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2.2 Introduction to Fractions 1 A fraction is a number that can express values that are not necessarily whole numbers. They are used to represent values that come between the whole numbers. By using a fraction, the portion can be described exactly as to its size in comparison to the whole. For example: 4/5 implies that if the whole were divided into five equal parts that 4/5 would comprise of four of the five parts. The top number of the fraction is called the called the numerator and the bottom number is called the denominator. Numerator & Denominator Next Slide Therefore the top number of the fraction is representative of the portion to be described or expressed and the bottom number is representative of the number of divisions, sections, or portions in which the whole has been divided. For example:

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2.2 Introduction to Fractions 2 Proper Fractions A proper fraction is a fraction whose numerator is smaller than its denominator. All proper fractions are less than 1. Improper Fractions An improper fraction is a fraction whose numerator is larger than or equal its denominator. Mixed Numbers A mixed number is a number written as a whole number and a fraction. It represents the sum of the whole number and the fraction. All mixed numbers are greater than 1. Next Slide

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2.2 Introduction to Fractions 3 Fractions Equaling "1" Any fraction with a non-zero denominator that is equal to its numerator is equal to one. If portion is the same as the number of divisions, then the amount described is the whole. For example: Fractions Equaling "0" Any fraction with a numerator of zero and a non-zero denominator is equal to zero For example: Undefined Fractions Since division by zero is undefined, any fraction with a denominator of zero is undefined. For example: Next Slide

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2.2 Introduction to Fractions 4 Converting an Improper Fraction to a Mixed Number To change an improper fraction to a mixed number: Divide the denominator into the numerator. The quotient is the whole number. The fraction part of the mixed number as the remainder in the numerator and the original denominator as the denominator. Example 1. Convert 9/4 to a mixed number: whole number denominator numerator Your Turn Problem #1 Convert 17/5 to a mixed number.

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2.2 Introduction to Fractions 5 Converting a Mixed Number to an Improper Fraction To change a mixed number into an improper fraction, multiply the denominator by the whole number and then add that product to the numerator. This sum becomes the numerator of the improper fraction and the denominator of the improper fraction is the same as the denominator from the mixed number. 1.Multiply whole number and denominator. 2.Add numerator. 3.Write result over original denominator. Your Turn Problem #2

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2.2 Introduction to Fractions 6 Equivalent Fractions Recall the Multiplicative Identity (also called the multiplication property of 1): a 1 = a. When we multiply a number by 1, we get the same number. Example: 5 1 = 5 Also, any fraction with the same numerator and denominator is equal to 1.,, If we take a fraction, such as , and multiply it by 1, we get the same number. Next Slide

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2.2 Introduction to Fractions 7 A fraction written in lowest terms is when the fraction is expressed using the lowest numerator and denominator possible. Reducing Fractions (Simplifying) represents that fraction written in lowest terms because it is impossible to find an equivalent fraction that represents that value using a smaller numerator or denominator. Reducing is the process of converting a fraction to an equivalent fraction in lowest terms. It is done by dividing the numerator and denominator by the largest number that divides evenly into both numbers. The largest number that divides evenly into a set of numbers is called the gcf, (greatest common factor). Actually, it is not absolutely necessary to use the gcf. It will just take more steps if we dont use the gcf. Next Slide

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2.2 Introduction to Fractions 8 Step 1. Find the gcf for both numerator and denominator. Step 2. Divide both numerator and denominator by the gcf. So 6 is the largest number that divides evenly into both numerator and denominator. Ideally, we want to divide both numerator and denominator by the gcf, but any common factor will work. The question you are asking yourself is what number divides evenly (goes into) both numbers? Some may think of 6. Others may think of 2 since the numbers are both even. Or some may think of 3. Any will work. However, if we use a number that is not the gcf, we need to continue the process until the numerator and denominator have no factors in common. Alternative method: Next Slide

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2.2 Introduction to Fractions 9 Your Turn Problem #3 Simplify the following: Step 1. Find the gcf for both numerator and denominator. Step 2. Divide both numerator and denominator by the gcf. So 15 is the largest number that divides evenly into both numerator and denominator. Alternative Method: What number divides evenly into both 30 and 75? Since the numbers end in a 0 and a 5, they are divisible by 5. Next, 6 and 15 are divisible by what number? Answer: 3 Answer

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2.2 Introduction to Fractions 10 Prime Factorization Method of Reducing This method can be more efficient when the fraction contain large numbers. This method requires writing the prime factorization of both numerator and denominator. Since any number divided by itself equals one, we will line out any like factors that are contained in both numerator and denominator. Your Turn Problem #4 Step 1. Find the prime factorization for both numerator and denominator. Step 2. Rewrite the fraction using the prime factorizations. Procedure for Reducing using Prime Factorization Step 1: Find the prime factorization of the numerator and denominator. Step 2: Rewrite the fraction using the prime factorizations. Step 3: Line out, one for one, like factors that are contained both in the numerator and denominator. Step 4: Multiply the remaining not lined-out factors together. Step 3. Line out like factors. Step 4. Multiply the remaining factors. Answer: The End B.R. 5-24-08