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Next Fractions: Simplification, Multiplication & Division Lesson 1e Fractions Next 3 12 , , 5 4 Numerals such as and called fractions. Fractions have two parts: A Numerator & A Denominator a b 2 8 are Numerator Denominator Since you cannot divide by zero, the denominator of a fraction can never equal zero. However, a numerator can equal zero. Fractions Next Any fraction with the same nonzero numerator and denominator equals 1. a 1 a 22 1 22 223 1 223 Fractions Next Any fraction with a denominator of 1 names the same number as its numerator. 22 22 a 1 1 a 0 0 1 Product Rule for Fractions Next Multiply the numerators. Multiply the denominators “•” is a symbol for multiplication (X). a c a c ac b d b d bd 2 5 2 5 10 3 7 3 7 21 Next Examples: Multiplying Fractions a c aa cc aac c Multiply the numerators. b d bb dd bbd d Multiply the denominators “•” is a symbol for multiplication (X). Multiply: Click here to check your answers! 3 1 1. 4 5 2 4 2. 3 5 3 7 3. 5 8 1 3 4. 2 8 5 1 5. 6 3 9 3 6. 10 7 ac bd Next Examples: Multiplying Fractions a c aa cc aac c Multiply the numerators. b d bb dd bbd d Multiply the denominators “•” is a symbol for multiplication (X). ac bd Multiply: 3 1 3 1. 4 5 20 2 4 8 2. 3 5 15 1 3 3 4. 2 8 16 5 1 5 5. 6 3 18 3 7 21 3. 5 8 40 9 3 27 6. 10 7 70 Writing Fractions in Lowest Terms 3 6 2 4 Next A fraction is in lowest terms when the numerator and denominator have no common prime factors. In each of the fractions below, what common prime factor do each the numerator and denominator share? The common factor is 3 The common factor is 2 5 10 9 12 The common factor is 5 The common factor is 3 Writing Fractions in Lowest Terms: Example 1 Next The first step in writing fractions in lowest terms is to write the numerator and denominator as a product of prime factors. 7 1 1 Write in lowest terms: Solution: Write the prime factorization. 42 Write as a product of two fractions. Any number multiplied by one is equal to itself. 6 6 7 7 42 2 3 7 7 1 7 23 1 1 1 6 6 Writing Fractions in Lowest Terms: Example 2 Next 12 2 Write in lowest terms: 30 5 Solution: Write the prime factorization. 2 5 12 2 2 3 30 2 3 5 1 Divide the numerator and the denominator by common factors. Write in lowest terms. 1 223 235 1 2 5 1 2 5 Practice Next Click here to check your answers! Write the following in lowest terms: 5 1. 20 9 2. 15 16 3. 36 Practice Next Write the following in lowest terms: 5 5 1. 20 2 2 5 1 9 33 2. 15 35 1 16 2 2 2 2 3. 36 2 2 3 3 1 1 5 225 33 35 1 2222 2233 1 4 3 5 4 9 1 1 1 Next More on Multiplying Fractions Sometimes after multiplying fractions your answer needs to be reduced to lowest terms. For example: 3 4 33 4 12 12 8 5 88 5 40 40 1 12 3 4 3 40 28 5 10 Next More on Multiplying Fractions To reduce fractions, you can often use a shortcut when multiplying. If possible reduce by dividing a denominator and a numerator by the same number(a common factor). Then multiply. For Example: 1 5 and 10 have a common factor of 5. 7 5 7 8 102 16 Next Examples: Multiplying Fractions Multiply and then write your answers in lowest terms. Click here to check your answers! 2 14 1. 7 15 6 5 2. 11 12 3 5 3. 10 6 6 14 4. 7 15 7 5. 36 8 5 3 4 6. 12 20 7 Next Examples: Multiplying Fractions Multiply and then write your answers in lowest terms. 2 2 14 4 1. 7 15 15 1 2 2 1 5 6 14 4 4. 7 15 5 1 6 5 5 2. 11 12 22 2 9 1 1 2 2 3 5 1 3. 10 6 4 1 1 7 36 63 1 15 3 4 1 5. 31 6. 8 1 2 2 412 204 7 28 2 1 Multiplying Mixed Numbers Next In order to multiply mixed numbers you must change all mixed numbers to fractions. For Example: 3 11 ? 2 4 2• 4 = 8 4 Multiply the 4 & 5 Now add the 8 to the numerator 3. Now add the 20 to the numerator 1. Multiply the 2 & 4 8 + 3 = 11 1 21 ? 4 4 4 • 5 = 20 5 20 + 1 = 21 Multiplying Mixed Numbers Next Now that we can change mixed numbers into fractions, we can multiply mixed numbers. First, we change the mixed number to a fraction. Next, we multiply the fractions as we have learned previously. Example 1: Example 2: 2 3 11 4 4 3 1 2 ? 4 6 11 1 11 4 6 24 1 1 2 5 3 6 1 7 2 3 3 ? 1 31 5 6 6 7 31 217 1 12 3 6 18 18 Next Examples: Multiplying Mixed Numbers Multiply and then write your answers in lowest terms. Click here to check your answers! 1 4 1. 9 2 7 2 5 2. 1 5 8 1 4. 5 4 3 3 1 5. 3 1 5 6 1 3 3. 3 7 4 4 4 6. 2 3 5 7 Next Examples: Multiplying Mixed Numbers Multiply and then write your answers in lowest terms. 2 1 4 19 4 1. 9 2 7 12 7 38 3 5 7 7 16 4 1 4. 5 4 3 1 3 64 1 21 3 3 1 2 5 7 5 2. 1 5 8 15 8 7 8 3 3 1 18 7 5. 3 1 5 6 5 61 21 1 4 5 5 11 1 3 22 3 3. 3 7 4 7 42 33 14 2 5 4 4 14 25 6. 2 3 5 7 1 5 71 10 10 1 Dividing Fractions Next Now that you can multiply fractions, you can learn to divide fractions. First, we change any mixed numbers to fractions. The second step is to write the reciprocal of the fraction that follows the division symbol (That fraction is called the divisor). Lastly, we change the division to multiplication, and multiply the fractions as we have learned previously. Example 1: 3 2 11 22 2 ? 4 5 4 55 7 11 52 55 6 8 8 4 25 Example 2: 4 4 8 8 ? 5 5 1 1 1 4 18 5 812 10 Next Examples: Dividing Fractions Multiply and then write your answers in lowest terms. Click here to check your answers! 5 2 1. 6 3 10 5 2. 17 8 7 3. 10 10 Next Examples: Dividing Fractions Multiply and then write your answers in lowest terms. 1 5 2 5 3 1. 6 3 26 2 5 4 2 10 5 10 8 2. 17 8 17 51 16 17 7 10 10 3. 10 10 1 7 100 2 14 7 7 BACK Definition: Reciprocal Two numbers whose product is 1 are reciprocals. The reciprocal of is . X = 1 The reciprocal of 7 is . 7 X = 1 The reciprocal of 2 is 3/7 2 = X 3/7 = 1