Lesson 5.1 Skills Practice
Name_________________________________________________________ Date__________________________
Is It a Bird or a Plane?
Rational Numbers
Vocabulary
Define each set of numbers in your own words.
1. Natural numbers Natural numbers consists of the numbers that you use to count objects.
2. Whole numbers Whole numbers are made up of the set of natural numbers and the number 0.
3. Integers Integers include all of the whole numbers and their additive inverses.
4. Closed A set of numbers is said to be closed when the operations could produce a defined
value that is also in the set.
5. Rational numbers Rational numbers are numbers that can be written in the form __
​ a ​, where a and
b
b are both integers and b is not equal to 0.
Problem Set
© 2011 Carnegie Learning
Write each fraction as a decimal.
1. __
​ 1 ​ 5 0.25
4
2. __
​ 5 ​ 5 0.625
8
23  ​ 5 0.23
3. ​ ____
100
2 ​ 5 1.4
4. 1​ __
5
7  ​ 5 3.4375
5. 3​ ___
16
15 ​ 5 12.46875
6. 12​ ___
32
Graph each pair of rational numbers on the number line. Use the graph and write ., ,, or 5 to
compare the numbers.
3  ​ . 0.25
7. ​ __
5
3
5
0.25
0
0.1
0.2 0.3
0.4
0.5
5  ​
8. 1.7 . 1​ __
8
0.6 0.7
1
1.0 1.1 1.2
1.3 1.4
1.5
0.8 0.9
1.0
1.8
2.0
5
1.7
8
1.6
1.7
1.9
Chapter 5 Skills Practice • 523
Lesson 5.1 Skills Practice
9. 0.09 , ___
​  9  ​ 
10
page 2
9
10
0.09
0
0.1
0.2 0.3
3 ​  . 4.34
10. 4​ __
4
0.4
0.5
0.6 0.7
4
4.34
4.0 4.1 4.2
4.3 4.4
4.5 4.6
11. ___
​ 11 ​ . 0.65
16
0.8 0.9
1.0
3
4
4.7 4.8
4.9 5.0
11
0.65 16
0
0.1
0.2 0.3
0.4
0.5
0.6 0.7
0.8 0.9
1.0
4
5
3.8
3
4  ​
12. 3.8 5 3​ __
5
3.0 3.1 3.2
3.3 3.4
3.5 3.6
3.7 3.8
3.9 4.0
Tell whether the numbers in each problem are natural numbers, whole numbers, or integers, and
identify whether those numbers are closed or open under the operation used.
13. 23 1 18 5 41
natural numbers, whole numbers, or integers; closed under addition
whole numbers or integers; closed under multiplication
15. 4 2 (23) 5 7
integers; closed under subtraction
16. 0 2 5 5 25
whole numbers are not closed under subtraction; closed under subtraction as integers
524 • Chapter 5 Skills Practice
© 2011 Carnegie Learning
14. 35 3 0 5 0
Lesson 5.1 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
17. 9 2 11 5 22
natural numbers or whole numbers are not closed under subtraction; closed under subtraction
as integers
2 ​ 
18. 212 4 (25) 5 2​ __
5
integers; not closed under division
© 2011 Carnegie Learning
Add, subtract, multiply, or divide the rational numbers in each.
19. 27 1 13 5 6
1  ​ 1 __
​ 3 ​ 5 ___
​  7   ​
20. ​ ___
10 5 10
3 ​ 5 2​__
1 ​ 2 2​ __
 1 ​
21. 4​ __
2
8
8
22. 21.44 4 0.12 5 212
23. 2.40 3 (23.75) 5 29
7
6  ​3 ​2___
24. ​ __
​    ​  ​5 2​___
 42  ​5 2​___
 7   ​
12
5
60
10
​ 1 ​ 5 ___
 2 ​
​ 20 ​ 5 6​__
25. __
​ 5 ​ 4 __
3 4
3
3
26. 12.7 2 39.6 5 226.9
27. 27.82 2 (25.09) 5 22.73
32
4
​   ​  ​5 __
​ 3 ​
28. 2​ __  ​4 ​2___
9
27
8
(  )
(  )
Chapter 5 Skills Practice • 525
© 2011 Carnegie Learning
526 • Chapter 5 Skills Practice
Lesson 5.2 Skills Practice
Name_________________________________________________________ Date__________________________
Sew What?
Irrational Numbers
Vocabulary
Match each term with the number that represents that term.
1 ​ 5 0.5
a. ​ __
2
1. Irrational number
c
__
2. Terminating decimal
b. 0.​3 ​ 
a
3. Repeating decimal
c. π
d
5 ​  5 0.555…
d. ​ __
9
4. Bar notation
b
Problem Set
Convert each fraction to a decimal and tell whether the fraction results in a terminating or repeating
decimal.
2. __
​ 7 ​ 
9
© 2011 Carnegie Learning
1. ___
​ 1  ​ 
25
0.04
_____
)
25​1.00 ​
 ;
 
5  ​ 
3. ​ ___
12
terminating __
0.416​6​ 
_______
12​)5.0000 ​
 
; repeating
 
13 ​ 
5. ​ ___
16
0.8125
________
16​)13.0000 ​
 
; terminating
 
__
0.7​7​ 
_____
)
9​7.00 ​
 ;
 
repeating
4. __
​ 5 ​ 
8
0.625
______
8​)5.000 ​
 
; terminating
 
6. ___
​ 8  ​ 
11
___
0.​72 ​ 
_____
11​)8.00 ​
 ;
 
repeating
Chapter 5 Skills Practice • 527
Lesson 5.2 Skills Practice
page 2
Write each repeating decimal as a fraction.
8. 0.888 …
10w 5 3.33...
10w 5 8.88...
2w 5 0.33...
2w 5 0.88...
9w 5 3
9w 5 8
​ 1 ​ 
w 5 __
​ 3 ​ 5 __
9 3
9. 0.0707 …
100w 5 7.07...
2w 5 0.07...
99w 5 7
w 5 ___
​  7  ​ 
99
11. 0.1515 …
8  ​
w 5 ​ __
9
10. 0.5454 …
100w 5 54.54...
2w 5 0.54...
99w 5 54
w 5 ___
​ 54 ​ 5 ___
​  6  ​ 
99 11
12. 0.2727 …
100w 5 15.15... 100w 5 27.27...
2w 5 0.15...
2w 5 0.27...
99w 5 15
​ 5  ​ 
w 5 ___
​ 15 ​ 5 ___
99 33
528 • Chapter 5 Skills Practice
99w 5 27
w 5 ___
​ 27 ​ 5 ___
​  3  ​ 
99 11
© 2011 Carnegie Learning
7. 0.333 …
Lesson 5.2 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
13. 0.298298 …
14. 0.185185 …
1000w 5 298.298...
1000w 5 185.185...
2w 5 0.298...
2w 5 0.185...
999w 5 298
 ​
w 5 ____
​ 298 
999
15. 0.67896789…
10,000w 5 6789.6789...
2w 5 0.6789...
9999w 5 6789
w 5 ____
​ 185 
 ​5 ___
​  5  ​ 
999 27
16. 0.0243902439 …
100,000w 5 2439.02439...
2w 5 0.02439...
99,999w 5 2439
2439 
w 5 ​ _______
 ​ 
5 ___
​  1  ​ 
99,999 41
© 2011 Carnegie Learning
 ​5 _____
​ 2263 ​ 
w 5 _____
​ 6789 
9999 3333
999w 5 185
Chapter 5 Skills Practice • 529
Lesson 5.2 Skills Practice
page 4
Tell whether each number is rational or irrational.
17. π
irrational
___
19. √
​ 18 ​ 
irrational
__
18. ​√4 ​ 
rational (2)
 
20. 
​3 27 ​
rational (3)
__
3
   
21.
​ 30 ​
√
​ 49 ​ 
(  )
​ 1  ​  ​
rational ​__
7
© 2011 Carnegie Learning
irrational
√1 ​ 
22. ____
​  ​___
  
530 • Chapter 5 Skills Practice
Lesson 5.3 Skills Practice
Name_________________________________________________________ Date__________________________
Worth 1000 Words
Real Numbers and Their Properties
Vocabulary
Write a description of each term in your own words.
1. real numbers: The set of real numbers is the combined set of rational numbers and irrational
numbers.
2. Venn diagram: A Venn diagram is a diagram used to show how sets overlap.
3. closed (under an operation): A set of numbers is closed under an operation if the result of the
operation on two numbers in the set is another member of the set.
Problem Set
Indicate whether each real number shown is rational, irrational, an integer, a whole number, a natural
number, or some combination.
1. 35
rational, integer, whole
___
2. ​√17 ​ 
irrational
number, natural number
3. 26
© 2011 Carnegie Learning
rational, integer
___
5.​√ 81  ​
rational, integer, whole
4. 5.25
rational
2
6. 2​ __ ​
3
rational
number, natural number
Chapter 5 Skills Practice • 531
Lesson 5.3 Skills Practice
7. ___
​ 14 ​ 
5
page 2
0  ​
8. ​ __
7
rational
rational, integer, whole number
___
3
 
9.
​ 9 ​
√
​ 16 ​ 
10. 2​ ____
 
3
 
​ 8 ​

irrational
rational, integer
Identify the property represented in each problem.
11. 18 3 1 5 18
multiplicative identity
13. 174 1 (2174) 5 0
additive inverse
15. 213 3 21 5 21 3 (213)
commutative property of
12. 87 1 (259) 5 259 1 87
commutative property of addition
(  )
(  )
1 ​1 __
​ 1 ​​5 1 then __
​ 1 ​1 __
​ 1 ​​
​ 1 ​5 1 and 4​__
​ 1 ​5 4​__
14. If ​ __
2 2
4
2 2
4
transitive property of equality
16. 2365 1 0 5 2365
additive identity
multiplication
19. 567 5 567
reflexive property of equality
21. [5 3 (23)] 3 12 5 5 3 [(23) 3 12]
associative property of multiplication
532 • Chapter 5 Skills Practice
18. 245 1 (232 1 87) 5 [245 1 (232)] 1 87
associative property of addition
3
3
20. If 2​ __  ​5 20.75 then 20.75 5 2​ __ ​ .
4
4
symmetric property of equality
4
22. 2​ __  ​1 0.8 5 0
5
additive inverse
© 2011 Carnegie Learning
1 ​ 3 0.3 5 1
17. 3​ __
3
multiplicative inverse