reciprocal).

Multiply the first fraction by that

reciprocal.

Simplify the

fraction.

DIVIDING FRACTIONS

1 5

2 3

1 5

2 3

5

6

22

3

Distribute the exponent into the

numerator as well as into the denominator.

Evaluate the numerator

and the denominator.

Simplify the

fraction.

POWER OF FRACTIONS

2

2

2

3

4

9

4

9

Distribute the root into the

numerator as well as into the denominator.

Evaluate the numerator

and the denominator.

Simplify the

fraction.

ROOTS OF FRACTIONS

4

9

2

3

TRICKY OPERATIONS

The reciprocal of a is .1

a

The reciprocal of 2 is .1

2

The reciprocal of is .a b1

a b

The reciprocal of is .3

4

4

3

a

a c a d a dbc b d b c b c

d

COMPLEX FRACTIONS

A fraction with fractions in the

numerator or denominator.

Proper Fractionfraction that is less than one, with the numerator

less than the denominator.

Improper Fraction

a fraction in which the numerator is greater

than the denominator.

Multiply the whole number

part by the denominator

Add the numerator

The result is the new numerator (over the same denominator)

25

7

MIXED NUMBER TO IMPROPER FRACTION

5 7 2

7

37

7

Divide the denominator into the numerator.

The quotient becomes the whole number.

The remainder becomes the new numerator.

7

2

IMPROPER FRACTION TO MIXED NUMBER

13

2

MIXED NUMBERS

Part fraction whole

3 3100 100 75

4 4of

25252525

252525

PART – FRACTION

Part fraction whole

1 1100 100 50

2 2of

50

50

PART – FRACTION

FRACTIONS

DECIMALS

2 decimal

places

1 decimal

place

3 decimal

places

Divide the top

of the fraction

by the bottom.

FRACTION TO DECIMAL

50.625

8

40.571428

7

FRACTION TO DECIMAL

Write down the decimal divided by 1.

Multiply both top and bottom by 10 for every number after the decimal

point.

Simplify (or reduce) the

fraction.

DECIMAL TO FRACTION

0.750.75 100

1 100

75

100

3

4

Terminating Decimals

When the denominator has only factors 2, 5, a

combination of both or of its powers.

TERMINATING DECIMALS

1.5

2

71.4

5

2

4

3 3.06

50 2 5

3 3.1875

16 2

Repeating Decimals

when the denominator has other factors than 2 and 5 or its powers.

REPEATING DECIMALS

10.333

3

120.1212

99

40.571428571428...

7

50.384615384615...

13

repeatingdecimals

repeatingdecimals

10.333

3

120.1212

99

40.571428571428...

7

50.384615384615...

13

repeatingdecimals

repeatingdecimals

LENGTH OF THE CLUSTER

2n

d

4th

6th

3rd

6th

9th

1 20.111... 0.222...

9 9

3 70.333... 0.777...

9 9

COMMON REP. DECIMALS

11 120.1111... 0.1212...

99 99

25 830.2525... 0.8383...

99 99

COMMON REP. DECIMALS

127 2150.127127... 0.215215...

999 999

853 6150.853853... 0.615615...

999 999

COMMON REP. DECIMALS

OPERATIONS

WITH

DECIMALS

ADDING DECIMALS

Line up decimal

points.

132.7

96.543

229.243

SUBTRACTINGDECIMALS

Line up decimal

points.

132.7

96.543

36.157

It is not necessary to align the

decimal points.

Add the number of digits to the right of the decimal points in the

decimals being multiplied.

MULTIPLYING DECIMALS

125.3

1.2

2506

1253

150.36

MULTIPLYING DECIMALS

12.53

1.2

2506

1253

15.036

Move the decimal point in the divisor to

the right until the divisor becomes an

integer.

Move the decimal point in the dividend the same number of

places.

Proceed with the division.

DIVIDING DECIMALS

1.6 128.32

80.2

160 12832

1280

320

320

0

DIVIDING DECIMALS

1.6 12.832

8.02

1600 12832

12800

3200

3200

0

DECIMALS

PERCENTS

Percents:

a percentage is a number or ratio expressed as a fraction of 100.

PERCENTS

Percent means

hundredths or

number out of

100.

1%

2%

20%

PERCENTS

Percent means

hundredths or

number out of

100.

%100

11%

100

22%

100

2020%

100

nn

PERCENT EQUIVALENTS

14

12

34

25%

0.25

50%

0.50

75%

0.75

PERCENT EQUIVALENTS

16

13

23

16.6%

0.1666

33.33%

0.333

66.66%

0.666

PERCENT EQUIVALENTS

110

15

12

10%

0.1

20%

0.20

50%

0.5

100

percentPart whole

PERCENTS FORMULA

PERCENTS FORMULA

1225

100x

45 9100

x

6015

100x

2540 160

100

Percent increaseIs the ratio of the increase of two

numbers divided by the original number multiplyied by 100.

100%increase

Percent increaseoriginal whole

100%

(100 + n)%

n %

100%increase

Percent increaseoriginal whole

PERCENT INCREASE

The price of a tour goes up from

$80 to $100. What is the percent

increase?

20100% 25%

80Percent increase

PERCENT INCREASE

The price of a tour goes up from $80 to

$100. What is the percent increase?

20100% 25%

80Percent increase

100%decrease

Percent decreaseoriginal whole

(100 – n) %

100 %

n %

100%decrease

Percent decreaseoriginal whole

PERCENT DECREASE

The price of a tour goes down from

$100 to $80. What is the percent

decrease?

20100% 20%

100Percent decrease

PERCENT DECREASE

COMBINED PERCENT INCREASE

A price went up 10% one year, and the

new price went up 20% the next year.

What is the combined percent

increase?

32% increase110 120

100 132100 100

COMBINED PERCENT DECREASE

A price went down 10% one year, and

the new price went down 20% the next

year. What is the combined percent

decrease?

28% decrease90 80

100 72100 100

COMBINED PERCENT INC/DEC

A price went down 20% one year, and

the new price went up 10% the next

year. What is the combined percent

decrease?

12% decrease80 110

100 88100 100

COMBINED PERCENT INC/DEC

INICIAL

AMMOUNT

%

INCREASE /

DECREASE

PARTIAL

RESULT

%

INCREASE /

DECREASE

FINAL

RESULT

100 + 10% 110 -10% 99

100 - 10% 90 + 10% 99

100 + 20% 120 - 20% 96

100 - 20% 80 + 20% 96

100 + 50% 150 - 50% 75

100 - 50% 50 + 50% 75

100 100100 100

100 100

n n

100 100100 100

100 100

n n

COMBINED PERCENT INC/DEC

INTEREST

Interestis a fee paid by a

borrower of assets to the owner as a form of

compensation for the use of the assets.

INTEREST

1

I P r n

F P I

F P rn

Simple interest (I) is determined

by multiplying the interest rate (r)

by the principal (P) by the

number of periods (n).

SIMPLE INTEREST

SIMPLE INTEREST

1

I P r n

F P I

F P rn

SIMPLE INTEREST

Carine deposits $ 1,000 into a special bank account

which pays a simple annual interest rate of 5% for 3

years. How much will be in her account at the end of the

investment term?

P = 1,000

r = 5% = 0.05

n = 3

1

1,000 1 0.05 3

1,150

F P rn

F

F

55% 1,000 1,000 50

100of

SIMPLE INTEREST

55% 1,000 1,000 50

100of

55% 1,000 1,000 50

100of

Simple Interest on 1,000.00 after:

SIMPLE INTEREST

Interest (I) calculated on the initial

principal (P) and also on the

accumulated interest of previous

periods of a deposit or loan.

1n

F P r

I F P

COMPOUND INTEREST

COMPOUND INTEREST

1n

F P r

I F P

Annual rate = 12%, compounded:0 12 months6

6% 6%

COMPOUND INTEREST

Principal = $ 100, Annual rate = 12%,

Time = 1 year, compounded:

COMPOUND INTEREST

1

100 1 0.12F

2

100 1 0.06F

4

100 1 0.03F

12

100 1 0.01F

COMPOUND INTEREST

Carine deposits $ 1,000 into a special bank account

which pays a compound annual interest rate of 5% for 3

years. How much will be in her account at the end of the

investment term?

P = 1,000

r = 5% = 0.05

n = 3

3

1

1,000 1 0.05

1,157.625

nF P r

F

F

COMPOUND INTEREST

55% 1,000.00 1,000.00 50.00

100of

55% 1,050.00 1,050.00 52.50

100of

55% 1,102.50 1,102.50 55.13

100of

Compound Interest on 1,000.00 after:

COMPOUND INTEREST

SIMPLE INTEREST vs

COMPOUND INTEREST

Prin

cip

al

Co

mp

ou

nd

Inte

rest

Sim

ple

Inte

rest

Co

mp

ou

nd

Inte

rest

Sim

ple

Inte

rest

Co

mp

ou

nd

Inte

rest

Sim

ple

In

tere

st1,0

50

.00

1,1

00

.00

1,1

50

.00

P = 1,000.00

r = 5% = 0.05

n = 3 years

PERCENTS

SUMMARY

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