PRESENTATION 6 Decimal Fractions

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PRESENTATION 6Decimal Fractions
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DECIMAL FRACTIONS

• A decimal fraction is written with a decimal
  point
• Decimals are equivalent to common fractions
  having denominators which are multiples of 10
• The chart below gives the place value for each
  digit in the number 1.234567

1 2 3 4 5 6 7
UNITS TENTHS HUNDREDTHS THOUSANDTHS TEN THOUSANDTHS HUNDRED THOUSANDTHS MILLIONTHS
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READING DECIMAL FRACTIONS

• To read a decimal fraction, read the number as a
  whole number
• Say the name of the decimal place of the last
  digit to right
• Example 0.532 is read five hundred thirty-two
  thousandths

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READING DECIMAL FRACTIONS

• To read a mixed decimal (a whole number and a
  decimal fraction), read the whole number, read
  the word and at the decimal point, and read the
  decimal
• Example 135.0787 is read one hundred
  thirty-five and seven hundred eighty-seven
  ten-thousandths

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ROUNDING DECIMAL FRACTIONS

• Rounding rules
• Determine the place value to which the number is
  to be rounded
• Look at the digit immediately to its right
• If the digit is less than 5, drop it and all
  digits to its right
• If the digit is 5 or more, add 1 to the digit in
  the place to which you are rounding, then drop
  all digits to its right
• Note The sign means approximately equal to

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ROUNDING DECIMAL FRACTIONS

• Example In determining rivet hole locations, a
  sheet metal technician computes a dimension of
  1.5038 inches. 0.01 precision is needed for
  laying out the hole locations. Round the
  dimension to two decimal places.
• Locate the digit in the second decimal place (0)
• The third-decimal place digit, 3, is less than 5
  and does not change the value, 0
• Therefore, 1.5038 inches rounds to 1.50 inches
• 1.5038 inches 1.50 inches

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EXPRESSING FRACTIONS AS DECIMALS

• Fractions can be converted to decimals by
  dividing the numerator by the denominator
• Example Express 3/8 as a decimal fraction
• Place a decimal point after the 3 and add zeroes
  to the right of the decimal point
• Place the decimal point for the answer directly
  above the decimal point in the dividend. Divide.
• The fraction 3/8 equals 0.375

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EXPRESSING DECIMALS AS FRACTIONS

• To change a decimal to a fraction, use the number
  as the numerator and the place value of the last
  digit as the denominator
• Example Change 0.065 to a common fraction
• The number 0.065 is read as sixty-five
  thousandths
• Write the denominator as 1,000 and 65 as the
  numerator

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ADDITION AND SUBTRACTION OF DECIMALS

• To add and subtract decimals, arrange numbers so
  that decimal points are directly under each other
• Add or subtract as with whole numbers
• Place decimal point in answer directly under the
  other decimal points

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ADDITION AND SUBTRACTION OF DECIMALS

• ExampleAdd 8.75 231.062 0.7398 0.007 23
• Arrange the numbers so that the decimal points
  are directly under each other
• Add zeroes so that all numbers have the same
  number of places to the right of the decimal
  point

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ADDITION AND SUBTRACTION OF DECIMALS

• Add each column of numbers
• Place the decimal point in the sum directly under
  the other decimal points

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ADDITION AND SUBTRACTION OF DECIMALS

• Example Subtract 44.6 27.368
• Arrange the numbers so that the decimal points
  are directly under each other
• Add zeroes so that the numbers have the same
  number of places to the right of the decimal point

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ADDITION AND SUBTRACTION OF DECIMALS

• Subtract each column of numbers
• Place the decimal point in the difference
  directly under the other decimal points

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MULTIPLYING DECIMALS

• Multiply decimals using the same procedure as
  with whole numbers
• Count the number of decimal places in both the
  multiplier and multiplicand
• Begin counting from the last digit on the right
  of the product and place the decimal point the
  same number of places as there are in both the
  multiplicand and multiplier

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MULTIPLYING DECIMALS

• Example Multiply 60.412 ? 0.53
• Align the numbers on the right
• Multiply as with whole numbers

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MULTIPLYING DECIMALS

• Since 60.412 has 3 digit to the right of the
  decimal and 0.53 has 2 digits to the right of the
  decimal, the answer should have 5 digits to the
  right of decimal point
• Move the decimal point 5 places from the right

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DIVIDING DECIMALS

• Divide using the same procedure as with whole
  numbers
• Move the decimal point of the divisor as many
  places as necessary to make it a whole number
• Move the decimal point in the dividend the same
  number of places to the right
• Divide and place the decimal point in the answer
  directly above the decimal point in the dividend

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DIVIDING DECIMALS

• Example Divide 0.3380 by 0.52
• Move decimal point 2 places to the right in the
  divisor
• Move the decimal point 2 places to the right in
  the dividend
• Place decimal point in the quotient directly
  above the dividend and divide

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DIVIDING BY POWERS OF 10

• Since division is the inverse of multiplication,
  dividing by 10 is the same as multiplying by
  or 0.1
• Dividing a number by 10, 100, 1,000, and so on is
  the same as multiplying by 0.1, 0.01, 0.001
• To divide by 10, 100, 1,000, move the decimal
  point in the dividend as many places to the left
  as there are zeroes in the divisor

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POWERS OF DECIMALS

• Two or more numbers multiplied to produce a given
  number are factors of the given number
• A power is the product of two or more equal
  factors
• An exponent shows how many times a number is
  taken as a factor. It is smaller than the number,
  above the number, and to the right of the number

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POWERS OF DECIMALS

• Example Evaluate 0.83
• The power 3 means to multiply 0.8 by itself 3
  times
• It is read 0.8 to the third power or 0.8
  cubed
• 0.8 0.8 0.8 0.512

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POWERS OF DECIMALS

• Example Evaluate (1.4 0.3)2
• Perform the operation in parentheses first
• 1.4 0.3 0.42
• Raise the result to the power of 2
• 0.42 0.42 0.1764

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ROOTS

• The root of a number is a quantity that is taken
  two or more times as an equal factor of a number
• Finding a root is the opposite or inverse
  operation of finding a power
• The radical symbol (?) is used to indicate the
  root of a number
• Index indicates the number of times a root is to
  be taken as an equal factor to produce the given
  number
• Note Index 2 for square root is usually omitted

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ROOTS

• Example Evaluate
• This means to find the number that can be
  multiplied by itself to equal 144
• Since 12 12 144, the is 12
• Example Evaluate
• This means to find the number that can be
  multiplied by itself three times to equal 125
• Since 5 5 5 125, the is 5

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ORDER OF OPERATIONS

• Order of operations including powers and roots
  is
• Parentheses
• Fraction bar and radical symbol are used as
  grouping symbols
• For parentheses within parentheses, do innermost
  parentheses first
• Powers and Roots
• Multiplication and division from left to right
• Addition and subtraction from left to right

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ORDER OF OPERATIONS

• Example
• Multiply
• 8.14 3.6 x 0.8 1.37 8.14 2.88 1.37
• Add
• 8.14 2.88 1.37 11.02 1.37
• Subtract
• 11.02 1.37 9.65

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PRACTICAL PROBLEMS

• A certain 6-cylinder automobile engine produces
  1.07 brake horsepower for each cubic inch of
  piston displacement
• Each piston displaces 28.94 cubic inches
• Find the total brake horsepower of the engine to
  the nearest whole horsepower

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PRACTICAL PROBLEMS

• Determine the total number of cubic inches for
  the 6 cylinders
• Determine the total horsepower
• The total horsepower is 186