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Multiplying and Dividing Real Numbers CONFIDENTIAL 1 Warm Up Add or subtract: 1) -6 2) 12 3) 8 4)3 1) 12 - 18 2) -6 + 14 3) 3 - (-5) 4) 11 + (-8) CONFIDENTIAL 2 Multiplying and Dividing Real Numbers When you multiply two numbers, the signs of the numbers you are multiplying determine whether the product is positive or negative. This is true for division also. CONFIDENTIAL 3 Multiplying and Dividing Signed Numbers CONFIDENTIAL 4 Multiplying and Dividing Signed Numbers Find the value of each expression. A) -12 · 5 = -60 The product of two numbers with different signs is negative. B) 8x for x = -5 4 8x = 8 x -5 4 = 8(-5) 4 = -40 = -10 4 First substitute - 5 for x. 4 Multiply. The quotient of two numbers with different signs is negative. CONFIDENTIAL 5 Now you try! Find the value of each expression. 1a) 35 ÷ (-5) 1b) -11 (-4) 1a) -7 1b) 44 1c) -42 1c) -6x for x = 7 CONFIDENTIAL 6 Two numbers are reciprocals if their product is 1. A number and its reciprocal are called multiplicative inverses . To divide by a number, you can multiply by its multiplicative inverse. Dividing by a nonzero number is the same as multiplying by the reciprocal of the number. CONFIDENTIAL 7 Dividing by Fractions Divide. To divide by - 8 , multiply by -15 . 15 8 Multiply the numerators and multiply the denominators. - 4 , multiply by -8 have the same sign, 5 15 so the quotient is positive. CONFIDENTIAL 8 Write 4 as a fraction with a denominator of 1. Write 9 1 as an improper fraction. 4 To divide by 37 , multiply by 4 . 4 37 - 4 and 9 1 have different signs, 4 so the quotient is negative. CONFIDENTIAL 9 Now you try! Divide. 2a) 2b) 2c) 2a) 1/12 2b) -1/4 2c) -1/2 CONFIDENTIAL 10 No number can be multiplied by 0 to give a product of 1, so 0 has no reciprocal. Because of this, division by 0 is not possible. We say that division by 0 is undefined. CONFIDENTIAL 11 Properties of Zero CONFIDENTIAL 12 Multiplying and Dividing with Zero Multiply or divide if possible. A) 0 ÷ 16.568 =0 B) 63 7 ÷ 0 8 = undefined C) 1 · 0 =0 Zero is divided by a nonzero number. The quotient of zero and any nonzero number is 0. A number is divided by zero. Division by zero is undefined. A number is multiplied by zero. The product of any number and 0 is 0. CONFIDENTIAL 13 Now you try! Multiply or divide. 3a) 3b) 3c) 3a) 0 3b) undefined 3c) 0 CONFIDENTIAL 14 Recreation Application A hot-air balloon is taken for a 2.5-hour trip. The wind speed (and the speed of the balloon) is 4.75 mi/h. The balloon travels in a straight line. How many miles away from the liftoff site will the balloon land? Find the distance traveled at a rate of 4.75 mi/h for 2.5 hours. To find distance, multiply rate by time. rate times time 4.75 · 2.5 = 11.875 The hot-air balloon will land 11.875 miles from the liftoff site. CONFIDENTIAL 15 Now you try! 4) On another hot-air balloon trip, the wind speed is 5.25 mi/h. The trip is planned for 1.5 hours. The balloon travels in a straight line. How many miles away from the liftoff site will the balloon land? 4) 7.875 mi/h. CONFIDENTIAL 16 Assessment Find the value of each expression. 1) -72 ÷ (-9) 2) 11 (-11) 3) -7.2 ÷ x for x = 3.6 1) 8 2) -121 3) -2 CONFIDENTIAL 17 Divide: 4) 5) 6) 5) 7 6) -1/2 7) -2 CONFIDENTIAL 18 Multiply or divide if possible. 7) 3.8 ÷ 0 8) 0 (-27) 9) 0 ÷ 2 3 7) undefined 8) 0 9) 0 CONFIDENTIAL 19 10) It is estimated that 7 million people saw off-Broadway shows in 2002. Assume that the average price of a ticket was $30. How much money was spent on tickets for off-Broadway shows in 2002? 10) $210 million CONFIDENTIAL 20 Let’s review Multiplying and Dividing Real Numbers When you multiply two numbers, the signs of the numbers you are multiplying determine whether the product is positive or negative. This is true for division also. CONFIDENTIAL 21 Multiplying and Dividing Signed Numbers CONFIDENTIAL 22 Multiplying and Dividing Signed Numbers Find the value of each expression. A) -12 · 5 = -60 The product of two numbers with different signs is negative. B) 8x for x = -5 4 8x = 8 x -5 4 = 8(-5) 4 = -40 = -10 4 First substitute - 5 for x. 4 Multiply. The quotient of two numbers with different signs is negative. CONFIDENTIAL 23 Two numbers are reciprocals if their product is 1. A number and its reciprocal are called multiplicative inverses . To divide by a number, you can multiply by its multiplicative inverse. Dividing by a nonzero number is the same as multiplying by the reciprocal of the number. CONFIDENTIAL 24 Dividing by Fractions Divide. To divide by - 8 , multiply by -15 . 15 8 Multiply the numerators and multiply the denominators. - 4 , multiply by -8 have the same sign, 5 15 so the quotient is positive. CONFIDENTIAL 25 Properties of Zero CONFIDENTIAL 26 Multiplying and Dividing with Zero Multiply or divide if possible. A) 0 ÷ 16.568 =0 B) 63 7 ÷ 0 8 = undefined C) 1 · 0 =0 Zero is divided by a nonzero number. The quotient of zero and any nonzero number is 0. A number is divided by zero. Division by zero is undefined. A number is multiplied by zero. The product of any number and 0 is 0. CONFIDENTIAL 27 Recreation Application A hot-air balloon is taken for a 2.5-hour trip. The wind speed (and the speed of the balloon) is 4.75 mi/h. The balloon travels in a straight line. How many miles away from the liftoff site will the balloon land? Find the distance traveled at a rate of 4.75 mi/h for 2.5 hours. To find distance, multiply rate by time. rate times time 4.75 · 2.5 = 11.875 The hot-air balloon will land 11.875 miles from the liftoff site. CONFIDENTIAL 28 You did a great job today! CONFIDENTIAL 29