Unit 3 Decimal Fractions

DECIMAL FRACTIONS Written with a decimal point 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 HUNDRED MILLIONTHS

READING DECIMAL FRACTIONS To read a decimal, read number as whole number. Say name of place value of last digit to right. 0.567 is read “five hundred sixty-seven thousandths” To read a mixed decimal (a whole number and a decimal fraction), read whole number, read word and at decimal point, and read decimal. 45.00753 is read “forty-five and seven hundred fifty-three hundred thousandths”

ROUNDING DECIMAL FRACTIONS Rounding rules: Determine place value to which number is to be rounded Look at digit immediately to its right If digit is less than 5, drop it and all digits to its right If digit is 5 or more, add 1 to digit in place to which you are rounding. Then drop all digits to its right

ROUNDING EXAMPLES Round 14.763 to the nearest hundredth 6 is in the hundredths place value, so look at 3. Since 3 is less than 5, leave 6 alone and drop 3. Ans: 14.76 Round 0.0065789 to the nearest ten thousandth 5 is in the ten thousandths place value, so look at 7. Since 7 is greater than 5, raise 5 to 6 and drop all digits to its right. Ans: 0.0066

CONVERTING FRACTIONS TO DECIMALS Fractions can be converted to decimals by dividing the numerator by the denominator Express 5/8 as a decimal fraction: Ans Place a decimal point after the 5 and add zeros to the right of the decimal point. Bring the decimal point straight up in the answer. Divide. 8 20 16 40

CONVERTING DECIMALS TO 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 Change 0.015 to a common fraction: 0.015 is read as fifteen thousandths

ADDITION AND SUBTRACTION 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 Perform the following operations: 13.475 + 6.367 3.537 – 1.476 19.842 Ans 2.061 Ans

MULTIPLICATION Multiply decimals using same procedures as with whole numbers Count total number of digits to right of decimal points in both numbers being multiplied Begin counting from last digit on right in answer and place decimal point same number of places as there are total in both of the numbers being multiplied

MULTIPLICATION Multiply 62.4  1.73: Since 62.4 has 1 digit to right of decimal and 1.73 has two points to right of decimal, answer should have 3 digits to right of decimal point 62.4 1.73 1 872 43 68 62 4 107 952 = 107.952 Ans

DIVISION 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

DIVISION Divide 2.432 by 6.4: Move decimal point 1 place to right in 6.4 Move decimal point 1 place to right in 2.432 Place decimal point straight up in the answer Divide 2 5 12 .38 Ans

POWERS Product of two or more equal factors Appear slightly smaller Located above and to right of number being multiplied

The power 3 means to multiply .4 by itself 3 times POWERS Evaluate each of the following powers: .43 (2.5 × 3)2 The power 3 means to multiply .4 by itself 3 times .43 = .4 × .4 × .4 = .064 Ans Parentheses first: 2.5 × .3 = .75 (.75)2 = .75 × .75 = .5625 Ans

ROOTS A quantity that is taken two or more times as an equal factor of a number Finding a root is opposite operation of finding a power Radical symbol () is used to indicate root of a number Index indicates 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

FINDING ROOTS Determine the following roots: This means to find the number that can be multiplied by itself to equal 64. Since 8 × 8 = 64, the = 8 Ans This means to find the number that can be multiplied by itself three times to equal 27. Since 3 × 3 × 3 = 27, = 3 Ans Note: Roots that are not whole numbers can easily be computed using a calculator

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 Parentheses and grouping symbols (square root) first: (1.2)2 + 6 ÷ 2 Powers next: 1.44 + 6  2 Divide: 1.44 + 3 Add: 4.44 Ans

PRACTICE PROBLEMS Write the following numbers as words. a. 0.0027 Round 10.2364579 to each of the following place values: TENTHS HUNDREDTHS THOUSANDTHS TEN HUNDRED MILLIONTHS

PRACTICE PROBLEMS (Cont) Express each of the following as decimal fractions: 4. Express each of the following as fractions in lowest terms: a. 0.16 b. 0.1204 c. 0.635 5. Perform the indicated operations: a. 0.0027 + 0.249 + 0.47 b. 6.45 + 2.576 c. 3.672 – 1.569 d. 45.3 – 16.97

PRACTICE PROBLEMS (Cont) f. 25.63 × 3.46 g. 0.12  .4 h. 15.325  2.5 i. 0.33 j. (12.2 × .2)2

Solutions Writing Rounding Twenty-seven ten thousandths One hundred forty-three and forty-five hundredths One and seven thousand three hundred sixty-eight millionths Rounding 10.2 10.24 10.236 10.2365 10.23646 10.236458

Solutions Convert to decimal Decimal to Fraction 0.5 0.875 0.9375 a b

Solutions Order of operations 0.7217 9.026 2.103 28.33 4.158 88.6798 0.03 6.13 0.027 5.9536 4 5.32 3.82