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Fractions!!. Prime Numbers and Factorization. How many positive prime numbers are less than 100?. Answer: 25. What is the sum of the prime numbers between π and 10 π ?. Answer: 155. List the positive prime numbers less than 100 that have the units digit equal to 3.

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Fractions!!

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Fractions!!


Prime NumbersandFactorization


How many positive prime numbers are less than 100?


Answer:25


What is the sum of the prime numbers between π and 10π?


Answer:155


List the positive prime numbers less than 100 that have the units digit equal to 3.


Answer:3, 13, 23, 43, 53, 73, 83


Prime Factorization


Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).The number 12 is a multiple of 3, because it can be divided evenly by 3.3 · 4 = 123 and 4 are both factors of 123 · 2 · 2 (prime factorization of 12)12 is a multiple of both 3 and 4.


Write the prime factorization of:

100


Answer:2∙2∙5∙5or2² ∙ 5²(Prime-Power Factorization)


Write the prime factorization of:

972


Answer:2∙2∙3∙3∙3∙3∙3 or2² ∙ 3⁵(Prime-Power Factorization)


List the prime factorization of the following terms (includes variables):

25n²


Answer:5·5·n·n


List the prime factorization of the following terms (includes variables):

92xy


Answer:2·2·23·x·y


Factors


List all the positive factors of

30


Answer:1, 2, 3, 5, 6, 10, 15, 30


List all the positive factors of

87


Answer:1, 3, 29, 87


Greatest Common Factor (GCF)


Greatest Common FactorThe highest number that divides exactly into two or more numbers. If you find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.Abbreviated "GCF". Also called "Highest Common Factor"Example: the GCF of 12 and 30 is 6, because 1, 2, 3 and 6 are factors of both 12 and 30, and 6 is the greatest.


Find the GCF of

24 & 28


Answer:4


Find the GCF of

39v & 30uv


Answer:3v


Find the GCF of

35n²m & 21m²n


Answer:7nm


Find the GCF of

36xy³ & 24y²


Answer:12y²


Find the GCF of

105x, 30yx & 75x


Answer:15x


Least Common Multiple (LCM)


Least Common MultipleThe smallest (non-zero) number that is a multiple of two or more numbers.Least Common Multiple is made up of the words Least, Common and Multiple:What is a "Multiple" ?The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table. Here are some examples: The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ... The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc... What is a "Common Multiple" ?When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. For example, when you write down the multiples of 4 and 5, the common multiples are those that are found in both lists:The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...Notice that 20 and 40 appear in both lists? So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too) What is the "Least Common Multiple" ?It is simply the smallest of the common multiples. In our previous example, the smallest of the common multiples is 20 ...... so the Least Common Multiple of 4 and 5 is 20.


Finding the Least Common MultipleIt is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.Example: Find the least common multiple for 3 and 5:The multiples of 3 are 3, 6, 9, 12, 15, ..., and the multiples of 5 are 5, 10, 15, 20, ..., like this:As you can see on this number line, the first time the multiples match up is 15. Answer: 15More than 2 NumbersYou can also find the least common multiple of 3 (or more) numbers.Example: Find the least common multiple for 4, 6, and 8Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...Multiples of 8 are: 8, 16, 24, 32, 40, ....So, 24 is the least common multiple (I can't find a smaller one !)


Find the LCM of

35 & 25


Answer:175


Find the LCM of

30, 25 & 10


Answer:150


Find the LCM of

18 & 6v


Answer:18v


Find the LCM of

3x² & 10


Answer:30x²


Find the LCM of

20y & 14y²


Answer:140y²


Find the LCM of

16x²y & 32x


Answer:32x²y


Find the LCM of

8y², 16xy & 16y


Answer:16y²x


What is the negative difference of the least common multiple (LCM) and the greatest common factor (GCF) of:

80, 140, and 200


Answer:-2780


Types of Fractional Numbers

  • A proper fraction is a fraction whose value is less than 1 (numerator < denominator)

  • An improper fraction is a fraction whose value is greater than or equal to 1 (numerator > denominator)

  • A mixed number is a number whose value is greater than 1 made up of a whole part and a fraction part


Improper FractionMixed Number

  • Denominator: tells us how many parts make up a whole

  • Numerator: tells us how many parts we have

  • How many wholes can we make out of the parts we have?

  • Divide the numerator by the denominator  the quotient is the whole part

  • How many parts do we have remaining?

  • The remainder (over the denominator) makes up the fraction part


Simplest Form of a Fraction

  • A fraction is in simplest form when there are no common factors in the numerator and the denominator.


Ex: Simplest Form

Ex: 6/8 and 3/4 are equivalent

The fraction 6/8 is written in simplest form as 3/4

=

=

=

1 x

Magic one


Ex: Write 12/42 in simplest form

  • First prime factor the numerator and the denominator:

  • 12 = 2 x 2 x 3 and 42 = 2 x 3 x 7

  • Look for Magic Ones

  • Simplify

=

=

=

1 x 1 x

=

Notice: 2 x 3 = 6 = GCF(12, 42)

 factoring (dividing) out the GCF will simplify the fraction


Ex: Write 7/28 in simplest form

  • What is the GCF(7, 28)?

    • Hint: prime factor 7 = 7

    • prime factor 28 = 2 x 2 x 7

= 7

=

=

=

1 x

=

Dividing out the GCF from the numerator and denominator simplifies the fraction.


Ex: Write 27/56 in simplest form

  • What is the GCF(27, 56)?

    • Hint: prime factor 27 = 3 x 3 x 3

    • prime factor 56 = 2 x 2 x 2 x 7

= 1

There is no common factor to the numerator and denominator (other than 1)

Therefore, 27/56 is in simplest form.


Rational and Irrational Numbers


Rational and Irrational Numbers Essential Question

How do I distinguish between rational and irrational numbers?


Real Numbers

Rational numbers

Irrational numbers

Integers

Whole

numbers

The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.


Rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

⅔, ⅕, ¼ or

144 = 12


Caution!

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Irrational numberscan be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, the square root of 2 is not a perfect square, so the square root of 2 is irrational. Also, π is irrational.


Make a Venn Diagram that displays the following sets of numbers:

Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.

Reals

Rationals

-2.65

Integers

-3

-19

Wholes

0

Irrationals

Naturals

1, 2, 3...


Classifying Real Numbers

Write all classifications that apply to each number.

A.

5 is a whole number that is not a perfect square.

5

irrational, real

B.

–12.75

–12.75 is a terminating decimal.

rational, real

16

C.

whole, integer, rational, real


9 = 3

Write all classifications that apply to each number.

9

A.

whole, integer, rational, real

–35.9

–35.9 is a terminating decimal.

B.

rational, real

81

C.

whole, integer, rational, real


A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.


Adding and SubtractingFractions


Evaluate the expression:6- ¹⁄₆


Answer:³⁵/₆


Evaluate the expression:⅕+ ¾


Answer:¹⁹/₂₀


Evaluate the expression:(-4/5) – 7/8


Answer:- ⁶⁷/₄₀


Evaluate the expression:(-10/7) + 1/6


Answer:- ⁵³/₄₂


Evaluate the expression:2 – ¹³/₈


Answer:3/8


Evaluate the expression:(-4/3) – (-3/2)


Answer:1/6


Evaluate the expression:- 3 ⅗ - 4 ⅖


Answer:-8


Evaluate the expression:(-2⁷/₈) + (-1 ¹/₂)


Answer: -4 ³/₈


Evaluate the expression:(-2⁵/₆) - (-1 ¹/₄)


Answer: -1 ⁷/₁₂


Evaluate the expression:2⁴/₅ - ⁵/₈


Answer: 2 ⁷/₄₀


Multiplying Fractions


Evaluate the expression:-⁵/₄ · ¹/₃


Answer: -⁵/₁₂


Evaluate the expression:-2 · ³/₇


Answer: -⁶/₇


Dividing Fractions


Evaluate the expression:- ¹/₅ ÷ ⁷/₄


Answer: - ⁴/₃₅


Evaluate the expression:- ¹/₂ ÷ ⁵/₄


Answer: - ²/₅


Evaluate the expression:- ³/₂ ÷ ⁻¹⁰/₇


Answer: ²¹/₂₀


Evaluate the expression:- ⁹/₅ ÷ 2


Answer: -⁹/₁₀


Multiplying and DividingMixed Numbers


Evaluate the expression:-2²/₃ · 4 ¹/₁₀


Answer:-10 ¹⁴/₁₅


Evaluate the expression:-2 ¹/₅ · (-1 ³/₄)


Answer: 3 ¹⁷/₂₀


Evaluate the expression:-2 ÷ (-3 ⁴/₅)


Answer:+ ¹⁰/₁₉


Evaluate the expression:- 3 ⁷/₁₀ ÷ 2 ¹/₄


Answer:-1 ²⁹/₄₅


Challenge Problems


3_________ 3___ 2 + 3/2

2 +

2 +


Answer:3 ¹/₂₀


Simplify completely:

4x²y

2x


Answer:2xy


Simplify completely:

y¯¹

y¯²


Answer:y


Simplify completely:

16x⁴y¯¹

4x²y¯²


Answer:4x²y


Simplify completely:

36x³y⁶z¹²

4x¯¹y³z¹⁰


Answer:9x⁴y³z²


Simplify completely:

21x³y⁷z¹⁴30x³z¯⁵

18x⁴y⁶ y¹²z¯⁶

·


Answer:35x²z¹⁵y¹¹


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