DECIMAL COMPETENCY PACKET

DECIMAL COMPETENCY PACKET Developed by: Nancy Tufo Revised: Sharyn Sweeney 2004 Student Support Center North Shore Community College

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In this booklet arithmetic operations involving decimal numbers are explained If you have not already reviewed the Fraction booklet, please do so before working through this one Calculators are not allowed when taking the Computerized Placement Test (CPT), nor in Fundamentals of Mathematics, Pre- Algebra, and Elementary Algebra; therefore, do not rely on a calculator when working the problems in this booklet To use this booklet, review the glossary, study the examples, then work through the exercises, and check your answers at the end of the booklet When you find an unfamiliar word, check the glossary for a definition or explanation The last several pages are Place Value Charts that will be helpful to you Remove those pages for easier use If you have difficulty understanding any of the concepts, come to one of the Tutoring Centers located on the Lynn, Danvers Main and Danvers Hathorne Campuses Hours are available at (978) 762-4000 x 5410 Additional Tutoring Center information can be found on the NSCC website at wwwnorthshoreedu/services/tutoring The Centers are closed when school is not in session, and Summer hours are limited 3

Table of Contents Glossary 5 General Decimal Information 6 Place Value 7 Writing and Reading Decimals 8 Translating Numerical Expressions 9 Decimal Fractions 10 Comparing Decimals 11 Rounding 12 Addition 14 Subtraction 15 Multiplication 16 Multiplication by Multiples of 10 17 Division by Multiples of 10 18 Division by Whole Numbers 19 Division by Decimals 20 Converting Fractions to Terminating Decimals 21 Converting Fractions to Repeating Decimals 22 Converting Decimals to Fractions 23 Decimal Word problems 24 Answers 27 Place Value Charts 30 4

Decimal or Decimal Number Decimal Point Denominator Difference Digit Division Factor Multiple Non-Repeating Decimal Glossary Any number that includes place value to the right of a decimal point A dot or point that separates the decimal value from the integral value of a number The bottom number of a fraction It represents the number of pieces needed to make one whole (See Fraction booklet for more information) The result when two numbers are subtracted The order of subtraction is important A B means that the number represented by B is subtracted from A, that is A is the top number in a vertical subtraction The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are the digits of the Base ten number system Arithmetic operation: Dividend Divisor = Quotient Dividend Expressed as a fraction: = Quotient Divisor quotient or using a "division house" divisor dividend Each of the numbers that are multiplied, ie in the product 7 x 9=63 the numbers 7 and 9 are factors A number which is the product of a given number and another factor; Multiples are equal to or larger than the given number, ie the multiples of 3 are: 3,6,9,12,15, A decimal representation that has no pattern of repetition in the digits after the decimal point Numerator Place Value The top number of a fraction It represents the number of pieces of a certain size considered for the expression (See fraction booklet for more information) The position a digit holds in a number It tells us the 5

value of the digit (See chart page 7) Product Quotient Repeating Decimal Sum Terminating Decimal The result when two numbers are multiplied The result when two numbers are divided A numerical representation that has a pattern of digits that repeats to infinity 0323232 is a repeating decimal, and can be written as 0 32 The result when two numbers are added Every fraction can be written as a decimal by dividing the denominator into the numerator if there no remainder, the decimal will terminate Whole Numbers Positive numbers with no fractional or decimal portion General Decimal Information Our number system is a decimal system using the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, called the Base Ten Number System Only positive numbers will be used in this booklet Whole numbers are decimal numbers with no fractional part; a decimal point is assumed to be to the right of the whole number ( ie, 15 = 150) Numbers with no whole number component will be written with a zero preceding the decimal point ( ie, 023) Digits to the right of a decimal point represent fractional parts with a denominator of a multiple of ten Digits to the left of the decimal point are separated into groups of three using commas 6

Place Value The position of a digit in a number reflects the "place value" of that digit In the following table, the number represented has value according to the place the digit "1" holds in each case (Note the use of commas) In the following chart, note the similarity of place value names on both sides of the decimal Those places to the right of the decimal end in "ths" indicating that they are fractional Whole Numbers Decimal Fractions Etc Ten Millions Millions, Hundred Thousands Ten Thousands Thousands, Hundreds Tens Units (Ones) Decimal Point (and) Tenths Hundredths Thousandths Ten-Thousandths Hundred-Thousandths Millionths 1, 0 0 0, 0 0 0 0 0 0 0 0 1 1 0 0, 0 0 0 0 0 0 0 1 1 0, 0 0 0 0 0 0 1 1, 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 Ten Millionths Etc In a spoken or written number, the word "and" reflects placement of a decimal point Although each number uses the same digits, (ones and zeros), the value of each number in the chart above is very different The numbers, in order of the chart, are read: one million and one millionth one hundred thousand and one hundred-thousandth ten thousand and one ten-thousandth one thousand and one thousandth one hundred and one hundredth ten and one tenth one and no tenths, or more commonly, one one tenth 7

Writing Decimals Place value is reflected when writing and reading decimal numbers in words In writing the decimal is represented by the word "and" 47 is written "four and seven tenths" 70024 is written "seventy and twenty-four thousandths" Write the following in words as you would write the number (Use the chart at the end of the booklet to aid with number placement) 1) 2015 6) 405 2) 4521 7) 27854 3) 15196 8) 70007 4) 2,049009 9) 11 5) 0005 10) 192807

Translating Numerical Expressions To translate written numerical expressions, place the last written number in the correct place value Twenty and ninetysix thousandths 20096 Six, (6) the last digit belongs in the thousandths place (Third place to the right from the decimal point) 20096 Zero must be entered in the tenths place Write the following using digits (Use a chart if needed) 1) four and five tenths 2) fourteen hundredths 3) one thousand nine hundred seventy-two ten thousandths 4) four hundred seven and three hundred twenty-eight thousandths 5) one tenth 6) seven and nine hundredths 7) one hundred seventy-two ten-thousandths 8) twenty-two and five tenths 9) twenty and four hundred ninety-six thousandths 10) three hundred and three hundredths 9

Decimal Fractions A decimal number is another way to write a fraction with a denominator of a multiple of ten, (ie, denominators equal to 10; 100; 1,000; 10,000; etc) To convert a fraction with a denominator of a multiple of ten to a decimal, read the fraction and write as a decimal number 7 3 10 is read " three and seven tenths" 37 234 15 1000 15234 5 100 005 expressed with digits is read " fifteen and two hundred thirty-four thousandths" expressed with digits is read " five hundredths" expressed with digits Note the zero placement Write as a decimal number 1) 19 72 100 4) 3 1276 10 2) 7 301 1000 5) 6 1 100 3) 17 6) 100 24 1000 10

Comparing Decimals To compare decimals, write the decimal numbers with the same number of decimal places and decide which is larger Which is greater: 09 or 091? 090? 091 Write the following from smallest to largest: 078006, 07845, 07851, 0785, 078 078006, 078450, 078510, 078500, 078000 078000, 078006, 078450, 078500, 078510 To compare write both numbers with two decimal places Note zeros may be added or deleted from the right and after the decimal point Compare digits in hundredths place 1 is greater than 0; therefore, 091 is greater (hint: Consider money) Write the list adding zeros to hundred thousandths place as needed Since the digits in the tenths and hundredths places are the same, compare the digits in the thousandths place first Then compare the digits in the remaining places Re-write the list from smallest to largest Write from smallest to largest: 1) 1234, 1234, 01234 5) 0935, 12, 06, 056 2) 01, 001, 1001 6) 012, 016, 02, 048, 0054 3) 31, 0031, 0331 7) 5038, 50382, 50382, 05382 4) 006, 04, 09 8) 008, 808, 8808, 8888, 0088, 08 11

Rounding To round numbers for estimation: 1 Identify the place value to be rounded All digits to the left of that place remain the same 2 Check the number to the immediate right of the place to be rounded: a If the digit in that place is 5 or greater, add one to the digit in the place to be rounded OR b If the digit in that place is 4 or less, do not change the digit in the place to be rounded 3 Fill in the remaining place values to the right of the place to be rounded with zeros, or drop the digits after the decimal point Round 1792 to the hundreds place 18 18 0 0 Round 7364 to the tenths place 736 _ 7360 = 736 Round 49897 to the hundredths place 49 8 1 0 _ 49(8+1) 0 _ 4990 Identify the place value to be rounded, (7 hundred) Write the digit(s) to the left (1) Identify the number to the right (9) 9 is greater than 5; add one to 7, (7+1=8), enter 8 in the hundreds place Fill in all the places to the right with zeros Identify the place value to be rounded, (6 tenths) Write the digits to the left (73) Identify the number to the right (4) 4 is less than 5, 6 remains in the tenths place It is not need to fill in all the places to the right with zeros; rounding to tenths place Identify the place value to be rounded, (9 hundredths) Write the digits to the left (498) Identify the number to the right, (7) 7 is greater than 5, add one to 9 Since 9 + 1 = 10, a zero is entered in the hundredths place, and the 1 is carried to the tenths place The 1 is added to 8 The zero is needed to represent the hundredths place 12

Round these numbers as indicated 1) Tenths 6287 9) Units 3397 2) Units 1445 10) Hundredths 49995 3) Ten thousandths 356906 11) Thousandths 50074 4) Tenths 31416 12) Thousandths 06739 5) Hundreds 459326 13) Tenths 198 6) Tenths 1977 14) Ten thousandths 001704 7) Thousandths 00067 15) Hundredths 001011 8) Tens 38988 16) Thousandths 00007 13

Addition To add decimals, write the numbers vertically with the decimal points directly under each other, then add the digits Note: When the decimal points are lined up, the digits are automatically lined up in the correct place value 132 + 157 1320 + 157 1477 $437 + $4156 + $018 $43700 4156 + 018 $47874 Write the problem vertically Line up the decimal points Note the additional zero Adding zeros to the right of the final digit after the decimal does not change the value of the number Dollar values are the most familiar decimal values Write the problem vertically Line up the decimal points The additional zeros are optional, but help with placement Note dollar sign use Find the Sum (Add): 1) 003 + 04 6) 48 + 084 2) 03 + 003 + 0003 7) 10 + 96 + 376 + 8451 3) 205 + 0561 + 439 + 1732 8) $306 + $213 + $489 4) $4 + $1401 9) 2,13407 + 3065 + 2109 5) 80632 + 0234 + 081 + 0064 10) 563701 + 0268 + 42 14

Subtraction To subtract decimals, write the numbers vertically with decimal points directly under each other, and add zeros when needed, then subtract the digits Note: When the decimal points are lined up, the digits are automatically lined up in the correct place value 4263-18275 42630-18275 24355 $23 - $013 Write the problem vertically Line up the decimals Remember: always write the first number on the top Add zeros to the number with fewer places to the right of the decimal point Subtract Write the problem vertically Line up the decimals $2300-013 $2287 Insert the decimal point and two zeros Subtract; borrow if necessary Find the Difference (Subtract): 1) 84-735 5) 4355-1647 2) 125-87 6) 6054-0928 3) $1750 - $625 7) 89-5846 4) $18 - $563 8) 104003-2178 Find the Sum and Difference as indicated, (in the order indicated): 9) 146-198 + 37 11) 019 + 234-1003 10) 567 + 034-205 12) $2190 - $045 - $ 234 15

Multiplication To multiply decimals, write the problem and multiply as you would a whole number multiplication problem The product (answer) of two decimal numbers has the same number of decimal places after the decimal point as the total number of decimal places in the two numbers being multiplied 019 x 04 019 x 04 0076 708 x 032 1416 21240 22656 Find the Product (multiply): Write vertically (The decimal points do not have to line up) 2 decimal places (Decimal points not lined up) + 1 decimal place 3 decimal places Count from right to left; add a zero before the decimal point 0 decimal places (Decimal points not lined up) + 2 decimal places 2 decimal places Count from right to left to place decimal point 1) 032 x 06 4) 5048 x 203 7) 0075 x 54 2) 19 x 005 5) 015 x 015 8) 99 x 11 3) 400 x 017 6) 24 x 013 9) 2029 x 108 16

Multiplication by Multiples of 10 To multiply by a multiple of ten, move the decimal point RIGHT as many places as there are zeros in the multiplier 246 x 10 = 2460 There is one zero in the multiplier (10); therefore, the decimal point moves right one place 0048 7 x 1000 = 487 246_ x 100 = 2,4600 There are three zeros in the multiplier (1000); therefore, the decimal point movers right three places There are two zeros in the multiplier, (100); therefore, the decimal point moves right two places Note the additional zeros Multiply: 1) 483 x 10 = 7) 35961 x 100 = 2) 835 x 1000 = 8) 826 x 1000 = 3) 902 x 100 = 9) 7007 x 100 = 4) 1037 x 10 = 10) 72953 x 10 = 5) 076 x 1000 = 11) 0987 x 1000 = 6) 008 x 10 = 12) 476098 x 10,000 = 17

Division by Multiples of 10 To divide by a multiple of ten, (10; 100; 1,000; etc), move the decimal point to the LEFT as many places as there are zeros in the divisor 7 82 10 = = 782 _032 1000 = 000032 There is one zero in the divisor (10), therefore the decimal point moves left one place There are three zeros in the divisor (1000), therefore the decimal point moves left three places Note the additional zeros Divide: 1) 825 100 = 6) 78567 10 = 2) 9238 1000 = 7) 5487 1000 = 3) 0754 10 = 8) 2035 10 = 4) 0845 100 = 9) 5408 100 = 5) 638 100 = 10) 6200 10,000 = 18

Division by Whole Numbers To divide a decimal by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend to ensure the correct place value Divide as with whole numbers 5 5 5 = 5 5 5 5 1 1 5 5 5 5 5 0 Write the problem with a "division house," placing the quotient's (answer's) decimal point directly over the decimal point of the dividend 22 5 3 = 3 22 5 A fraction is another way to express a division problem The divisor is the denominator and the dividend is the numerator 3 75 22 5 21 1 5 1 5 0 Write the problem with a "division house," placing the quotient's (answer's) decimal point directly over the decimal point of the dividend Divide: 1) 1 8 6 = 4) 0 264 4 = 7) 0 32 5 = 2) 084 5) 4 396 8) 9 34 5 5 3) 0096 6) 0 016 2 = 9) 8 149 2 19

Division by Decimals In division, the divisor must be a whole number To convert a decimal divisor to a whole number, multiply the divisor and the dividend by a multiple of ten Then divide as usual 49 07 (49 x 10) (07 x 10) 49 7 = 7 8505 100 8505 x = 005 100 5 1701 5 8505 5 35 35 00 5 5 0 The divisor (07) has one decimal place To change the divisor to a whole number, multiply the divisor and the dividend by 10 Divide as usual The divisor (005) has two decimal places To change the divisor to a whole number, multiply the divisor and the dividend by 100 Divide as usual Place the decimal point for the quotient (1701) directly above the decimal point in the dividend (8505) Divide: 1) 574 0 0 7 4) 35 1 2 7 7) 82 8 0 03 2) 0 4 6 988 5) 2 4 77 04 8) 0 41 205 3) 00144 6) 12 0132 9) 0011 06832 0004 20

Converting Fractions to Terminating Decimals To convert a fraction to a decimal, divide Some fractions will convert to a decimal representation with a remainder of zero, called a terminating decimal Convert to a Decimal 3 = 12 025 12 300 24 60 60 0 Divide 3 by 12 The decimal equivalent to three twelfths is twentyfive hundredths 3 = 025 12 Convert to a Decimal 020 5 11 = 11+ 25 500 25 50 0 5 11 = 1120 25 The whole number portion of the number will remain the same The fraction will convert to a decimal Divide 5 by 25 The decimal equivalent to eleven and five twentyfifths is eleven and two tenths Convert to a Decimal: 1) 9 6) 18 19 40 2) 15 7) 30 48 32 3) 6 8) 16 2 5 20 4) 9 9) 20 7 77 40 5) 13 10) 50 37 47 50 21

Converting to Repeating Decimals To convert a fraction to a decimal, divide Some fractions will convert to a decimal representation with pattern, called a repeating decimal 2 = 3 0666 3 2000 18 20 20 20 0 666 = 06 30909 34 = 11 34 0000 11 33 100 99 100 99 1 3 0909 = 309 Divide two by three Note that the remainder will continue to be two; therefore, the decimal answer is a repeating decimal Repeating decimals are written with a bar over the repeating digits in the pattern Divide 34 by 11 Since 11 does not divide 10, there is a need to bring down an additional zero Note that there is a portion of the quotient that does not repeat The bar indicates that only the 09 repeats Convert: 1) 1 6) 11 1 1 3 2) 1 7) 33 1 8 6 3) 4 8) 9 7 33 4) 1 9) 3 7 42 5) 3 10) 22 2 4 3 22

Converting Decimals to Fractions To convert a terminating decimal to a fraction, write the decimal with the place value multiple of ten as a denominator and reduce to simplest terms 2 3 2 = 3 10 The decimal fraction portion of the number terminates in the tenths place; therefore the denominator will be 10 2 1 3 = 3 10 5 This fraction is not in lowest terms, therefore must be reduced Divide numerator and denominator by 2 To convert a repeating decimal to a fraction, use a value of 9 as the denominator 9 The repeating pattern ends in the hundredths place, 309 = 3 99 therefore the denominator will have two nines, or be 99 9 1 3 = 3 99 11 This fraction is not in lowest terms, therefore must be reduced Divide numerator and denominator by 9 Convert: 1) 7 85 6) 34 0102 2) 10 3 7) 7 7 3) 2 08 8) 10425 4) 0 45 9) 0006 5) 0 360 10) 2 360 23

Word Problems To solve a word problem, read the problem and express what you are trying to learn in your own words Identify the operation to be used, (addition, subtraction, multiplication, or division) Translate the problem from words to math symbols, (ie write an equation) Solve the equation Carlos bought one pair of shoes for $1995, two neckties for $395 each, three pairs of socks for $125 a pair, and one suit for $8995 What was the total cost? Trying to learn total cost of the items $ 1995 790 375 + 8995 $12155 Total cost usually implies addition One pair of shoes 395 x 2 Two neckties 125 x 3 Three pair of socks One suit Add with decimal points lined up Nora made 18 equal monthly payments on her new stereo set If the total cost of the set was $35500, what was her monthly payment? (Round off to the nearest cent) Trying to learn Nora's monthly payment $35500 18 1972 18 35500 18 175 162 130 126 40 36 4 The monthly payment is $1972 Total and number of months are given Division is implied Total divided by number of months gives each month's payment The answer must be to the nearest cent; that is rounded to the hundredths place Divisor is a whole number To continue the division, another zero may be added However, the next digit in the quotient will be a 2 and by the rules of rounding, will not effect the current quotient

Solve the following 1) The $14635 cost of a party was shared by 10 people How much did each person have to pay? (Be sure to round your answer to the nearest cent) 2) 537 people attended a $100 dollar a plate fund raising dinner for the NSCC Foundation How much money did this dinner raise? 3) At the beginning of the month, Jim's bank balance was $27538 During the month he wrote the following checks: $17489, $68, and $5776 He made deposits of $250 and $350 Find his bank balance at the end of the month 4) Rudy drove his car 9,600 miles last year His total car expenses were $625 for the year Find the average cost per mile (Round off your answer to the nearest hundredth) 25

5) A garden is 3375 feet long and 216 feet wide Draw a diagram of the garden with the lengths written on all four sides What is the total distance around the garden? 6) A car traveled at 50 miles an hour for 25 hours How far did it go? 7) A can of ham weighing 775 pounds costs $ 1186 What does the ham cost per pound? (Round to the nearest cent) 8) A park is 46 miles long and 27 miles wide a What is the total distance around the park? b If a racecar drove 50 times around the park, how far will it have to go? 26

1 Twenty and fifteen hundredths 2 Forty-five and twenty-one hundredths 3 Fifteen and one hundred ninety-six thousandths 4 Two thousand, forty-nine and nine thousandths 5 Eight thousandths Answers to Exercises Page 8 Page 9 Page 13 1 45 2 014 3 01972 4 407328 5 01 6 709 7 00172 8 225 9 20496 10 30003 6 Four and five hundredths 7 Two hundred seventyeight and fifty-four hundredths Page 10 1 7219 2 301007 3 017 4 12763 5 106 6 0024 1 629 2 14 3 35691 4 31 5 500 6 198 7 0007 8 390 9 34 10 5000 11 5007 12 0674 13 20 14 00170 15 001 16 0001 8 Seven and seven tenthousandths 9 One and one tenth 10One thousand nine hundred twenty-eight and seven hundredths Page 11 1 01234, 1234, 1234 2 001, 01, 1001 3 0031, 0331, 31 4 006, 04, 09 5 056, 06, 0935, 12 6 0054, 012, 016, 02, 048 7 05382, 5038, 50382, 50382 8 008, 0088, 08, 808, 8808, 8888 Page 14 1 043 2 0333 3 63831 4 $1801 5 91712 6 4884 7 31811 8 $1008 9 2442679 10 608381 27

Page 15 Page 18 Page 21 1 105 1 0825 1 05 2 38 2 09238 2 05 3 $1125 3 00754 3 0375 4 $1237 4 000845 4 045 5 2708 5 0638 5 026 6 59612 6 78567 6 0475 7 3054 7 005487 7 15 8 82223 8 2035 8 51 9 1632 9 5408 9 77175 10 396 10 06200 10 4774 11 1527 12 $1911 Page 16 Page 19 Page 22 1 0192 1 03 1 0 09 2 0095 2 021 3 6800 3 0012 2 0 03 4 1024744 4 0066 3 0 4 5 00225 5 044 4 0 3 6 00312 6 0008 7 04050 7 0064 5 0 136 8 1089 8 69 9 219132 9 0745 6 1 3 Page 17 Page 20 1 483 2 83,500 3 9,020 4 1037 5 760 6 08 7 3,5961 8 82,600 9 7007 10 72953 11 987 12 4,760,980 1 820 2 1747 3 0012 4 13 5 321 6 12 7 2,760 8 500 9 1708 7 0 16 8 0 21 9 0 16 10 4 6 28

1 2 3 4 5 6 7 8 9 10 Page 23 Page 25-26 17 1 $1464 7 20 2 $53,700 1 3 $57473 10 4 $007 per mile 3 5 1107 feet 2 2 3375 ft 25 5 216 ft 216 ft 11 9 3375 ft 25 34 34 6 125 miles 3333 7 $153 per pound 7 8 a 146 miles 7 9 b 7300 miles 17 10 40 3 500 40 2 111 29

Place Value Chart Hundred Million Ten Millions Millions, Hundred Thousands Ten Thousands Thousands, Hundreds Tens Units (Ones) Decimal Point (and) Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths Ten millionths Hundred Millionths

Place Value Chart Hundred Million Ten Millions Millions, Hundred Thousands Ten Thousands Thousands, Hundreds Tens Units (Ones) Decimal Point (and) Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths Ten millionths Hundred Millionths

Place Value Chart Hundred Million Ten Millions Millions, Hundred Thousands Ten Thousands Thousands, Hundreds Tens Units (Ones) Decimal Point (and) Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths Ten millionths Hundred Millionths

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