**Fractions and Decimals**

## Lesson 3 Reducing Fractions

**Welcome to Lesson 3 Reducing Fractions**

Objectives:In this lesson you will revise what are Prime Numbers and Composite numbers.

Objectives:

You will also learn how to find the Greatest Common Factor (GCF)

You will know how to reduce the Fraction.

**Prime Numbers:**A prime number is a whole number that only has two factors which are itself and one.

**Composite Numbers:**A composite number has factors in addition to one and itself.

The numbers 0 and 1 are neither prime nor composite.

All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

All numbers that end in five are divisible by five. Therefore all numbers that end with five and are greater than five are composite numbers.

The prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

**Factors:**

A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number. All numbers can be divided by themselves to produce the number one. Therefore, we normally ignore one and the number itself as useful factors.

The number fifteen can be divided into two factors which are three and five.

The number twelve could be divided into two factors which are 6 and 2. Six could be divided into two further factors of 2 and 3. Therefore the factors of twelve are 2, 2, and 3.

If twelve was first divided into the factors 3 and 4, the four could be divided into factors of 2 and 2. Therefore the factors of twelve are still 2, 2, and 3.

There are several clues to help determine factors.

· Any even number has a factor of two

· Any number ending in 5 has a factor of five

· Any number above 0 that ends with 0 (such as 10, 30, 1200) has factors of two and five.

To determine factors see if one of the above rules applies (ends in 5, 0 or an even number). If none of the rules apply, there still may be factors of 3 or 7 or some other number.

**Greatest Common Factor:**

The Greatest Common Factor (GCF) is the largest number that is a common factor of two or more numbers.

How to find the greatest common factor:

· Determine if there is a common factor of the numbers. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.

· Divide all of the numbers by this common factor.

· Repeat this process with the resulting numbers until there are no more common factors.

· Multiply all of the common factors together to find the Greatest Common Factor

Example:

1. To find the GCF of 4 and 14

Factors of 4 = 2 * 2

Factors of 14 = 2 * 7

Common Factor = 2

GCF = 2

2. To find GCF of 8 and 28

Factors of 8 = 2 * 2 * 2

Factors of 28 = 2 * 2 * 7

Common Factors = 2 * 2

GCF = 4

**Reducing Fractions**

Fractions may have numerators and denominators that are composite numbers (numbers that has more factors than 1 and itself).

How to simplify a fraction:

· Find a common factor of the numerator and denominator. A common factor is a number that will divide into both numbers evenly. Two is a common factor of 4 and 14.

· Divide both the numerator and denominator by the common factor.

· Repeat this process until there are no more common factors.

· The fraction is simplified when no more common factors exist.

Another method to simplify a fraction

· Find the Greatest Common Factor (GCF) of the numerator and denominator

· Divide the numerator and the denominator by the GCF

Example:

Reduce/Simplify following Fraction.

1. 4/14

Finding Common factor of Numerator and Denominator: = 2 * 2/ 2* 7

Divide Numerator and Denominator by the common factor: = 2/7

Reduced/ Simplified Fraction: 2/7

To find the factors of any number, you can make use of the rules of Divisibility.

**Rules of divisibility:**

· Numbers are divisible by 2 if the ones digit is evenly divisible by 2. This means that even numbers are divisible by 2.

· Numbers are divisible by 3 if the sum of all the individual digits is evenly divisible by 3. For example, the sum of the digits for the number 3627 is 18, which is evenly divisible by 3 so the number 3627 is evenly divisible by 3.

· Whole numbers are divisible by 4 if the number formed by the last two individual digits is evenly divisible by 4. For example, the number formed by the last two digits of the number 3628 is 28, which is evenly divisible by 4 so the number 3628 is evenly divisible by 4.

· Numbers are evenly divisible by 5 if the last digit of the number is 0 or 5.

· Numbers are evenly divisible by 6 if they are evenly divisible by both 2 AND 3. Even numbers are always evenly divisible by 2. Numbers are evenly divisible by 3 if the sum of all the individual digits is evenly divisible by 3. For example, the sum of the digits for the number 3627 is 18, which is evenly divisible by 3 but 3627 is an odd number so the number 3627 is not evenly divisible by 6.

· To determine if a number is divisible by 7, take the last digit off the number, double it and subtract the doubled number from the remaining number. If the result is evenly divisible by 7 (e.g. 14, 7, 0, -7, etc.), then the number is divisible by seven. This may need to be repeated several times.

· Numbers are divisible by 8 if the number formed by the last three individual digits is evenly divisible by 8. For example, the last three digits of the number 3624 is 624, which is evenly divisible by 8 so 3624 is evenly divisible by 8.

· Numbers are divisible by 9 if the sum of all the individual digits is evenly divisible by 9. For example, the last sum of the digits of the number 3627 is 18, which is evenly divisible by 9 so 3627 is evenly divisible by 9.

· A number is divisible by 10 only if the last digit is a 0.

## Video: Greatest Common Factor

## Practice Problems

Remember you will simply solve these problems on your own; It is

Reduce following Fractions:

1. 9/15

2. 25/100

3. 26/66

**not**needed to email the answers to the instructor.Reduce following Fractions:

1. 9/15

2. 25/100

3. 26/66

## Assignment 3

**Please email the answers to this Assignment 3 to the email address provided at the bottom of the page. Do not forget to write the Assignment number.**

1. Find the GCF of following numbers

1. 30 and 15

2. 15 and 9

3. 24 and 60

4. 24 and 16

2. Reduce/Simplify following Fractions:

1. 8/16

2. 15/27

3. 18/54

## Assessment 3

**Please email the answers to this Assessment 3 to the email address provided at the bottom of the page. Do not forget to write the Assessment number.**

Reduce following Fractions

1. 4/6

2. 5/15

3. 12/18

4. 24/36

5. 30/45

## Objectives of next lesson

Brief Objective of next Lesson: In the next lesson we will study the Mathematical Operations on Fractions.

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**Please email your assignments and assessments to: dskarkha@ncsu.edu**