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**Adding, Subtracting, Multiplying, and Dividing Fractions 3-5, 3-6, 3-7**

Notes 8 Adding, Subtracting, Multiplying, and Dividing Fractions 3-5, 3-6, 3-7

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Vocabulary Reciprocal: A fraction’s opposite (or flipped upside down). The two numbers should equal 1. Multiplicative Inverse: Same as reciprocal: A fraction’s opposite (or flipped upside down). The two numbers should equal 1.

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**Adding & Subtracting Fractions**

Additional Example 1A: Adding or Subtracting Fractions with Like Denominators Add. Write the answer in simplest form. 5 8 1 8 + 5 8 1 8 5 + 1 Add the numerators and keep the denominator. + = 8 6 8 3 4 = = Simplify.

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**Subtract. Write the answer in simplest form.**

Additional Example 1B: Adding or Subtracting Fractions with Like Denominators Subtract. Write the answer in simplest form. 9 11 4 11 – Subtract the numerators and keep the denominator. 9 11 4 11 9 – 4 – = 11 = 5 11 The answer is in the simplest form.

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**Two Ways to Find a Common Denominator**

To add or subtract fractions with different denominators, you must rewrite the fractions with a common denominator. Two Ways to Find a Common Denominator Find the LCM (least common multiple) of the denominators. Multiply the denominators. The LCM of two denominators is the lowest common denominator (LCD) of the fractions. Helpful Hint

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**Additional Example 2A: Adding and Subtracting Fractions with Unlike Denominators**

Add. Write the answer in simplest form. 5 6 7 8 + 5 6 7 8 5 · 4 6 · 4 7 · 3 8 · 3 + = + The LCM of the denominator is 24. 20 24 21 24 41 24 17 24 Write equivalent fractions. Add = + = = 1 is a reasonable answer. 1 17 24 Estimate = 2

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**Additional Example 2B: Adding and Subtracting Fractions with Unlike Denominators**

Subtract. Write the answer in simplest form. 2 3 3 4 – 2 3 3 4 2 · 4 3 · 3 – = – Multiply the denominators. 3 · 4 4 · 3 = 8 12 – 9 = – 1 12 Write equivalent fractions. Subtract. is a reasonable answer. - 1 12 Estimate 1 – 1 = 0

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**Add. Write the answer in simplest form.**

Additional Example 2C: Adding and Subtracting Fractions with Unlike Denominators Add. Write the answer in simplest form. 2 7 1 3 – + 2 7 + 1 3 = – 2 · 3 7 · 3 + 1 · 7 3 · 7 – Multiply the denominators. = + – 7 21 6 1 21 = Write equivalent fractions. Add Estimate = 0 1 2 is a reasonable answer. 1 21

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**Multiplying Fractions**

To multiply fractions, multiply the numerators to find the product’s numerator. Then multiply the denominators to find the product’s denominator.

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**Additional Example 1A: Multiplying Fractions**

Multiply. Write the answer in simplest form. 3 –12 · 4 3 12 3 –12 · = – Write –12 as a fraction. 4 1 4 3 12 · 3 = – Simplify. 1 · 4 1 9 Multiply numerators. Multiply denominators. = – = –9 1 The product of two positive proper fractions is less than either fraction. Helpful Hint

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**Additional Example 1C: Multiplying Fractions**

Multiply. Write the answer in simplest form. 3 1 – 5 4 3 5 1 4 – The signs are different, so the answer will be negative. 3 5 1 4 – = = 3 20 – Multiply numerators. Multiply denominators.

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**Additional Example 2A: Multiplying Mixed Numbers**

Multiply. Write the answer in simplest form. 2 2 1 5 3 2 2 2 5 Write the mixed number as an improper fraction. 1 = 3 5 3 5 1 2 5 = Simplify. 5 3 1 2 Multiply numerators. Multiply denominators. = 3

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Dividing Fractions Reciprocals can help you divide by fractions. Two numbers are reciprocals or multiplicative inverses if their product is 1. The reciprocal of is 3 because 1 3 1 3 · 3 = 1. Dividing by a number is the same as multiplying by its reciprocal. 2 ÷ 1 3 = 2 · 3 = 6

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**Additional Example 1: Dividing Fractions**

Divide. Write each answer in simplest form. 3 7 2 5 ÷ A. 3 7 2 5 3 7 5 2 2 5 ÷ = Multiply by the reciprocal of . 3 7 5 = 2 15 14 14 1 = or 1 3 8 B. ÷ 12 3 8 1 3 8 Multiply by the reciprocal of 12. ÷ 12 = 12 1 3 8 1 Simplify. = 12 4 1 = 32

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**Additional Example 2: Dividing Mixed Numbers**

Divide. Write each answer in simplest form. 2 3 1 4 A. 5 ÷ 1 Write mixed numbers as improper fractions. 2 3 17 3 5 4 5 1 4 ÷ 1 = ÷ 17 3 4 5 5 4 = Multiply by the reciprocal of . 68 15 = or 4 8 15 3 4 1 2 B. ÷ 2 Write 2 as an improper fraction. 1 2 3 4 1 2 3 4 5 2 ÷ 2 = ÷ 5 2 3 4 2 5 = Multiply by the reciprocal of . 1 3 2 Simplify. = 4 5 2 3 10 =

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