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Session 1

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  1. Session 1 Whole Numbers

  2. Session 1 – Objectives After studying this unit you should be able to: • express the digit place values of whole numbers. • write whole numbers in expanded form. • round answers to a given place value. • arrange, add, subtract, multiply, and divide whole numbers. • solve practical problems using addition, subtraction, multiplication, and division of whole numbers. • solve problems by combining addition, subtraction, multiplication, and division. • solve arithmetic expressions by applying the proper order of operations.

  3. Session 1 1–1 Place Value 2 1–2 Expanding Whole Numbers 3 1–3 Estimating (Approximating) 4 1–4 Addition of Whole Numbers 5 1–5 Subtraction of Whole Numbers 7 1–6 Problem Solving—Word Problem Practical Applications 8 1–7 Adding and Subtracting Whole Numbers in Practical Applications 9 1–8 Multiplication of Whole Numbers 11 1–9 Division of Whole Numbers 15 1–10 Multiplying and Dividing Whole Numbers in Practical Applications 18 1–11 Combined Operations of Whole Numbers 20 1–12 Combined Operations of Whole Numbers in Practical Applications 22 UNIT EXERCISE AND PROBLEM REVIEW 25

  4. PLACE VALUE • The value of any digit depends on its place value • Place value is based on multiples of 10 as follows: units HUNDRED THOUSANDS TEN THOUSANDS THOUSANDS MILLIONS HUNDREDS TENS ONES 2 , 6 7 8 , 9 3 2

  5. EXERCISE 1-2 Page 3 tens hundreds ones 100 thousands tens 10 millions 100 thousands 10 thousands

  6. EXPANDED FORM • Place value held by each digit can be emphasized by writing the number in expanded form 382 can be written in expanded form as: 3 hundreds + 8 tens + 2 ones or

  7. ESTIMATING Used when an exact mathematical answer is not required A rough calculation is called estimating or approximating Mistakes can often be avoided when estimating is done before the actual calculation When estimating, exact values are rounded

  8. ROUNDING • Used to make estimates • Rounding Rules: • Determine place value to which the number is to be rounded • Look at the digit immediately to its right. • If digit to right is less than 5, replace that digit and all following digits with zeros • If digit to right is 5 or more, add 1 to the digit in the place to which you are rounding. Replace all following digits with zeros

  9. ROUNDING EXAMPLES • Round 612 to the nearest hundred The hundred place value is where the 6 is at Look to the right. The 6 is followed by a 1. Since 1 is less than 5, the 6 remains unchanged and the numbers to the right of 6 are replaced with zeros • Ans: 600

  10. ROUNDING EXAMPLES • Round 175,890 to the nearest ten thousand. The ten thousand place value is where the 7 is at. Look to the right. The 7 is followed by a 5. Since the number is greater than or equal to 5, change the 7 to an 8 and replace the rest of the numbers with zeros • Ans: 180,000

  11. EXERCISE 1-3A Page 4 2. 540 to the nearest hundred 500 4. 2587 to the nearest thousand 3,000 6. 32,403 to the nearest ten thousand 32,000 8. 53,738 to the nearest ten thousand 54,000 10. 949,500 to the nearest hundred thousand 900,000

  12. ROUNDING TO THE EVEN • Many technical trades use a process of rounding to even • Reduces bias when several numbers are added • Referred to as the “Odd/Even Rounding Rule”.

  13. ROUNDING TO THE EVEN • Rounding Rules: • Determine place value to which the number is to be rounded • This is the same as the previous method • The only difference is if the digit to the right is 5 followed by all zeros or nothing at all. • Increase the digits at the place value by 1 if it is an odd number (1, 3, 5, 7, or 9) • Do not change the digits place if it is an even number (0, 2, 4, 6, 8)

  14. ROUNDING TO EVENS EXAMPLES • Round 4,250 to the nearest hundred 2 is in the hundreds place so look at 5. 5 is followed by zeros and 2 is an even number so drop the 5 and leave the 2 • Ans: 4,200

  15. ROUNDING TO EVENS EXAMPLES • Round 673,500 to the nearest thousand 3 is in the thousands place so look at the 5. 5 is followed by nothing but zeros. Since and 3 is odd, change the 3 to a 4 and the 5 is replaced with a zero. • Ans: 674,000

  16. EXERCISE 1-3B Page 6 2. 785 to the nearest ten 780 4. 5450 to the nearest hundred 5400 6. 24,520 to the nearest thousand 25,000 8. 26,455 to the nearest ten 26,460

  17. ADDITION OF WHOLE NUMBERS • The result of adding numbers is called the sum • The plus sign (+) indicates addition • Numbers can be added in any order

  18. PROPERTIES OF ADDITION • Commutative property of addition: • Numbers can be added in any order • Example: 2 + 4 + 3 = 3 + 4 + 2 • Associative property of addition: • Numbers can be grouped in any way and the sum is the same • Example: (2 + 4) + 3 = 2 + (4 + 3)

  19. PROCEDURE FOR ADDING WHOLE NUMBERS • Example: Add 763 + 619 • Align numbers to be added as shown; line up digits that hold the same place value • Add digits holding the same place value, starting on the right, 9 + 3 = 12 • Write 2 in the units place value and carry the one

  20. PROCEDURE FOR ADDING WHOLE NUMBERS • Continue adding from right to left • Therefore, 763 + 619 = 1,382

  21. EXERCISE 1-4 Page 7 12. 953 + 38 = 991 14. 896 + 675 + 33 = 1604 16. 3653 + 8063 + 47 = 11,763 18. 9734 + 10,050 + 91,613 = 111,397 20. 17,392 + 2085 + 1670 + 13= 21,160 22. 18,768 + 3023 + 7,787,030 + 544 = 7,809,365

  22. SUBTRACTION OF WHOLE NUMBERS • Subtraction is the operation which determines the difference between two quantities • It is the inverse or opposite of addition • The minus sign (–) indicates subtraction

  23. SUBTRACTION OF WHOLE NUMBERS • The quantity subtracted is called the subtrahend • The quantity from which the subtrahend is subtracted is called the minuend • The result is the difference

  24. PROCEDURE FOR SUBTRACTING WHOLE NUMBERS • Example: Subtract 917 – 523 • Align digits that hold the same place value • Start at the right and work left: • 7 – 3 = 4

  25. PROCEDURE FOR SUBTRACTING WHOLE NUMBERS • Since 2 cannot be subtracted from 1, you need to borrow from 9 (making it 8) and add 10 to 1 (making it 11) • Now, 11 – 2 = 9; 8 – 5 = 3; Therefore, • 917 – 523 = 394

  26. EXERCISE 1-5 Page 9 12. 98 - 29 = 69 14. 312 - 97 = 215 16. 1570 - 988 = 582 18. 49,406 - 5498 = 43,908 20. 707,353 – 533,974 = 173,379

  27. EXERCISE 1-7 Pages 10-12 2. A sheet metal contractor has 124 feet of band iron in stock. An additional 460 feet are purchased. On June 2, 225 feet are used. On June 4, 197 feet are used. How many feet of band iron are left after June 4? 124 + 460 = 584 584 - 225 = 359 359 - 197 = 162 162 feet of band iron is left after June 4

  28. EXERCISE 1-7 Pages 10-12 4. Five stamping machines in a manufacturing plant produce the same product. Each machine has a counter that records the number of parts produced. The table below shows the counter readings for the beginning and end of one week’s production. a. How many parts are produced during the week by each machine? b. What is the total weekly production? M1 48,951 – 17,855 = 31,096 M2 42,007 – 13,935 = 28,072 M3 37,881 – 7,536 = 30,345 31,096 + 28,072 + 30,345 + 33,367 + 28,599 M4 72,302 – 38,935 = 33,367 M5 29,275 – 676 = 28,599 151,479

  29. EXERCISE 1-7 Pages 10-12 6. An electrical contractor has 5000 meters of BX cable in stock at the beginning of a wiring job. At different times during the job, electricians remove the following lengths from stock: 325 meters, 580 meters, 260 meters, and 65 meters. When the job is completed, 135 meters are left over and are returned to stock. How many meters are now in stock? 325 + 580 + 260 + 65 = 1230 5000 - 1230 = 3770 3770 + 135 = 3905

  30. EXERCISE 1-7 Pages 10-12 8. An electrical contractor uses the following amounts of cable during the first three months of the year: January: 8320 feet; February: 7650 feet; and March: 4972 feet. b. Find the actual amount of cable used during the first three months. 8320 + 7650 + 4972 = 20,942

  31. EXERCISE 1-7 Pages 10-12 10. The table lists various kinds of flour ordered and received by a commercial baker. a. Is the total amount of flour received greater or less than the total amount ordered? Ordered 3875 + 2000 + 825 + 180 + 210 + 85 = 7,175 lb Received 3650 + 2670 + 910 + 75 + 165 + 85 = 7,555 lb b. How many pounds greater or less? 7,555 – 7,175 = 380 pounds greater

  32. EXERCISE 1-7 Pages 10-12 12. A small business complex is shown in the diagram in Figure 1-6. To provide parking space, a paving contractor is hired to pave the area not occupied by buildings or covered by landscaping areas. The entire parcel contains 41,680 square feet. How many square feet of land are to be paved? 2450 + 11,875 + 5250 + 2800 + 3050 = 25,425 square feet 41,680 – 25,425 = 16,255 square feet of land to be paved

  33. EXERCISE 1-7 Pages 10-12 14. On a particular job, the contractor’s expenses were $794 for materials, $537 in carpenter labor, and $486 for taxes and insurance. The contractor is paid $1974. a. What is the total of the expenses? $794 + $537 + $486 = $1817 b. If the profit is the difference between the amount paid and the expenses, what was the profit for this job? $1974 - $1817 = $157 profit

  34. MULTIPLICATION OF WHOLE NUMBERS • Multiplication is a short method of adding equal amounts • The times sign (×) is used to indicate multiplication • 5 x 7 = • The times dot () is used to indicate multiplication • 5 7 = • The Parenthesis (..) is used to indicate multiplication • 5(7) = • (5)7 = • (5)(7) =

  35. MULTIPLICATION OF WHOLE NUMBERS • The number to be multiplied is called the multiplicand • The number by which the multiplicand is multiplied is called the multiplier • Factors are the numbers used in multiplying • The result of multiplying is called the product

  36. PROPERTIES OF MULTIPLICATION • Commutative property of multiplication: • Numbers can be multiplied in any order • Example: 2 x 4 x 3 = 3 x 4 x 2 • Associative property of multiplication: • Numbers can be grouped in any way and the product is the same • Example: (2 x 4) x 3 = 2 x (4 x 3)

  37. Example Multiply 386 × 7 • First, multiply 7 by the units; 7 ×6 = 42. Write 2 in the units position and 4 above tens position • Multiply the 7 × 8 = 56. Add the 4 tens from the product of the units; 56 + 4 = 60. Write the 0 in the tens position and 6 above tens position . • Multiply 7 × 3 = 21. Add the 6 hundreds from the • product of the tens; 21 + 6 = 27 • Write the 7 in the hundreds position and the 2 in • the thousands position • Therefore, 386 × 7 = 2,702

  38. EXERCISE 1-8 Pages 15-16 12. 775 x 5 14. 54,157 x 8 3,875 433,256 16. 3 x 1804 = 5,412 18. 4 x 456,900 = 1,837,600 20. 9 x 2,132,512 = 19,192,608

  39. EXERCISE 1-8 Pages 15-16 22. 914 x 67 24. 7816 x 513 61,238 4,009,600 26. 23,418 x 1147 28. 405,607 x 112 26,860,446 45,427,984 30. 423 x 63,940 = 27,046,620

  40. DIVISION OF WHOLE NUMBERS • In division, the number to be divided is called the dividend • The number by which the dividend is divided is called the divisor • The result is the quotient • A difference left over is called the remainder

  41. DIVISION OF WHOLE NUMBERS • Division is the inverse, or opposite, of multiplication • The symbol for division is ÷ • 36 ÷ 6 = • Division can also be expressed in fractional form such as • The long division symbol is .

  42. DIVISION WITH ZERO • Zero divided by a number equals zero • For example: 0 ÷ 5 = 0 • Dividing by zero is impossible; it is undefined • For example: 5 ÷ 0 is not possible

  43. PROCEDURE FOR DIVISION • Example: Divide 4,505 ÷ 6 • Write division problem with divisor outside long division symbol and dividend within symbol • Since, 6 does not go into 4, divide 6 into 45. 45  6 = 7; write 7 above the 5 in number 4505 as shown • Multiply: 7 × 6 = 42; write this under 45 • Subtract: 45 – 42 = 3

  44. PROCEDURE FOR DIVISION • Bring down the 0 • Divide 30  6 = 5; write the 5 above the 0 • Multiply: 5 × 6 = 30; write this under 30 • Subtract: 30 – 30 = 0 • Since 6 can not divide into 5, write 0 in the answer above the 5. Subtract 0 from 5 and 5 is the remainder • Therefore 4,505  6 = 750 r5

  45. EXERCISE 1-9 Page 19 12. 45 14. 218 16. 6,589 18. 1,905 r1 20. 12,368 r7

  46. EXERCISE 1-9 Page 19 22. 77835 r1 24. 13 26. 207 783 r105 28. 30. 216

  47. EXERCISE 1-10 Pages 20-22 2. A chef estimates that an average of 150 pounds of ground beef are prepared daily. How many pounds of ground beef should be ordered for a 4-week supply? The restaurant is closed only on Monday 150 x 6 x 4 = 3,600 pounds 4. A tractor-trailer operator totals diesel fuel bills for 185 gallons of fuel used in a week. The truck travels 1,665 miles during the week. How many miles per gallon does the truck average? 1665 ÷ 185 = 9 miles per gallon

  48. EXERCISE 1-10 Pages 20-22 6. In a commercial bakery, roll dividing machines produce 16,000 dozen rolls in 8 hours. Determine the number of single rolls produced per minute. 16,000 x 12 ÷ 8 ÷ 60 = 400 rolls per minute 8. An apartment complex is being built; it will have 318 apartments. Each workday heating and air-conditioning systems can be installed in 6 apartments. How many workdays are required to complete installations for the complete complex? 318 ÷ 6 = 53 workdays to complete

  49. EXERCISE 1-10 Pages 20-22 10. A cosmetologist determines that an average of 3 ounces of liquid shampoo are required for each shampooing application. The beauty salon has 9 quarts of shampoo in stock. How many shampooing applications are made with the shampoo in stock? 9 x 32 ÷ 3 = 96 applications 12. In estimating the time required to complete a proposed job, an electrical contractor determines that a total of 735 hours are needed. Three electricians each work 5 days per week for 7 hours per day. How many weeks are required to complete the job? 735 ÷ (3 x 5 x 7) = 7 weeks