Multiplying Rational Numbers 3-3 Multiplying Rational Numbers Warm Up Problem of the Day Lesson Presentation Pre-Algebra

Multiplying Rational Numbers Pre-Algebra 3-3 Multiplying Rational Numbers Warm Up Write each number as an improper fraction. 1 7 3 7 15 8 2 17 5 1. 2 2. 1 3. 3 3 8 5 2 20 3 3 43 8 4. 6 5. 5 3 8

Problem of the Day The sum of three consecutive integers is 168. What are the three integers? 55, 56, and 57

Learn to multiply fractions, decimals, and mixed numbers.

Kendall invited 36 people to a party Kendall invited 36 people to a party. She needs to triple the recipe for a dip, or multiply the amount of each ingredient by 3. Remember that multiplication by a whole number can be written as repeated addition. 1 4 3 4 + = Repeated addition 3 4 = Multiplication 3 • 1 1 Notice that multiplying a fraction by a whole number is the same as multiplying the whole number by just the numerator of the fraction and keeping the same denominator.

RULES FOR MULTIPLYING TWO RATIONAL NUMBERS If the signs of the factors are the same, the product is positive. (+) • (+) = (+) or (–) • (–) = (+) If the signs of the factors are different, the product is negative. (+) • (–) = (–) or (–) • (+) = (–)

Additional Example 1A: Multiplying a Fraction and an Integer Multiply. Write the answer in simplest form. 6 7 To write as a mixed number, divide: Helpful Hint 12 5 12 5 = 2 R2 2 5 = 2 A. –8 –8 • 6 7 = –48 7 = Multiply –6 6 7 = Simplify

Additional Example 1B: Multiplying a Fraction and an Integer Multiply. Write the answer in simplest form. 1 3 B. 2 5 16 3 = 2 5(3) + 1 3 = 1 3 5 16 3 32 3 = Multiply 10 2 3 = Simplify

5 8 A. –3 –3 • 5 8 = –15 8 = –1 7 8 = Multiply Simplify Try This: Example 1A Multiply. Write the answer in simplest form. 5 8 A. –3 –3 • 5 8 = –15 8 = Multiply –1 7 8 = Simplify

Try This: Example 1B Multiply. Write the answer in simplest form. 2 5 B. 4 9 47 5 = 4 9(5) + 2 5 = 2 5 9 47 5 188 5 = Multiply 37 3 5 = Simplify

3 5 2 3 A model of is shown. Notice that to multiply fractions, you multiply the numerators and multiply the denominators. • 35 23 6 15 • = If you place the first rectangle on top of the second, the number of green squares represents the numerator, and the number of total squares represents the denominator.

To simplify the product, rearrange the six green squares into the first two columns. You can see that this is . 2 5 2 5 = 6 15 A fraction is in lowest terms, or simplest form, when the numerator and denominator have no common factors. Helpful Hint

1 8 6 7 1(6) 8(7) = 1(6) 8(7) = 3 28 = Multiply numerators. Additional Example 2A: Multiplying Fractions Multiply. Write the answer in simplest form. 1 8 6 7 A. = 1(6) 8(7) Multiply numerators. Multiply denominators. 3 4 1(6) 8(7) = Look for common factors: 2. 3 28 = Simplest form

Additional Example 2B: Multiplying Fractions Multiply. Write the answer in simplest form. 2 3 9 2 B. – = –2(9) 3(2) Multiply numerators. Multiply denominators. 1 –1 3 1 –2(9) 3(2) = Look for common factors: 2, 3. –3 = Simplest form

Additional Example 2C: Multiplying Fractions Multiply. Write the answer in simplest form. 3 7 1 2 C. 4 1 2 3 7 4 = 31 1 7 2 Write as an improper fraction. = 31(1) 7(2) Multiply numerators. Multiply denominators. = or 2 31 14 3 14 31 ÷ 14 = 2 R3

3 5 5 8 A. 3(5) 5(8) = 3(5) 5(8) = 3 8 = Multiply numerators. Try This: Example 2A Multiply. Write the answer in simplest form. 3 5 5 8 A. = 3(5) 5(8) Multiply numerators. Multiply denominators. 1 3(5) 5(8) = Look for common factors: 5. 3 8 = Simplest form

7 8 4 7 B. – = –7(4) 8(7) –1 –7(4) 8(7) = 1 2 = – Multiply numerators. Try This: Example 2B Multiply. Write the answer in simplest form. 7 8 4 7 B. – = –7(4) 8(7) Multiply numerators. Multiply denominators. 1 –1 1 2 –7(4) 8(7) = Look for common factors: 4, 7. 1 2 = – Simplest form

C. 2 2 = = = or 2 Write as an improper fraction. Multiply numerators. Try This: Example 2C Multiply. Write the answer in simplest form. 3 5 7 9 C. 2 7 9 3 5 2 = 13 7 5 9 Write as an improper fraction. = 13(7) 5(9) Multiply numerators. Multiply denominators. = or 2 91 45 1 45 91 ÷ 45 = 2 R 1

Additional Example 3: Multiplying Decimals 2(–0.51) Product is negative with 2 decimal places. 2 • (–0.51) = –1.02 B. (–0.4)(–3.75) Product is positive with 3 decimal places. (–0.4) • (–3.75) = 1.500 = 1.5 00 You can drop the zeros after the decimal point.

A. 3.1 (0.28) 3.1 • (0.28) = 0.868 B. Multiply. (–0.4)(–2.53) Try This: Example 3 Multiply. A. 3.1 (0.28) 3.1 • (0.28) = 0.868 Product is positive with 3 decimal places. B. (–0.4)(–2.53) Product is positive with 3 decimal places. (–0.4) • (–2.53) = 1.012

Additional Example 4A: Evaluating Expressions with Rational Numbers Evaluate –3 x for the value of x. 1 8 A. x = 5 –3 x 18 –3 (5) 18 = Substitute 5 for x. –25 8 = (5) Write as an improper fraction. –125 8 = = –15 5 8 –125 ÷ 8 = –15 R5

2 B. x = –3 x 7 = –3 = = = – Evaluate –3 x for the value of x. Additional Example 4B: Evaluating Expressions with Rational Numbers Continued Evaluate –3 x for the value of x. 1 8 2 B. x = –3 x 18 7 –3 = 18 27 Substitute for x. 2 7 27 –25 8 = Write as an improper fraction. –25 • 2 8 • 7 = 1 Look for common factors: 2. 4 = – 25 28

A. y = –5 y –5 = = = = – , or – 4 Evaluate –5 y for the value of y. Try This: Example 4A Evaluate –5 y for the value of y. 3 5 67 A. y = –5 y 35 Substitute for x. 6 7 35 –5 = 67 Write as an improper fraction. –28 5 = 67 –4 –28 • 6 5 • 7 = Look for common factors: 7. 1 = – 24 5 , or – 4 45

B. y = 3 –5 y –5 (3) = = (3) = = –16 Evaluate –5 y for the value of y. Try This: Example 4B Evaluate –5 y for the value of y. 3 5 B. y = 3 –5 y 35 35 –5 (3) = Substitute 3 for y. –28 5 = (3) Write as an improper fraction. –84 5 = = –16 4 5 –84 ÷ 5 = –16 R4

Lesson Quiz: Part 1 Multiply. 1 7 1 2 7 1. 9 2 3 5 8 2. – – 5 12 3. –0.47(2.2) –1.034 1 2 45 4. Evaluate 2 (x) for x = . 2

Lesson Quiz: Part 2 5. Teri is shopping for new shoes. Her mom has agreed to pay half the cost (and all the sales tax). The shoes that Teri likes are normally $30 a pair but are on sale for off. How much money does Teri need to buy the shoes? 1 3 $10