Our Lesson

Our Lesson

Simplify and

Equivalent Fractions

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Warm up

1) Is 8 a factor of 2832? Yes

2) List all factors of 49 1, 7 and 49

3) Solve 4x – 2y 3, where x = 2 and y =-1 10

4) Write the prime factorization of 45 5 x 32

5) Write the GCF of 24 and 14 2

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Lets Review FactorsLets Review Factors

A whole number that divides exactly into another whole number is called a factor of that number. For example 100 / 25 = 4

So, 25 is a factor of 100 as it divides exactly into 20.

20 / 4 = 5So, 4 is a factor of 20 as it divides exactly into 20.

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The exponent can also be referred to as the power

46BaseExponent

Powers and Exponents

46 means to multiply the base 4 by itself 6 times 46 = 4 x 4 x 4 x 4 x 4 x 4

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A Prime number is a positive integer A Prime number is a positive integer >1>1

A number that has exactly two factors, A number that has exactly two factors, 1 and itself1 and itself

A number that cannot be A number that cannot be factored .factored .

Prime NumbersPrime Numbers

Example:

7 is a Prime number as it has only two factors 1 and 7

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Composite Numbers

When a whole number greater than one has more than 2 factors it is called a

Composite Number.

10 is a composite number as it has 1, 2, 5 and 10 as its factors

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Expressing a composite number as a Expressing a composite number as a product of prime numbers is called Prime product of prime numbers is called Prime FactorizationFactorization

Prime Factorization Prime Factorization

When we express a number as a product of prime factors, we have actually factored it completely. We refer to this process as prime factorization

The number 60 is a composite number. It can be written as the product 2 x 2 x 3 x 5. Note that 2, 3 and 5 are factors of 60 and all these factors are prime numbers. We call them prime factors.

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Greatest Common Factor of two or more numbers can be defined as the greatest number that is a

factor of each number

Greatest Common Factor

List the factors of each number. Then identify the common factors. The greatest of these common factors is the GCF.

Method 1

Write the prime factorization of each number. Then identify all common prime factors and find their product.

Method 2

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The The least common multipleleast common multiple of the of the numbers a and b is the smallest numbers a and b is the smallest number that is divisible by both a and number that is divisible by both a and bb

We denote the least common multiple of We denote the least common multiple of a and b by lcm (a, b)a and b by lcm (a, b)

Least Common MultipleLeast Common Multiple

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Lets get started

Simplest form of Fraction

A fraction is in its simplest form when the GCF of the numerator and the denominator is 1

There is no other common factor except 1

Example 2 17 5 3 31 19

, ,

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Steps to expressing a fraction in its simplest form

1)Find the GCF of the numerator and denominator

2)Divide the numerator and the denominator by the GCF, and write the resulting fraction

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Lets take an Example

Write the 4 in its simplest form 16

Factors of 4: 1, 2, 4

Factors of 16: 1, 2, 4, 8, 16

The GCF of 4/16 is 4

4 16

= 416÷ 4

÷ 4 = 14

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If a and c are two fractions where c m x a b d d m x b

Then, a c b d

=

=

Method 1

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Method2

If the cross products of two fractions are equal then they are equivalent fractions

a cb d

If ad = bc then a/b = c/d

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Equivalent fractions

In this picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts

But if we cut the cake into smaller congruent pieces, we can see that

1 22 4=

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Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same

Therefore,

1 2 32 4 6

= =

Or we can cut the original cake into 6 congruent pieces,

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then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same

We can generalize this to

whenever n is not 0

Therefore,

1 2 32 4 6

= = = 48

12

= 1 x n2 x n

If you don’t like this, we can cut the original cake into 8 congruent pieces,

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Another example:

and equal?

reduce

reduce

This shows that these two fractions are not the same!

reduce

Are 2440

3042

2440

24 ÷ 240 ÷ 2

= 1220

12 ÷ 420 ÷ 4

= 35

3042

30 ÷ 642 ÷ 6

= 57

24 x 42 = 100840 x 30 = 1200 24 x 42 = 40 x30

Method 2

Method 1

3 55 7

=

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Questions

Write the following in the simplest form

1)18/20 9/10

2)4/16 ¼

3) 36/44 9/11

4)45/105 3/7

5)848/1240 101/155

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Questions

Which of these are equivalent fractions

6) 5/9 and 4/5 no

7) 4/5 and 16/20 yes

8) 9/16 and 5/9 no

Fill >,< or =

9) ½ __4/8 =

10) 4/50__5/75 >

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Lets play a Game

Click here to play a Game

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1) Explain how can you find a fraction in which the numerator and denominator are greater than 100 and they have a common factor of 13.

Multiply the numerator and denominator of a fraction by 13.

(Both numbers must be greater than 7)

Example 8 x 13 104 9 x 13 117

=

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2) Emily had 20 pencils, Sheena had 50 pencils and Ben had 80 pencils. After 4 months, Emily used up 10 pencils, Sheena used up 25 pencils and Ben used up 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of their pencils?

Emily = 10/20 = ½

Sheena = 50/25 = ½

Ben = 40/80 = ½

Yes! Each has used up equal fraction of their pencils

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3) A sitar is a South Asian instrument that has two sets of strings. One type of sitar has 6 top strings and 16 bottom strings. Express the number of top strings as a fraction of total number of strings.

6 6 3(16 +6) 22 11

= =Total number of strings

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Lets review what we have learned in this lesson

Simplest form of Fraction

A fraction is in its simplest form when the GCF of the numerator and the denominator is 1

There is no other common factor except 1

Example 2 17 5 3 31 19

, ,

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Steps to expressing a fraction in its simplest form

1)Find the GCF of the numerator and denominator

2)Divide the numerator and the denominator by the GCF, and write the resulting fraction

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If a and c are two fractions where c m x a b d d m x b

Then, a c b d

=

=

Method 1

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Method2

If the cross products of two fractions are equal then they are equivalent fractions

a cb d

If ad = bc then a/b = c/d

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You had a Great lesson Today!

Be sure to practice what you have learned

Our Lesson

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