Free Pre-Algebra Lesson 5 page 1

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1 Free Pre-Algebra Lesson 5 page 1 Lesson 5 Arithmetic Operations and Precedence This book assumes that you have previously learned the four basic arithmetic operations (addition, subtraction, multiplication, and division), and that you can do these operations with small whole numbers in your head or with pencil and paper, and with larger numbers using a calculator. In Lesson 4, you learned a fifth operation, exponentiation, or raising a number to a power. Below is a reminder about some key points about the basic operations: Addition and Subtraction Addition is an operation that combines groups. Two groups that were formerly separate are now considered together. combined with The answer to an addition problem is called the sum or total. becomes Subtraction is an operation that breaks up groups. A group that was formerly together is separated. remove The answer to a subtraction problem is called the difference. is now Addition and subtraction are related, because a subtraction undoes an addition. We consider them opposite operations. Example: Write the four related addition/subtraction equations using the numbers 5, 3, and = = = = 3 Addition is commutative, meaning that has the same answer as However, subtraction is not commutative, because 8 3 does not have the same answer as 3 8. Multiplication and Division Multiplication is an operation that combines duplicate groups. 3 copies of the group looks like, and combined becomes. The answer to a multiplication problem is called the product. Multiplication can be written with the x symbol, the symbol, or by putting one or both numbers in parentheses right next to the other. Division breaks a large group into duplicate groups. broken into 3 matching groups:, or. The answer to a division problem is called the quotient. Division can be written with the symbol or the / symbol.

2 Free Pre-Algebra Lesson 5 page 2 Multiplication and division are related, because a division undoes a multiplication. We consider them opposite operations. Example: Write the four related multiplication/division equations using the numbers 5, 3, and = = = = 3 Multiplication is commutative, meaning that 5 3 has the same answer as 3 5. However, division is not commutative, because 15 3 does not have the same answer as Division and Zero The related multiplication/division equations explain why division by zero is not allowed in mathematics. Let s try writing the equations that go with the problem 11 0 =?. 0? = 11 11? = 0? 0 = =? The related multiplication problems are impossible! Any number times zero is always zero, and never ever eleven. The problem 11 0 can t have an answer, because there is no number we can fill in for the?. We say division by zero is undefined. This is not a temporary restriction that will be removed in a higher mathematics class it is fundamentally not possible. On the other hand, the problem 0 11 is possible, and the answer is 0, because the related multiplication problems are just fine it is true that 0 11 = 0. The relate 11 0 = = 11 This one is a little odd, however. Is 0 0 really 11? NO, keep reading = = 0 Now 11 is an arbitrary example number, so it s strange that it would be the answer to 0 0. Let s try another number how about 42? 42 0 = = 42 OK, wait a minute. How can the answer to the same problem 0 42 = = 0 be 11 one minute and 42 the next? The answer is that division by zero (ANYTHING 0) is never never allowed. The problem 0 0 is UNDEFINED. Remembering the Zero Division Rule Write the division sign as a fraction bar, like this: not possible Picture this as a stick figure of a person. 0 is the head, and the 11 can be the legs. When 0 is on top, the person is right-side up, and the problem has an answer (the answer is 0). When the person is upside down, the problem has no answer. Some people remember it this way: 0/K and N/0. Or try the problems on your calculator, and see the results: 0/11= 0 11/0=ERROR

3 Free Pre-Algebra Lesson 5 page 3 Ranking the Operations The operations have a relative importance, based on how powerful they are. Exponents and Roots* are the most powerful operations. They have the highest rank. *The operation of raising a number to a power, also has an opposite operation, which is called taking the root. You don t have to worry about that now, we ll cover it later in the course, but you should know that the operations come in pairs of opposites. Multiplication and Division are the middle rank operations. They outrank the addition/subtraction pair, but are lower than the exponent/root pair. Addition and Subtraction are the least powerful operations and have the lowest rank. When more than one operation appears in an expression, higher ranking operations are done before lower ranking operations. This is usually called the order of operations. Sometimes it s necessary to do a lower-ranking operation before one that s ranked higher. This is indicated with parentheses around the numbers and operation. Parentheses override rank, and operations in parentheses* are done first. *Or other grouping symbols such as absolute value, which we ll learn later. You may have learned the acronym PEMDAS to help you remember the order of operations. Court Etiquette When you enter the court, greet each person you encounter. However, you MUST follow correct court precedence in your greetings: First, greet any members of the royal family, in the order you encounter them. Next, greet any knights or dragons, in the order you encounter them. Finally, greet any commoners, in the order you encounter them. But there is one more rule to remember: The gold medallion is a sign of great honor! Greet all medallion wearers first, in order of their rank. Then proceed to those not wearing the medallion. Example: Identify the operations and their rank, then evaluate The first operation you encounter is multiplication, and the second is exponentiation. First do exponentiation, the do multiplication = 3 16 = 48

4 Free Pre-Algebra Lesson 5 page 4 Example: Identify the operations and their rank, then evaluate The first operation you encounter is multiplication, next is addition, and the third is multiplication. First, do the first multiplication, then do the second multiplication, finally do the addition = = = 38 You may have noticed that the evaluation we did above is just like evaluating the perimeter formula P = 2L + 2W, where the length is 6 and the width is 13. When we derived that formula in Lesson 3, we thought about the rectangle as two lengths and two widths, then added to find the perimeter. The order of operations follows that train of thought. First double the length, then double the width, then add the two numbers to find the perimeter. + The next example is like the second version of the formula, P = 2(L + W). In deriving that formula, we thought of the perimeter of the rectangle as two copies of the length plus width. and This would look like (L + W) + (L + W), or more simply 2(L + W). Since addition is ranked lower than multiplication, we require parentheses to override the ranking and do the addition first. Example: Identify the operations and their rank, then evaluate 2(6 + 13). The first operation you encounter is multiplication, next is addition, BUT the addition is in parentheses. First, do the addition in parentheses, then do multiplication. 2(6 + 13) = 2(19) = 38 This is like the second version of the formula, P = 2(L + W). Again the length is 6 and the width is 13. Since this version of the formula only requires two operations instead of three, it is a bit quicker to evaluate. The answer, of course, is the same either way.

5 Free Pre-Algebra Lesson 5 page 5 Example: Evaluate = = = 502 One of the advantages of using a scientific calculator (rather than a four-function calculator) is that the correct order of operations is programmed into the scientific calculator. You can enter the expression in the order it is written, and so long as you do not press = or ENTER until the end, the calculator will do the operations in the correct order. Example: Evaluate using a scientific calculator. Enter x 5 ^ 3 = and read the answer 502. The calculator also has keys for parentheses: Example: Evaluate 4 + 3(17 8) 2 with pencil and paper and then check your answer using a scientific calculator (17 8) 2 = 4 + 3(9) 2 = = = 247 Calculator: Enter x ( 17 8 ) ^ 2 = and read the answer 247. Example: Evaluate 6 3(2) with pencil and paper and then check your answer using a scientific calculator. 6 3(2) The parentheses have no operation inside to evaluate first. It is just another way to write multiplication. You could rewrite the expression to make it clearer: Since multiplication and division have the same rank, we do them in the order encountered = 2 2 = 4 Calculator: Enter 6 3 x 2 = and read the answer 4.

6 Free Pre-Algebra Lesson 5 page 6 Lesson 5: Arithmetic Operations and Precedence Worksheet Name Write each step, then check your answer with a calculator. 1. Evaluate Operation Precedence P arentheses E xponents/roots M ultiplication / D ivision A ddition / S ubtraction 2. Evaluate 2 + 5( ) First do operations within parentheses in order of rank. 3. Evaluate 200 3( ) 4. Evaluate 5 2 (14 + 6)(7 3 2) 5. Evaluate (4 + 6) 3 (8 + 7) 2 6. Evaluate Evaluate (3 2 9) (3 2 8) Is the answer different if you do (4 2)? Evaluate (3 2 8) (3 2 9)

7 Free Pre-Algebra Lesson 5 page 7 Lesson 5: Arithmetic Operations and Precedence Homework 5A Name 1. Estimate the number of marbles in the plastic box, if each compartment contains two layers below similar to the top layer pictured (3 layers of marbles). 2. a. Write 13 8 as a multiplication. b. Evaluate c. Write the number in words. d. Round the number to 1 significant figure. 3. Find the perimeter (P) and area (A) of the rectangle with the given dimensions, length (L) and width (W). P = a. L = 19 feet, W = 12 feet. 4. Find the volume (V) of a box with the given dimensions. a. L = 42 inches, W = 38 inches, H = 18 inches A = b. Subtract 1 foot from each side of rectangle a for the dimensions of rectangle b. L = ft, W = ft P = A = b. L = 90 m, W = 65 m, H = 30 m c. Add 1 m to each side of box b for the dimensions of box c. L = m, W = m, H = m 5. Find the area and perimeter of the figure. Grid is cm a. Write the four related addition/subtraction equations using the numbers 4, 7, and 11. b. Write the four related multiplication/division equations using the numbers 4, 7, and 28. Area Perimeter

8 Free Pre-Algebra Lesson 5 page 8 7. Evaluate each expression. a b. (4 9) 2 c d. (5 + 6) 7 e f. (22 4) 2 + (18 3) 2 g h (5 2) 8. The twenty-foot equivalent unit (often TEU or teu) is an inexact unit of cargo capacity often used to describe the capacity of container ships and container terminals One TEU represents the cargo capacity of a standard intermodal container, 20 feet (6.1 m) long and 8 feet (2.4 m) wide. There is a lack of standardization in regards to height, ranging between 4.25 and 9.5 feet (1.30 and 2.9 m), with the most common height being 8.5 feet (2.6 m). a. Find the volume of a container with L = 20 feet, W = 8 feet, and H = 5 feet. 9. Round the numbers in the presentation using your own judgment. a. The largest container shipping company is APM-Maersk, which has a capacity of 2,031,886 TEU. The total world fleet capacity is 13,108,859 TEU. b. Emma Maersk is the the largest container ship ever built. Her cargo capacity is 14,875 TEU and the cost of building was $144,885,000. b. Find the volume of a container with L = 20 feet, W = 8 feet, and H = 9 feet. 10. Draw a line 2 cm long and a line 3 cm long. Measure each to the nearest inch:

9 Free Pre-Algebra Lesson 5 page 9 Lesson 5: Arithmetic Operations and Precedence Homework 5A Answers 1. Estimate the number of marbles in the plastic box, if each compartment contains two layers below similar to the top layer pictured (3 layers of marbles). 2. a. Write 13 8 as a multiplication b. Evaluate ,730,721 c. Write the number in words. 815 million, 730 thousand, 721 d. Round the number to 1 significant figure. 800,000, Find the perimeter (P) and area (A) of the rectangle with the given dimensions, length (L) and width (W). a. L = 19 feet, W = 12 feet. P = 2(19 ft + 12 ft) = 2(31 ft) = 62 feet 4. Find the volume (V) of a box with the given dimensions. a. L = 42 inches, W = 38 inches, H = 18 inches (42 in)(38 in)(18 in) = 28,728 in 3 28,728 cubic inches A = (19 ft)(12 ft) = 228 ft 2 b. Subtract 1 foot from each side of rectangle a for the dimensions of rectangle b. L = 18 ft, W = 11 ft P = 2(18 ft + 11 ft) = 2(29 ft) = 58 ft A = (18 ft)(11 ft) = 198 ft 2 b. L = 90 m, W = 65 m, H = 30 m (90 m)(65 m)(30 m) = 175,500 m 3 175,500 cubic meters c. Add 1 m to each side of box b for the dimensions of box c. L = 91 m, W = 66 m, H = 31 m (91 m)(66 m)(31 m) = 186,186 m 3 186,186 cubic meters 5. Find the area and perimeter of the figure. Grid is cm 2. Area 16 cm 2 Perimeter 22 cm 6. a. Write the four related addition/subtraction equations using the numbers 4, 7, and = = = = 7 b. Write the four related multiplication/division equations using the numbers 4, 7, and = = = = 7

10 Free Pre-Algebra Lesson 5 page Evaluate each expression. a = 4 81 = 324 b. (4 9) 2 (4 9) 2 = (36) 2 = 1296 c = = 47 d. (5 + 6) 7 (5 + 6) 7 = (11) 7 = 77 e = = = = 24 9 = 15 f. (22 4) 2 + (18 3) 2 (22 4) 2 + (18 3) 2 = (18) 2 + (18-3) 2 = (18) 2 + (15) 2 = (15) 2 = = 549 g = = 20 2 = 40 h (5 2) 10 2 (5 2) = 10 2 (10) = = The twenty-foot equivalent unit (often TEU or teu) is an inexact unit of cargo capacity often used to describe the capacity of container ships and container terminals One TEU represents the cargo capacity of a standard intermodal container, 20 feet (6.1 m) long and 8 feet (2.4 m) wide. There is a lack of standardisation in regards to height, ranging between 4.25 and 9.5 feet (1.30 and 2.9 m), with the most common height being 8.5 feet (2.6 m). a. Find the volume of a container with L = 20 feet, W = 8 feet, and H = 5 feet. (20 ft)(8 ft)(5 ft) = 800 cubic feet a. Find the volume of a container with L = 20 feet, W = 8 feet, and H = 9 feet. (20 ft)(8 ft)(9 ft) = 1440 cubic feet 9. Round the numbers in the presentation using your own judgment. a. The largest container shipping company is APM-Maersk, which has a capacity of 2,031,886 TEU. The total world fleet capacity is 13,108,859 TEU. about 2 million TEU over 13 million TEU b. Emma Maersk is the the largest container ship ever built. Her cargo capacity is 14,875 TEU and the cost of building was $144,885,000. about 15 thousand TEU, $145 million 10. Draw a line 2 cm long and a line 3 cm long. Measure each to the nearest inch: both lines are about 1 inch long

11 Free Pre-Algebra Lesson 5 page 11 Lesson 5: Arithmetic Operations and Precedence Homework 5B Name 1. Estimate the number of cannonballs in the whole stack (not just the ones you can see). Each layer is complete (the stack is not hollow) and it looks the same on the other side. 2. a. Write 12 7 as a multiplication. b. Evaluate c. Write the number in words. d. Round the number to 1 significant figure. 3. Find the perimeter (P) and area (A) of the rectangle with the given dimensions, length (L) and width (W). P = A = a. L = 48 inches W = 36 inches. 4. Find the volume (V) of a box with the given dimensions. a. L = 2 cm, W = 3 cm, H = 1 cm b. L = 20 cm, W = 30 cm, H = 10 cm b. Start with rectangle a (above). Add 1 inch to the length and subtract 1 inch from the width for the dimensions of rectangle b. L = in, W = in P = c. L = 200 cm, W = 300 cm, H = 100 cm A = 5. Find the area and perimeter of the figure. Grid is cm a. Write the four related addition/subtraction equations using the numbers 2, 6, and 8. b. Write the four related multiplication/division equations using the numbers 2, 6, and 12. Area Perimeter

12 Free Pre-Algebra Lesson 5 page Evaluate each expression. a b. (35 3) 3 c d. (6 + 10) 2 e f. (3 5) 4 g h. (85 9) 2 + (12 3) 2 8. Wikipedia: For larger volumes of liquid, one measure commonly used in the media in many countries is the Olympic-size swimming pool. A large Olympic swimming pool with dimensions 50 m 25 m 2 m (approx 164 ft 82 ft 6.6 ft) holds a. Find the volume of the pool in cubic meters. 9. Round the numbers in the presentation using your own judgment. a. a. Every day, the well is leaking almost 4 Olympic-sized swimming pools of oil, up to 105,387 gallons per hour. b. Find the volume of the pool in cubic feet. (Round 6.6 feet to 7 feet, if you like). b. the combined purchases of London based organisations signed up to the Green Procurement Code diverted 72,490 tonnes of waste from landfill. This is the equivalent of 187 Olympic sized swimming pools and represents a saving of 22,643 tonnes of CO 2. c. Use this volume converter to convert the cubic meters to gallons (U.S. liquid). How many gallons of water in an Olympic-sized swimming pool? 10. Draw a line 11 cm long and a line 9 cm long, then measure each to the nearest inch: