Introductions to decimal numbers. The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.
1. Introduction to Decimals Reading and Writing Decimals: Note: There is a relationship between fractions and numbers written in decimal notation. Three-tenths Three-hundredths Three-thousandths 3 3 3 0. 3 0. 03 0. 003 10 100 1000 1 zero 1 decimal place 2 zeros 2 decimal places 3 zeros 3 decimal places A number written in decimal notation has three parts. 351 . 7089 Whole-number part Decimal Point Decimal Part A number written in decimal notation is often called simply a decimal. The position of a digit in a decimal determines the digit’s place value. In the decimal 6,584,791.0324, the position of the digit 2 determines that its place value is thousandths. When writing a decimal in words, write the decimal part as if it were a whole number; then name the place value of the last digit. 0.6481 Six thousand four hundred eighty-one ten-thousandths 549.238 Five hundred forty-nine and two hundred thirty-eight thousandths (The decimal point is read as “and”) To write a decimal in standard form, zeros may have to be inserted after the decimal point so that the last digit is in the given place-value position. Five and thirty-eight hundredths 5.38 (8 is in the hundredths’ place.) Nineteen and four thousandths 19.004 (Insert two zeros so that the 4 is in the thousandths’ place.) Seventy-one ten-thousandths 0.0071 (Insert two zeros so that 1 is in the ten-thousandths’ place.) Example 1 Write 307.4027 in words. Solution Three hundred seven and four thousand twenty-seven ten-thousandths Example 2 Write six hundred seven and seven hundred eight hundred-thousandths in standard form. Solution 607.00708
2. Rounding Decimals: Rounding decimals is similar to rounding whole numbers except that the digits to the right of the given place value are dropped instead of being replaced by zeros. If the digit to the right of the given place value is less than 5, drop the digit and all digits to the right. If the digit to the right of the given place value is greater than or equal to 5, increase the number in the given place value by 1 and drop all digits to its right. Given place value Round 26.3799 to the nearest hundredth. 26.3799 9 > 5 Increase 7 by 1 and drop 26.3799 rounded to the nearest hundredth is 26.38. all digits to the right of 7. Example 3 Round 0.39275 to the nearest ten-thousandth. Solution Given place value 0.39275 5=5 0.3928 Example 4 Round 42.0237412 to the nearest hundred-thousandth. Solution Given place value 42.0237412 1<5 42.02374 Addition and Subtraction of Decimals To add or subtract decimals, write the numbers so that the decimal points are on a vertical line, placing zeros as place holders if necessary. Add or subtract as for whole numbers, and write the decimal point in the sum or difference directly below the decimal points in the addends or subtrahends. Add: 0.237 4.9 27.32 0.237 4.900 27.320 32.457 Subtract: 21.532 9.875 21.532 9.875 11.657 Subtract: 4.3 1.7942 4.3000 1.7942 2.5058 Note: By placing the decimal points on a vertical line, we make sure that digits of the same place value are added or subtracted. Multiplication of Decimals To multiply decimals, multiply the numbers as in whole numbers. Write the decimal point in the product so that the number of decimal places in the product is the sum of the decimal places in the factors. Multiply: 21.4 0.36
3. 21.4 1 decimal place 0.36 2 decimal places 1284 642 7.704 3 decimal places Multiply: 0.037 0.08 0.037 3 decimal places 0.08 2 decimal places 0.00296 5 decimal places Note: Two zeros must be inserted between the 2 and the decimal point so that there are 5 decimal places in the product. Multiplying by Powers of 10: To multiply a decimal by a power of 10 (10, 100, 1000, …), move the decimal point to the right the same number of places as there are zeros in the power of 10. 3.8925 10 38.925 3.8925 10 38.925 3.8925 100 389.25 3.8925 10 389.25 3.8925 1000 3892.5 3.8925 10 3892.5 3.8925 10,000 38,925. 3.8925 10 38,925. 3.8925 100,000 389,250. 3.8925 10 389,250. Note: If a power of 10 is written in exponential notation, the exponent indicates how many places to move the decimal point. Division of Decimals To divide decimals, move the decimal point in the divisor to the right to make the divisor a whole number. Move the decimal point in the dividend the same number of places to the right. Place the decimal point in the quotient directly over the decimal point in the dividend, and then divide as in whole numbers. Divide: 3.25 15.275 Move the decimal point 2 places to the right in the divisor and then in the dividend. Place the decimal point in the quotient. 4 .7 325 1527.5 1300 227 5 227 5 0 Moving the decimal point the same number of decimal places in the divisor and dividend does not change the value of the quotient, because this process is the same as multiplying the numerator and denominator of a fraction by the same number. In the example above, . . . 3.25 15.275 .
.
325 1527.5 Dividing by Powers of 10: To divide by powers of 10 (10, 100, 1000, …), move the decimal point to the left the same number of places as there are zeros in the power of 10. 34.65 10 3.465 34.65 10 3.465 34.65 100 0.3465 34.65 10 0.3465
4. 34.65 1000 0.03465 34.65 10 0.03465 34.65 10,000 0.003465 34.65 10 0.003465 Note: If the power of 10 is written in exponential notation, the exponent indicates how many places to move the decimal point. Comparing and Converting Fractions and Decimals Convert Fractions to Decimals: Every fraction can be written as a decimal. To write a fraction as a decimal, divide the numerator of the fraction by the denominator. The quotient can be rounded to the desired place value.
Convert to decimal.
0.42857
7 3.00000 rounded to the nearest hundredth is 0.43.
rounded to the nearest thousandth is 0.429.
rounded to the nearest ten-thousandth is 0.4286.
Convert 3 to a decimal. Round to the nearest thousandth.
3.2222
3 9 29.0000 3 rounded to the nearest thousandth is 3.222.
Convert Decimals to Fractions: To convert a decimal to a fraction, remove the decimal point and place the decimal part over a denominator equal to the place value of the last digit in the decimal. Reduce if possible. hundredths hundredths 47 45 9 0.47 7.45 7 7 100 100 20 thousandths hundredths 275 11 2 0.275 2 16 3 2 50 1 1 1000 40 0.16 16 100 3 100 3 3 100 6 Identify Order Relations Between 2 Decimals or a Decimal & a Fraction: Decimals, like whole numbers and fractions, can be graphed as points on the number line. The number line can be used to show the order of decimals. A decimal that appears to the right of a given number is greater than the given number. A decimal that appears to the left of a given number is less than the given number. 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 Note: 3, 3.0, and 3.00 represent the same number.
Find the order relation between and 0.38.
3 0.375 0.38 0.380 8 0.375 0.380 3 0.38 8
5. Practice Problems
1. Find the quotient of 3.6515 and 0.067. 11. Convert to a decimal. Round to the
2. Find the sum of 369.41, 88.3, 9.774, and nearest thousandth. 366.474. 12. Convert 0.66 to a fraction.
3. Place the correct symbol, or , 13. Subtract: 27.31 4.4465 between the two numbers. 0.055 0.1 14. Round 7.93704 to the nearest hundredth. 4. Write 22.0092 in words. 15. Find the product of 3.08 and 2.9. 5. Round 0.05678235 to the nearest 16. Write 342.37 in words. hundred-thousandth. 17. Write three and six thousand seven 6. Convert 2 to a decimal. Round to the hundred fifty-three hundred-thousandths nearest hundredth. in standard form. 7. Convert 0.375 to a fraction. 18. Multiply: 34.79 0.74 8. Add: 3.42 0.794 32.5 19. Divide: 0.053 0.349482 9. Write thirty-four and twenty-five thousandths in standard form. 20. What is 7.796 decreased by 2.9175? 10. Place the correct symbol, or ,
between the two numbers. 0.62
21. You had a balance of $895.68 in your checking account. You then wrote checks of $145.72 and $88.45. Find the new balance in your checking account. 22. The state income tax on the business you own is $560 plus 0.08 times your profit. You made a profit of $63,000 last year. Find the amount of income tax you paid last year. 23. A car costing $5944.20 is bought with a down payment of $1500 and 36 equal monthly payments. Find the amount of each monthly payment. 24. You have $237.44 in your checking account. You make deposits of $56.88, $127.40, and $56.30. Find the amount in your checking account after you make the deposits. 25. Place the correct symbol, or , 33. Round 7.0954625 to the nearest between the two numbers. 0.66 0.666 thousandth. 26. Subtract: 13.027 8.94 34. Divide: 0.006 1.392 27. Write 45.0302 in words. 35. Add: 270.93 97 1.976 88.675 28. Convert to a decimal. Round to the 36. Find the missing dimension. nearest thousandth. ? 29. Convert 0.825 to a fraction. 4.86 in 30. Round 0.07395 to the nearest ten- thousandth. 6.23 in 31. Find 0.0569 divided by 0.037. Round to the nearest thousandth. 37. Multiply: 1.37 0.004 32. Find 9.23674 less than 37.003. 38. What is the total of 62.3, 4.007, and 189.65? 39. Write two hundred nine and seven thousand eighty-six hundred-thousandths in standard form. 40. A car was bought for $6392.60, with a down payment of $1250. The balance was paid in 36 monthly payments. Find the amount of each monthly payment. 41. You received a salary of $363.75, a commission of $954.82, and a bonus of $225. Find your total income. 42. A long-distance telephone call costs $.85 for the first 3 minutes and $.42 for each additional minute. Find the cost of a 12-minute long-distance telephone call.
6. The table below shows the postal rates for selected weights for single-piece and presorted mail. Use this table for Exercises 43 and 44. Weight (not over) Single piece Presorted 1 ounce $.32 $.295 2 ounces .55 .525 3 ounces .78 .709 4 ounces 1.01 .939 5 ounces 1.24 1.69 43. Find the total cost of mailing the following: 3 single pieces weighing 2.3 ounces each, 2 single pieces weighing 3.4 ounces each, and 50 presorted pieces weighing 0.3 ounce each. 44. Find the total cost of mailing the following: 7 single pieces weighing 0.4 ounce each, 30 presorted pieces weighing 0.7 ounce each, and 75 presorted pieces weighing 1.4 ounces each. Round to the nearest cent. 45. Write 129.04 in words. 48. Round 3792.6974 to the nearest 46. Write in standard form: six and two hundredth. thousand fifty-three ten-thousandths. 49. Add: 42.97 162.9 12.876 1.093 47. Round 47.28351 to the nearest 50. Find the sum of 8962.51, 54.3, and thousandth. 123.017. 51. You have $692.46 in your checking account. You make deposits of $41.30, $122.79, and $98.42. You write a check for $219.67. Find the amount in your checking account after making the deposits and writing the check. 52. Subtract: 20.092 17.9263 53. Find 25.367 less than 108.44. 54. An electrician earns $3129.60. Deductions from the check are $630.24 for federal tax, $180.92 for state tax, and $43.12 for insurance. Find the electrician’s take-home pay. 55. Multiply: 8.35 0.017 56. Find the product of 0.93 and 0.027. 57. A car travels 32.3 miles on each gallon of gas. How far does a car travel on 11.8 gallons of gas? Round to the nearest tenth of a mile. 58. Find 2.192 divided by 0.004. 59. Divide: 34 5.072 Round to the nearest thousandth. 60. You work 40 hours and receive $302. Find your hourly rate of pay.
61. Convert to a decimal. Round to the 65. Write 4.00632 in words.
nearest thousandth. 66. Write twelve and six hundredths in 62. Convert 0.025 to a fraction. standard form. 63. Place the correct symbol, , 67. Round 37.295 to the nearest hundredth. between the two numbers. 0.017 0.17 68. Round 43.0973 to the nearest tenth. 64. Place the correct symbol, , 69. Add: 69.94 9.807 994.673 95.06 between the two numbers.
0.6255 70. Find the total of 34.92, 6.817, and 152.9.
71. Kara Brady bought a math textbook for $39.43, a history text for $43.95, a biology text for $35.61, and an English workbook for $19.45. Find the total cost of the four books. 72. Subtract: 109.63 94.984 73. What is 84.5903 decreased by 9.953? 74. Your lunch costs $4.87. An additional cost of $.29 is added for tax. Find the change you receive from a $10 bill. 75. Multiply: 0.086 0.07 76. Find the product of 16.5 and 0.24 77. A motorist averages 56.4 miles per hour while traveling for 3.5 hours. Find the total distance traveled during the 3.5 hours. 78. Find the quotient of 0.042 and 14. 79. Divide: 0.076 0.02904 Round to the nearest thousandth. 80. A $563.68 Sony stereo system is bought for a down payment of $100 and 18 equal monthly payments. Find the amount of each monthly payment.
81. Convert to a decimal. 82. Convert 0.16 to a fraction.
83. Place the correct symbol, , between 84. Place the correct symbol, , between
the two numbers. 0.72 0.702 the two numbers. 0.7
7. Practice Problems Key 1. 54.5 42. $4.63 2. 833.958 43. $19.11 3. 0.055 0.1 44. $50.47 4. Twenty-two and ninety-two ten- 45. one hundred twenty-nine and four thousandths hundredths 5. 0.05678 46. 6.2053 6. 2.33 47. 47.284
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