# Full text of "ERIC ED327701: Rounding Decimal Numbers to a Designated Precision. Fundamentals of Occupational Mathematics. Module 2."

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DOCUMENT RESUME ED 327 701 CE 056 763 AUTHOR TITLE INSTITUTION SPONS AGENCY PUB DATE CONTRACT NOTE PUB TYPE Engelbrecht, Nancy; And Others Rounding Decimal Numbers to a Designated Precision. Fundamentals of Occupational Mathematics. Module 2. Central Community Coll., Grand Island, NE. Office of Vocational and Adult Education (ED) , Washington, DC. 90 V199A90067 lOp.; For related modules, see CE 056 762-773. Guides - Classroom Use - Materials (For Learner) (051) EDRS PRICE MFOl/PCOl Plus Postage. DESCRIPTORS Arithmetic; Community Colleges; ^Decimal Fractions; Individualized Instruction; Learning Modules; ^Mathematical Applications; ^Mathematics Instruction; ^Measurement; Number Concepts; Numbers; Pacing; Two Year Colleges; Vocational Education IDENTIFIERS *Job Related Mathematics; ^Precision (Mathematics) ABSTRACT This module is the second in a series of 12 learning modules designed to teach occupational mathematics. Blocks of informative material and rules are followed by examples and practice problems. The solutions to the practice problems are found at the end of the module. Specific topics covered include rounding off, precision of measurement, and the concept of least precise measurement . ( YLB) * Reproductions supplied by EDRS are the best that can be made * from the original document. Ptx^0ctDlr9ctor Ron Vofd^strasse Prol9ct30emtMry Jan Wisiakjwski T$ehnle*t ContuiUnt Ray PfaMtton Technical Wrttm Nancy EnQotbrecht LyrmeGmt Ann Hunter StaceyOekes ^opynght. Central Community Coilege 3 1 Module 2 — ROUNDING DECIMAL NUMBERS TO A DESIGNATED PRECISION Rounding off a decimal niimber means rewriting the nximber in a shorter form to represent an approximation of the number. The symbol used for values that are approximately equal, but not exactly equal is In measuring the length of an object, it could be measured to the nearest 1 mm, the nearest 0.1 mm, the nearest 0.01 mm or even closer. Measurements are never exact. The precision with which the technician needs a measurement depends upon the particular application. The measurement a person records is a nxamber which has been read only as close as needed and therefore, represents a rounded off value. The number pi = jc = 3.141592653589793238 is a never ending decimal. It is necessary to round it off somewhere. Various possibilities includa: nearest ^ 3 nearest tenth ^ 3. 1 neazest hundredth ^ 3. 14 nearest thousandth ^ 3. 142 nearest ten-thousandth . . . 7C ^ 3. 1416 The proper value to select depends upon the demand for pr€.cision. The process called rounding off will follow the commonly used stops in what is called the rule--of-5. For example, when the number 748.537 is rounded to the units place, it is written as 749. Rounded to the nearest tenths, 748.537 becomes 748.5, and rounding to the nearest hundredth makes it 748.54. The following is the usual rule-of-5 for rounding a number written out in detail. 4 2 RULE FOR ROUNDING DECIMAlL NUMBERS ROUNDING TO TEN OR HIGHER PIACE To round a number to a particular place value, called the rounding place, that is in the ten (10) place or greater 1. If the digit immediately to the right of the rounding place is less than 5 (0,1,2,3,4), then (a) do not change the digit in the rounding place or any digits to the left of it. (b) replace, with zeros, all the digits to the right of the rounding place until the decimal point location is reached. (c) do not write the decimal point. (d) drop all digits which were to the right of the decimal point (do not replace them with zeros) . 2. If the digit iimnediately to the right of th^ rounding place is 5 or greater (5,6,7,8,9), then (a) add 1 (one) to the digit in the rounding place. When adding 1 to a 9 digit, add the 1 to the two digit number which ends in that 9. This has the effect of changing the 9 into 0 and the digit to its left will become 1 greater. (b) replace, with zeros, all the digits to the right of the rounding place until the decimal point location is reached. (c) do not write the decimal point, (d) drop all digits which wera to the right of the decimal point (ao not replace them with zero) . To round a number to a particular place value that is in the units place or to the right of the decimal point: 3. If the digit immediately to the right of the rounding place is less than 5 (0,1,2,3,4), then (a) do not change the digit in the rounding place or any digits to the left of it. (b) drop all the digits to the right of the rounding place. (c) do not replace dropped digits with zeros. 4. If the digit immediately to the right of the rounding place is 5 or greater (5,6,7,8,9), then (a) add 1 (one) to the digit in the rounding place. When adding 1 to a 9 digit, add the 1 to the two di jit number which ends in that 9. This has the effect of changing the 9 into 0 and the digit to its left will become 1 greater. (b) drop all the digits to the right of the rovnding place . (c) do not replace dropped digits with zeros. ERIC 5 3 EXAMPLE i: Round each number in the left-hand column to the place value (precision) indicated in each column heading . decimal ten unit tenth hundredth thousandth 275.8103 280 276 275.8 275.81 275.810 43. 9618 40 44 44.0 43.96 43.962 27.8205 30 28 27.8 27. 82 27.821 7.0261 10 7 7.0 7.03 7.026 A special case arose when 7.0261 was rounded to the nearest ten. The original tens digit is considered to be 0 so that 7.0261 = 07.0261. Since the digit immediately to the right of the ten digit is 7 (5 or more), then the tens digit of 0 is to be increased by 1 to form 10. Rounding 7.0261 to the nearest ten produced lU. PRACTICE PROBLEMS: Round each number in the left-hand column to the place value indicated in each column heading . Decimal hundred ten unit tenth 1. 685.31 2. 1728.47 3. 475.296 4. 88.972 Round each number in the left-hand column to the place value indicated in each column heading. Deciraal tenth hundredth thousandth ten-thousandth 5. 1.70062 6. 0.39491 7. 3.06177 8. 6.07939 9. 0.17316 10. 0.05592 6 4 A ruler, caliper and micrometer are three different devices used to make a length measurement. The biggest difference in these three instruments is the precision with which you are able to determine size. A person trained to use these might record the following for the measurement of the length of a pin. The PRECISION of each measurement is the place value of the last digit recorded by the measuring instrument and the unit of measure used by the instrument. The last digit of the metric caliper reading is the 3 of 17.3 mm. This 3 is in the tenths or 0.1 place value position. The precision of the 17.3 mm caliper reading is stated as 0.1 mm. An English caliper reading for length of the same pin is 0,68 in. The last digit 8 of 0.68 in. is in the hundredths or 0,01 place value position. The precision of the 0.68 in. English caliper reading is stated as 0.01 in. EXAMPLE 2: The precision of the measurements given for the pin length by the various types of instruments is summarized in the table. Type of Metric Metric English English Device Measure Precision Measure Precision ruler 17mm 1mm 0.7 in. 0.1 in. caliper 17.3 mm 0.1 mm 0.68 in. 0.01 in. micrometer 17.274 mm 0.001 mm 0.6801 in. 0.0001 in. As one learns production skills of machining, it is necessary to match the quality demanded of the craftsmanship to the precision of the measurements required to meet those quality standards . ruler, metric 17 mm caliper, metric 17.5 mm micrometer, metric 17.274 mm ruler, English 0.7 in. caliper, English 0.68 in. micrometer, English 0.6801 in. ERIC PRACTICE PROBLEnS: State the precision of the following measurements. Do not forget to include the unit of measurement with your niimber value. 11. 2.4 in. 13. 1.931 in IS. 7.09 in. 17. 91.0 mm 19, 0.81 in. 12. 6.08 mm 14. 12.280 mm 16. 91 mm 18. 91.000 mm 20. 0.810 in. The English length of pin measurements of Example 2 show that different instruments have different precisions. The least precise English measure is achieved by the English ruler as 0.7 in., while the most precise English measure is 0.6801 in. obtained by the English micrometer. The LEAST PRECISE of two or more measurements is the measure whose last recorded digit is farthest left (higher place value) . EXAMPLE 3: The diameter of a pin has been recorded as 0.25 in. and as 0.248 in. What is the precision of each measurement and which measurement is the least precise? Solution: 0.25 in. has precision 0.01 in. 0.248 in. has precision 0.001 in. The least precise is 0.25 in. EXAMPLE 4: The thickness of a steel plate has been recorded as 9.78 mm, 9.780 mm and as 9.8 mm. What is the precision of each measurement and which measurement is the least precise? Solution: 9.78 mm has precision 0.01 mm 9.780 mm has precision 0.001 mm 2^.8 mm has precision 0.1 mm The least precise is the 9.8 mm. 8 6 PRACTICE PROBLEMS: Identify the least precise of each set of measurements : 21. 2.181 in., 2.18 in., 2.1814 in. 22. 7.23 mm, 7.2346 mm, 7.2 mm, 7.235 mm. 23. 1230 mm, 1200 mm, 1232.1 mm, 1232 mm 24. 6.26 in., 6 in., 6.3 in., 6.277 in. 25. 4.00 in., 4.0 in., 4.000 in., 4 in. The widi-h of 3 separate steel plates are recorded below. Identify the least precise of each set of measurements. 26. 0.3 in., 2.56 in., 1.772 in. 27. 12.3 mm, 10.44 mm, 13 mm 28. 6.002 mm, 10.92 mm, 15.8 mm. 29. 14.125 in., 3.25 in., 0.875 in. 30. 9.40 mm, 27.900 mm, 316.0 mm ERIC 9 7 SOLUTIONS TO PRACTICE PROBLEMS — MODULE 2 Decimal hundred ten unit tenth 1. 685.31 700 690 685 685.3 2. 1728.47 1700 1730 1728 1728.5 3. 475.296 500 480 475 475.3 4. 88.972 100 90 89 89.0 Decimal tenth hundredth thousandth ten-thou 5. 1.70862 1.7 1.71 1.709 1.7086 6. 0.39491 0.4 0.39 0.395 0.3949 7. 3.06177 3.1 3.06 3.062 3.0618 8. 6.07939 6.1 6.08 6.079 6.0794 9. 0.17316 0.2 0.17 0.173 0.1732 10. 0.05592 0.1 0.06 0.056 0.0559 11. 0.1 iu. 14. 0.001 mm 17. 0.1 mm 20. 0.001 in 23. 1200 mm 26. 0.3 in. 29. 3.25 in. 12. 0.01 mm 15. 0.01 in. 18. 0.001 mm 21 . 2.18 in. 24. 6 in. 27. 13 mm 30. 316.0 mm 13. 0.001 in 16. 1 mm 19. 0.01 in. 22. 7.2 mm 25. 4 in. 28. 15.8 mm ERIC 10 1/ U.S. Dept. of Education Office of Educational Research and Improvement (OER!) Date Filmed July 17, 1991