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LAHPA11FLCRB_c05_225-234.qxd
LESSON
1/23/09
1:52 PM
Page 229
Name
Date
5.1 Review for Mastery
For use with pages 221–226
GOAL
Write fractions as decimals and vice versa.
VOCABULARY
EXAMPLE
Lesson 5.1
A rational number is a number that can be written as a quotient of
two integers. In a terminating decimal, the division ends because
you obtain a final remainder of zero. In a repeating decimal, a digit
or block of digits in the quotient repeats without end.
1 Identifying Rational Numbers
Show that the number is rational by writing it as a quotient of two integers.
2
7
5
12
c. 8 b. 3 a. 9
Solution
9
1
a. Write the integer 9 as .
41
12
5
12
b. Write the mixed number 3 as the improper fraction .
2
7
2
7
2
7
58
7
c. Think of 8 as the opposite of 8 . First write 8 as . Then you can
2
7
58
7
58
7
write 8 as . To write as a quotient of two integers, you can
assign the negative sign to either the numerator or the denominator. You can
58
7
58
7
write or .
Exercises for Example 1
Show that the number is rational by writing it as a quotient of two integers.
1. 84
EXAMPLE
2. 12
8
17
2
9
4. 5 3. 2 2 Writing Fractions as Decimals
7
8
a. Write as a decimal.
Solution
a. 0.875
87.0
00
64
60
56
40
40
0
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4
11
b. Write as a decimal.
b.
The remainder is 0, so
the decimal is a
terminating decimal:
7
0.875.
8
0.3636. . .
114.0
000
33
70
66
40
33
70
Use a bar to show the
repeating digits in the
repeating decimal:
4
0.36
ww.
11
Pre-Algebra
Chapter 5 Resource Book
229
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5.1 Review for Mastery
Continued
For use with pages 221–226
Exercises for Example 2
Write the fraction or mixed number as a decimal.
4
9
4
5
Lesson 5.1
5. EXAMPLE
8
11
8. 3 Writing Terminating Decimals as Fractions
9
10
a. 0.9 31
100
b. 0.31 EXAMPLE
9
16
7. 2 6. 1 Place value of 9 is tenths, so denominator is 10.
Place value of 1 is hundredths, so denominator is 100.
4 Writing a Repeating Decimal as a Fraction
To write 0.45
ww as a fraction, let x 0.45
ww.
ww has 2 repeating digits, multiply each side of x 0.45
ww by 102,
(1) Because 0.45
or 100. Then 100x 45.45
ww.
ww
100x 45.45
(2) Subtract x from 100x.
(x 0.45
ww )
99x 45
(3) Solve for x and simplify.
99x
45
99
99
5
11
x 5
11
Answer: The decimal 0.45
ww is equivalent to the fraction .
Exercises for Examples 3 and 4
Write the decimal as a fraction or mixed number.
9. 0.25
230
Pre-Algebra
Chapter 5 Resource Book
10. 0.32
ww
11. 3.1w
12. 7.325
Copyright © Holt McDougal.
All rights reserved.
FOCUS
ON
Name
Date
5.1 Review for Mastery
For use with pages 227–228
GOAL
Determine whether mathematical statements are true or false.
VOCABULARY
EXAMPLE
5.1 Focus On Reasoning
When you make a conclusion based on several examples, you are
using inductive reasoning.
A conclusion reached using inductive reasoning is a conjecture. A
conjecture is a statement that is thought to be true but not yet shown
to be true.
The process of starting with one or more given facts and using rules,
definitions, or properties to reach a conclusion is called deductive
reasoning.
1 Using Inductive Reasoning
Consider fractions whose denominators are powers of 3, such as 3, 9, and 27.
Make a conjecture about the decimal form of such fractions.
Solution
Find a pattern using a few examples.
⎯
⎯
1
1
0.3
0.1
3
9
⎯⎯⎯
1
0.037
27
Conjecture: The decimal form of any fraction whose denominator is a power of
3 is a repeating decimal.
EXAMPLE
2 Using Deductive Reasoning
Give a convincing argument to show that the conjecture in Example 1 is true.
Solution
The decimal system is based on powers of 10, and the decimal form of any
fraction whose denominator is a power of 10 is a terminating decimal. A fraction
whose denominator is a power of 3 cannot be rewritten as a fraction whose
denominator is a power of 10. Powers of 10 and powers of 3 do not share
common factors. So, no power of 10 is divisible by a power of 3. Thus, the
decimal form of any fraction whose denominator is a power of 3 is a repeating
decimal.
1
729
1
3
For instance, because 6 , its decimal form is repeating.
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Pre-Algebra
Chapter 5 Resource Book
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FOCUS
ON
1/23/09
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Date
5.1 Review for Mastery
Continued
For use with pages 227–228
Exercise for Examples 1 and 2
5.1 Focus On Reasoning
1. Make a conjecture about the sum of two odd integers. Give
a convincing argument to show that the conjecture is true.
EXAMPLE
3 Using a Counterexample
Show that a conjecture is false by finding a counterexample.
Conjecture: The decimal form of any fraction whose denominator is a prime
number is a repeating decimal.
Solution
Write fractions whose denominators are prime as decimals.
⎯
1
1
0.3
0.2
3
5
Because a counterexample exists, the conjecture is false.
Exercises for Example 3
Show that the conjecture is false by finding a counterexample.
2. All even numbers are composite.
3. The difference of any two positive numbers is positive.
4. All powers of 3 are divisible by 9.
234
Pre-Algebra
Chapter 5 Resource Book
Copyright © Holt McDougal.
All rights reserved.
Answers
Answers
36. Decimal form. Using fractions, you would
Lesson 5.1
have to change each fraction to an equivalent form
with the same denominator before you could
compare the values. Using decimals, you can just
compare place values.
Practice A
1. repeating decimal
2. terminating decimal
3. terminating decimal 4. repeating decimal
38
5
7
14
38
14
5. or 6. 7. 8. or 9
1
3
9
1
1
45
9
58
1
58
9. 10. or 11. or 1
100
20
1
10
9
9
12. or 13. 0.75 14. 0.8w 15. 7.6
14
14
16. 2.16w
11
25. 5 2
0
99
7
11
32. 27. 9
368
4. 15
24. 1 100
8
28. 33
29. >
1
1
33. 0.02, 5, 0.25, 2, 2 2
7
34. 3, 3.6, 2, 3, 0.3
w
2
5
7
Practice B
19.
23.
27.
30.
31.
32.
34.
A32
6. 0.516
7. 8.28
w
10. 64.75
11. 5.84
ww
12. 17.325
13. 9.2w
14. 0.897
www
15. 10.111
16. 7.108
w
17. 200
18. 1 200
101
18
1
41
19. 143 9
31
21. 400
20. 99
22. 6 125
1
428
24. 9 495
23. 990
111
157
9 5 8
25. 2, 2 1
, , , , 2.8
0 4 2 3
1
11
26. 1
, 0, 0.19, 4, 0.25
ww, 4
0
0
22
7
25
3
27. 3, 6 8, 6.34, 4, 6, 5 4
28. 10
29. 2.5
30. 6
12. 8.15
16. 2.225
31. Sample answer: 2
5
69
10
121
32. Sample answer: 100
63
555
555
6. or 1
1
33. 0.8631; 8
; 0.8631 is closer to 1.
1
0.32 10. 0.27
w
Review for Mastery
13. 11.3
w
14. 7.25
17. 0.38
w
18. 1.5
1
5.93w 20. 10.194
w
22. 2
5
4
1
7
16
25. 8 26. 5 25 24. 14 9
200
99
23
27
6
2
15 28. 3 30 29. 0.5
w, 0.55, 11 , 50
1
13 17
, 0.6, , , 0.69
2
20 25
11
7
8
6, 4, 1.7, 5, 1.5, 0.8
w
1 17 43
4.02, 4.1, 4 5, 4, 10, 4.41 33. Monday
7
2
0.28, terminating; 0.4, terminating;
25
5
1
0.07, terminating; 12 0.083
w, repeating;
7
100
1
6
133
133
3. or 9
9
92
86
273
86
1. 2. or 3.
1
1
1
3400
109
109
4. or 5. 1
12
12
47
47
67
7. or 8. 9.
20
20
14
15. 6.37
9. 21.2
1
35. 3 0.6
w, 8 0.625, 10 0.7; Sam lives the
farthest from school. 36. chemistry book
11. 3.18
ww
5. 0.7625
8. 3.483
w
43
4
23. 3 5
100
100
2. or 1
1
3691
1. 1
19. 1.53
w
22. 50
26. 100
31. <
18. 2.5
19
3
21. 2
0
20. 4.05
30. >
17. 3.6
Practice C
13
21. 2
0
0.16
w, repeating 35. banana
Pre-Algebra
Chapter 5 Resource Book
84
84
1. or 1
1
5. 0.4
w
1
9. 4
6. 1.8
32
12
2. 1
42
3. 17
7. 2.5625
1
10. 9
9
11. 3 9
47
47
4. or 9
9
8. 0.72
ww
13
12. 7 4
0
Challenge Practice
8
1. 9
54
2. 2 9
9
1.13w, 1.03
ww, 1.3
6. >
7. <
901
4
4. , 1.30
3. 6 ww,
999
3
5. 0.06, 0.06
w, 0.60
ww, 0.606
www, 0.6
w
8. <
2
9. Sample answer: 3
234
10. Sample answer: 100
Focus On Reasoning 5.1
Practice
1. The product of two prime numbers is a
composite number; A composite number is
divisible by at least one number other than itself
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All rights reserved.
Lesson 5.1 continued
5. 1 centimeter = 100 millimeters. To a convert a
length in centimeters to a length in millimeters, the
centimeter length is multiplied by 100. Any whole
number multiplied by 100 is a whole number.
(x 1) = 7 and (x 3) = 9. Their sum is 2x 4,
or 16, which is an even integer.
2. 2
3. 1 2 1
4. 30 9 0.1
Lesson 5.2
Practice A
1. 12
2. 4
1
7
8. 3
4
4
12. 1
1
1
1
4
Practice B
11
7. 2 3 5
6. 25
4
13
2. 5
1
3. 1
6
21
3
2
5
11. 4 3
12. 11 9
5
15. 2 16. 5 17.
8
7
20. x 21. 22.
x
5
3
8. 40
7. 5
3
2
5. 15
4. 1 4
3
1
10. 1 7
9. 12 5
3
6
13. 8 1
14. 4 5
1
4
6x
5x
18. 19. 7x
5
7
1
4
3
23. 24. 12x
3x
x
1
27. 14
1
31. 3 1
4 12 7 12 7 3;
2
26. 1 8
1
4
28. 1 7
12. (3)3 27
25. 1 1
2
13. 27 6 4.5
29. 1 9
14. 1
because 7 3 < 7 4, you have enough wood.
15. (1)5 1
32. 2 sec faster
16. no; 1 0 1 0
Review for Mastery
1. The sum of two odd integers is an even
integer; For any even integer x, the integers (x 1)
and (x 3) are consecutive odd integers. The sum
of (x 1) and (x 3) = (2x 4). The integer
(2x 4) is even: 2x is even because any multiple
of 2 is even; adding 4 is the same as adding 2 2,
so 2x 4 is a multiple of 2. For instance, if x 6,
Copyright © Holt McDougal.
All rights reserved.
5
7. 7
14. 4 2 15. 5 16. 10 7 17. 5 3
x
1
2x
18. 9 19. 5 20. 1 21. 22. 2
5
5
x
5
4x
10
4
23. 24. 25. 26. 27. 5
3x
9
x
x
15
3
1
28. 29. ft 30. 4 mi 31. 1 mi
8x
4
2
13. 8 5
6. (2)(2) 4
11. 3 p 1 3
3
6. 5
11. 1 1
3
10. 1 4
1
1
5. 2
4. 4
3
9. 8
1. 1 13
8. 10 4 2.5
2
9. 2
10. 3 p 4 12
3. 7
4
30. 4
2
7
1
8
2
3
1
3
33. 3 1
lb
6
Practice C
12
2
1
3. 1 4.
2. 125
29
1
13
14
6. 51 7. 8. 33 5
50
33
4x
4
11. 0 12. 13. 14.
19
3x
1. 5
2
25
18. 28
17. 5
16. 0
1
21. 5 5
4
22. 9
2
32
125
9.
3
2x
13
19. 40
8
19
5. 4 25
x
2x
10. 10
3
17
15. 5x
29
20. 33
2
24. 1
25. 7
3
Pre-Algebra
A33
Chapter 5 Resource Book
23. 5
Answers
and 1. Suppose p is the product of two prime
numbers m and n. Besides being divisible by itself
and 1, p must also be divisible by m and n.
2. The product of two even numbers is always
even; The product of an even number and any
number will contain 2 as part of its prime
factorization and thus be divisible by 2, or even.
3. The factors of an odd product must both be
odd; Every even number is a multiple of 2. The
product of an even number and any number will
contain 2 as part of its prime factorization and thus
be divisible by 2, or even. So, the factors of an odd
product must both be odd.
1 2 3
4
4. The decimal forms of , , , and form
8 8 8
8
an arithmetic sequence with a common difference
of 0.125. This is because the decimal form of 18 is
0.125, and each successive number in the
sequence is 18 more than the number before it.