PRESENTATION 6Decimal Fractions

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

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”

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”

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”

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

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

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

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

ADDITION AND SUBTRACTION OF DECIMALS

•Example:Add 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

ADDITION AND SUBTRACTION OF DECIMALS

•Add each column of numbers•Place the decimal point in the sum

directly under the other decimal points

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

ADDITION AND SUBTRACTION OF DECIMALS

•Subtract each column of numbers•Place the decimal point in the

difference directly under the other decimal points

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

MULTIPLYING DECIMALS

•Example: Multiply 60.412 0.53

•Align the numbers on the right

•Multiply as with whole numbers

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

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

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

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

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

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

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

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

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

144

144

3 125

3 125

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

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

Solve 8.14 3.6 0.8 1.37

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

PRACTICAL PROBLEMS

•Determine the total number of cubic inches for the 6 cylinders

•Determine the total horsepower

•The total horsepower is 186