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

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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

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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