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

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.

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.

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.

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

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 1 + 1 = 2

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

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

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

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

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.

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

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

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

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 =