DECIMAL FRACTIONS

Introduction to Decimal Numbers A number written in decimal notation has 3 parts: Whole # part The decimal comma Decimal part The position of the digit in the decimal number determines the digit’s value.

Place Value Chart Whole number part Decimal part Decimal comma , 103 102 101 100 10-1 10-2 10-3 10-4 10-5 tens ones tenths thousands hundreds hundredths thousandths ten-thousandths Hundred-thousandths Whole number part Decimal part Decimal comma

Writing a Decimal Number in Words Write the whole number part The decimal comma is written “and” Write the decimal part as if it were a whole number Write the place value of the last non-zero digit Ex: Write 6,32 in words Six and thirty-two hundredths

Ex: Write 0,276 in words Zero and two hundred seventy-six thousandths Or two hundred seventy-six thousandths Ex: Write 10,0304 in words Ten and three hundred four Ten-thousandths

Writing Decimal Number Standard Form Write the whole number part Replace “and” with a decimal comma Write the decimal part so that the last non-zero digit is in the identified decimal place value Note: if there is no “and”, then the number has no whole number part.

Ex: Write in standard form “seven hundred sixty-two thousandths” Ex: Write in standard form “eight and three hundred four ten-thousandths” 8 3 0 4 , Ex: Write in standard form “seven hundred sixty-two thousandths” Note: no “and”  no whole part 0 , 7 6 2

To write a decimal as a fraction, write the fraction as you would say the decimal Hundredths Ten-thousandths tenths 0,12345 Hundred-thousandths Thousandths

Converting Decimal to Fractions To convert a decimal number to a fraction, read the decimal number correctly. Simplify, if necessary. Ex: Write 0,4 as a fraction 0,4 is read “four tenths”  Ex: Write 0.05 as a fraction 0,05 is read “five hundredths” 

0,007 is read “seven thousandths”  Ex: Write 0,007 as a fraction 0,007 is read “seven thousandths”  Note: the number of decimal places is the same as the number of zeros in the power of ten denominator Ex: Write 4,2 as a fractional number Note: there’s a whole and decimal part  Mixed number 4,2 is read “four and two tenths”  4

Examples Continue: Convert decimal to base 10 fractions and simplify. 0,35 is read “thirty-five hundredths”  0,8 is read “eight tenths”  7,28 is read “seven and twenty-eight hundredths”  0,375 is read “three hundred seventy-five thousandths” 

Your turn to try a few

Converting Fractions to Decimal Numbers (base 10 denominator) When the fraction has a power of 10 in the denominator, we read the fraction correctly to write it as a decimal number Ex: Write as a decimal number The fraction is read “three tenths” Note: no “and”  no whole part = 0, 3

, Ex: Write as a decimal number The fraction is read “twenty-seven hundredths” Note: no “and”  no whole part 0 , 2 7 Ex: Write as a decimal number The mixed number is read “five and thirty-three thousandths” , 5 3 3

1 25 , = = 4 100 Ex: Write as a decimal number Setup an equivalent fraction with a denominator in closest powers of ten x 25 1 25 , 2 5 = = 4 100 x 25 Check to see how you would convert the denominator to the closest power of 10 – 100 (102).

3 6 , = = 5 10 Ex: Write as a decimal number Setup an equivalent fraction with a denominator in closest powers of ten x 2 3 6 , 6 = = 5 10 x 2 Check to see how you would convert the denominator to the closest power of 10 – 10 (101).

3 375 , = = 8 1000 Ex: Write as a decimal number Setup an equivalent fraction with a denominator in closest powers of ten x 125 3 375 , 3 7 5 = = 8 1000 x 125 Check to see how you would convert the denominator to the closest power of 10 – 1000 (103).

Denominator goes outside Using Long Division Converting fractions to decimals, take the numerator and divide by the denominator. If the fraction is a mixed number, put the whole number before the decimal. Rewrite as long division. Denominator goes outside Numerator goes inside

Ex: Write as a decimal number , 3 7 5 Begin Long Division 8 3 , 0 - 2 , 4 Add decimal and zeros as needed 6 - 5 6 Align decimal and divide 4 - 4 0 Answer will Terminate or repeat = 0,375

Ex: Write as a decimal number , 7 5 Begin Long Division 4 3 , 0 - 2 , 8 Add decimal and zeros as needed 2 - 2 0 Align decimal and divide Answer will Terminate or repeat = 0,75

3 75 , = = 100 4 Ex: Write as a decimal number Create equivalent fraction on the fraction part with a denominator in closest powers of ten x 25 3 75 , 7 5 = = 4 100 x 25 Check to see how you would convert the denominator to the closest power of 10 – 1000 (103).

Ex: Write as a decimal number , 8 3 3 6 5 , 0 Place a bar over the part that repeats. - 4 , 8 2 - 1 8 = 0,83 2 - 1 8 2 Is there an echo? This will repeat  repeating decimal number

Examples: A Terminating decimal The division problem goes on forever…. Repeating decimal

Repeating Decimals A single digit might repeat…. 0,3333…. Or a group of digits might repeat… 0,275275275….

Show repeating decimals by placing a line over the digit or group of digits that repeats 0,33333…. Becomes 0,3 And 0,275275….becomes 0,275

Ex: Convert to a decimal Notice the mixed number – whole & fraction part  The decimal number will have a whole & decimal part The whole part is 2  2 . ________ Now convert the fraction 5/8 to determine the decimal part: , 6 2 5 = 2.625 8 5 , 0 - 4 , 8 2 - 1 6 4 - 4 0

5 625 , 2 = = 8 1000 Ex: Write as a decimal number Create equivalent fraction on the fraction part with a denominator in closest powers of ten x 125 5 625 , 2 2 6 2 5 = = 8 1000 x 125 Check to see how you would convert the denominator to the closest power of 10 – 1000 (103).

Ex: Write as a decimal number 5 , 1 2 5 Set Long Division on the Fraction Part 8 1 , 0 - 0 , 8 Add decimal and zeros as needed 2 - 1 6 4 Align decimal and divide - 4 0 Place decimal in front of Decimal = 5,125

Ex: Write as a decimal number 12 , 6 2 5 Set Long Division on the Fraction Part 8 5 , 0 - 4 , 8 Add decimal and zeros as needed 2 - 1 6 4 Align decimal and divide - 4 0 Place decimal in front of Decimal = 12,625

Your turn to try a few

Rounding Decimal Numbers Rounding decimal numbers is similar to rounding whole numbers: Look at the digit to the right of the given place value to be rounded. If the digit to the right is > 5, then add 1 to the digit in the given place value and zero out all the digits to the right (“hit”). If the digit to the right is < 5, then keep the digit in the given place value and zero out all the digits to the right (“stay”).

Ex: Round 7,359 to the nearest tenths place Identify the place to be rounded to: Tenths Look one place to the right. What number is there? Compare the number to 5: 5 > 5  “hit” (add 1) 3 + 1 = 4 in the tenths place, zero out the rest 7,359 rounded to the nearest tenths place is 7,400 = 7,4

Ex: Round 22,68259 to the nearest hundredths place Identify the place to be rounded to: Hundredths Look one place to the right. What number is there? Compare the number to 5: 2 < 5  “stay” (keep) Keep the 8 and zero out the rest 22,68259 rounded to the nearest hundredths place is 22,68000 = 22,68

Ex: Round 1,639 to the nearest whole number Identify the place to be rounded to: ones Look one place to the right. What number is there? Compare the number to 5: 6 > 5  “hit” (add 1) 1 + 1 = 2 in the ones place, zero out the rest 1,639 rounded to the whole number is 2,000 = 2

Your turn to try a few

Decimal Addition & Subtraction To add and subtract decimal numbers, use a vertical arrangement lining up the decimal places (which in turn lines up the place values.) Ex: Add 16,113 + 15,21 + 2,0036 Put in 0 place holders 16,113 15,21 + 2,0036 3 3 , 3 2 6 6

Put in the decimal comma - 9,6413 Put in 0 place holders 6 , 3 5 8 7 Ex: Subtract 24,024 – 19,61 1 1 3 1 Put in 0 place holders 24,024 - 19,61 4 , 4 1 4 Ex: Subtract 16 – 9,6413 1 9 9 9 5 1 16 , 0000 Put in the decimal comma - 9,6413 Put in 0 place holders 6 , 3 5 8 7

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Decimal Multiplication Multiply by Powers of 10 When multiplying by 10, 100, 1000, … Move the decimal in the number to the right as many times as there are zeros. 2,345 times 10, move the decimal one place to the right, 23,45

Ex: Multiply 1,2345 x 10 Think 12345 x 10  12345 x 10 = 123450 1,2345 has 4 decimal place 10 has 0 decimal places Therefore the product of 1,2345 and 10 will have 4 + 0 = 4 decimal places 12 3450 ,  1,2345 x 10 = 12,3450 = 12,345

Ex: Multiply 1,2345 x 100 Think 12345 x 100  12345 x 100 = 1234500 1,2345 has 4 decimal place 100 has 0 decimal places Therefore the product of 1,2345 and 100 will have 4 + 0 = 4 decimal places 1234500 ,  1,2345 x 100 = 123,4500 = 123,45

Ex: Multiply 1,2345 x 1000 Think 12345 x 1000  12345 x 1000 = 12345000 1,2345 has 4 decimal place 1000 has 0 decimal places Therefore the product of 1,2345 and 1000 will have 4 + 0 = 4 decimal places 12345000 ,  1,2345 x 1000 = 1234,5000 = 1234,5

So what have we seen? 1,2345 x 10 = 12,345 1 zero  move decimal comma 1 place to the right 1,2345 x 100 = 123,45 2 zeros  move decimal comma 2 places to the right 1,2345 x 1000 = 1234,5 3 zeros  move decimal comma 3 places to the right To multiply a decimal number by a power of 10, move the decimal comma to the right the same number of places as there are zeros.

Ex: Multiply 34,31 x 1000 How many zeros are there in 1000? 3  Move the decimal comma in 34,31 to the right 3 times 34 , 31 ,  34,31 x 1000 = 34 310

Ex: Multiply 21 x 100 How many zeros are there in 100? 2  Move the decimal comma in 21 to the right 2 times 21 , ,  21 x 100 = 2100

Decimal Multiplication Continued Decimal numbers are multiplied as if they were whole numbers. The decimal comma is placed in the product so that the number of decimal places in the product is equal to the sum of the decimal places in the factors.

Ex: Multiply 1,2 x 0,04 Think 12 x 4  12 x 4 = 48 1,2 has 1 decimal place 0,04 has 2 decimal places Therefore the product of 1,2 and 0,04 will have 1 + 2 = 3 decimal places 4 8 ,  1,2 x 0,04 = 0,048

Ex: Multiply 3,1 x 1,45 Think 31 x 145  31 x 145 =4495 3,1 has 1 decimal place 1,45 has 2 decimal places Therefore the product of 3,1 and 1,45 will have 1 + 2 = 3 decimal places 4 4 9 5 ,  3,1 x 1,45 = 4,495

Your turn to try a few