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Unit 3
Decimal Fractions
2
1 • 2 3 4 5 6 7
UNITS TENTHS HUNDREDTHS THOUSANDTHS TEN
THOUSANDTHS
HUNDRED
THOUSANDTHS
MILLIONTHS
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
3
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”
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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
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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
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CONVERTING FRACTIONS TO DECIMALS
Fractions can be converted to decimals by dividing the numerator by the denominator
Express 5/8 as a decimal fraction:
• 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.
62500058..
Ans
8 20 16 40 40
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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
Ans200
3
1000
15
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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
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Perform the following operations:
13.475 + 6.367
19.842 Ans
3.537 – 1.476
2.061 Ans
ADDITION AND SUBTRACTION
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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
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MULTIPLICATION
• 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
Multiply 62.4 1.73:
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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
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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
32.2464 2 5 12 5 12
.38 Ans
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POWERS
Product of two or more equal factors
Appear slightly smaller
Located above and to right of number being multiplied
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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
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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
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FINDING ROOTS
• Determine the following roots:
64– This means to find the number that can be multiplied by
itself to equal 64. Since 8 × 8 = 64, the = 8 Ans 3 27
– This means to find the number that can be multiplied by itself three times to equal 27. Since 3 × 3 × 3 = 27, = 3 Ans 3 27
Note: Roots that are not whole numbers can easily be computed using a calculator
64
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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
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
236)24.2(Solve 2
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PRACTICE PROBLEMS
1. Write the following numbers as words. a. 0.0027 b. 143.45 c. 1.0073682. Round 10.2364579 to each of the
following place values: TENTHS HUNDREDTHS THOUSANDTH
STEN
THOUSANDTHS
HUNDRED
THOUSANDTHS
MILLIONTHS
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PRACTICE PROBLEMS (Cont)
3. 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. 0.6355. 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
16
15.c
8
7.b
2
1.a
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PRACTICE PROBLEMS (Cont)
e. 1.54 × 2.7f. 25.63 × 3.46g. 0.12 .4h. 15.325 2.5i. 0.33
j. (12.2 × .2)2
2)6(.28.m
22.14.164.4.l64.k
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2
3
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Solutions
1. Writinga. Twenty-seven ten
thousandths
b. One hundred forty-three and forty-five hundredths
c. One and seven thousand three hundred sixty-eight millionths
2. Rounding1. 10.2
2. 10.24
3. 10.236
4. 10.2365
5. 10.23646
6. 10.236458
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Solutions
3. Convert to decimala. 0.5
b. 0.875
c. 0.9375
4. Decimal to Fractiona. a
b. b
c. c
25
4
2500
301
200
127
25
Solutions
5. Order of operationsa. 0.7217
b. 9.026
c. 2.103
d. 28.33
e. 4.158
f. 88.6798
g. 0.03
h. 6.13
i. 0.027
j. 5.9536
k. 4
l. 5.32
m.3.82