DOCUMENT RESUME

ED 327 706 CE 086 768

AUTHOR Engelbrecht, Nancy; And Others

TITLE Fractions and Conversion to Decimal. Fundamentals of
Occupational Mathematics. Module 7.
INSTITUTION Central Community Coll., Grand Island, NE.
SPONS AGENCY Office of Vocational and Adult Education (ED),
Washington, DC.
PUB DATE 90
CONTRACT V199A90067
NOTE 17p.; For related modules, see CE 056 762-773.
PUB TYPE Guides Classroom Use Materials (For Learner)
(051)

EDRS PRICE MF01/PC01 Plus Postage.

DESCRIPTORS *Arithmetic; *Calculators; Community Colleges;
Decimal Fractions; *Fractions; Individualized
Instruction; Learning Modules; *Mathematical
Applications; *Mathematics Instruction; Measurement;
Number Concepts; Pacing; Two Year Colleges;
Vocational Education
IDENTIFIERS *Job Related Mathematics

ABSTRACT
This module is the seventh in a series of 12 learning
modules designed to teach occupational mathematics. Blocks of
informative material and rules are followed by examples and practice
problems. The solutions to the practice problems are found at the end
of the module. Specific topics covered include fractions, fraction to
decimal conversion, reducing and raising fractions, measuremert,
mixed numbers, and improper fractions. (YLB)

********************A*************************************************
* Reproductions supplied by EDRS are the best that can be made *

* from the original document. *

********2**************************************************************
 Project Director
Ron Vorderstrasse

Project So:Mary
Jan Wlslakrriskl

Toohnkal Coneuftant
Ray Pfankkiton

Technke: Writers
Nancy Engelbreck
Lyrre Graf
Ann Hunter
Stacey Oakes

4:Copyrigh4 Central Comnerity College

 1

Module 7 Fractions and Conversion To Decimal

A fraction is a number of the form 121 used to indicate that some whole thing has been divided into
n
n equal parts. The m of the fraction shows how many of the equal parts are being considered. The
fraction 7
32
suggests that 7 of 32 equal parts are under consideration. A fraction like 12 is a single
16
number which is written using two whole numbers and a separation line. The separation line is called
the fraction bar. This fraction bar serves as a grouping symbol to separate the two whole numbers.
The whole number below the fraction bar is called the DENOMINATOR of the fraction. The
denominator tells how many parts a whole thing has been divided into. The whole number above the
fraction bar is called the NUMERATOR of the fraction. The numerator reports how many parts were
counted for consideration.

m numerator
Fraction fraction bar
n denominator

PROPER fractions have numerators which are less than their denominator.

Two fractions which have the same value are called equal fractions. Equal fractions often look dif-
ferent because of unequal denominators. A quick method of testing two fractions to determ;ne if they
are equal is to use cross-multiplication.

3 12 3 12
To test -ti and 71-6- for equal value, consider the cross-multiplication 7t Because
16'
the values (3)(16) = 48 and (4)(12) = 48 are equal, the fractions are Nual: 3 12

The values of two fractions can be compared using the following rule.

Rule for Comparing Two Fractions

c
When comparing the two fractions -4 and -ii compute the cross-multipHcations
b '
a c
b d
1) If ad = bc, then they are equal fractions. II
c
d
2) If ad is less than bc, then the fractions are not equal

Follow the use of numerator a to determine how to compara the values of the fractions.
The numerator a was used in the cross-product ad, resulting in the smaller of the two
cross-products. The numerator a is then in the smaller fraction. Therefore, 1-) is less
c a c
than -a and this is written as -6 <

4
 2

3) If ad is greater than bc, then the fractions are not equal.

Here the numerator a is part of the greater cross- product ad and a will be the
numerator of the larger fraction. When -: is greater than fi , the comparison

is written b
a
< ;
c
a

EXAMPLE 1: Compare the values of the following pairs of fractions. Describe

their relative size using one of the symbols =, < , or > .

1 8
a)
4 32

7 23
'I' 16 ' 64

3 13
c)
8 32

d) L
64
1

'
5
-8-

Solution:

(a) The cross-products are (1)(32) and (4)(8).

(1)(32) = 32 and (4)(8) = 32.
Since the cross-products are equal, the fractions are equal.
1 8
4 32
Solution:

(b) The cross-products are (7)(64) and (16)(23).

(7)(64) = 448 and (16)(23) = 368.
The cross-products are not equal.
The numerator 7 is used in the larger cross-product (7)(64)=448 and
is the numerator of the largest fraction.
7 23
16 > 54

Solution:

(c) The cross-products are (3)(32) and (8)(13).

(3)(32) = )6 and (fo)(13) = 104
The cross-products are not equal.
The numerator 3, used in 3(32) = 96 produced the smaller
cross-product and is from the smallest fraction.
3 13
8 < 32
 3

Solution:
(d) The cross-products are (41)(8) and (64)(5).
(41)(8) = 328 and (64)(5) = 320
The fractions are not equal.
The 41 numerator is part of the larger cross-product and is the
numerator of the larger fraction.
41 5
64 > 8
PRACTICE PROBLEMS: Compare the values of the following pairs
of fractions. Describe
their relative size using one of the symbols = , < , or > .
3 7 7 27
1. 2.
8 16 16 ' 64

7 1 3 48
3' 4. -4
TE '

c 3 6 1 11
6. 74, ,
' 32

9 1 13 7
8.
1' 32 4 64 ' 32

15 60 27 3
16 64
1.
0
32 4

Testing the ftactions

6 12 48 3 24
and will show that they are all of the same value. That
o 16' 64'
= 12 = = 3 = 24 The best fraction to use from this list will be a different selection for dif-
48
'8 16 64 4 32
ferent situations. One of these fractions is considered to be the
simplest fraction of all the fractions
which have this value. A fraction is said to be in LOWEST TERMS
when the numerator and
denominator cannot both be divided exactly by a whole number greater than
1. The simplest fraction
is 3 for the above set of equal fractions. Each of the other fractions
4 can be reduced to 3
4

Reducing a fraction is the process of dividing both the numerator and

denominator of the fraction
by the same whole number (greater than 1). The quotients obtained
must be whole numbers so they
can be used as the numerator and denominator of the new fraction. The goal of
reducing a fraction is
to reach lowest terms.

EXAMPLE 2: Reduce the following fractions to lowest terms.

a) Simplify 12
16

30
b) Reduce -z

. 90
c) Simplify
6
 4
1
Solution:
a)
1 4 will divide into both 12 and 16.
12
16
12 -;'-- 4
16 --i-- 4
3
I 4
The only whole number which divides into both 3 and 4 is 1 (one), so

II 3 is the lowest terms.

4
Solution:
b) 2 will divide into both 30 and 48
30 30 2
48 48 + 2
I 24
15

Both 15 and 24 can be divided by 3

I 15
24
15
24
4-
--
3
3
5
I 8
The only whole number which divides both 5 and 8 is 1, so 5 is in lowest
8
terms. The reduced form solution can also be achieved in one step if a
person notices that 6 divides into both 30 and 48.
30 30 --:- 6
48 48 + 6
5
8
Solution:
c) Find some whole number, preferably the largest possible, which
divides both 90 and 150.
90 90 4- 10
150 150 10
= -- 9
15
Look tor some whole number which divides 9 and 15.
9 9 --:- 3
15 15 3
3
= """.
5
The only whole number which divides both 3 and 5 is 1, so
2 is the simplest form.
5

The process of raising fractions to higher terms is just as important as that of reducing fractions.
Raising a fraction to higher terms results from designing P^ equal fraction that has larger whnie num-
bers in :he numerator and denominator than were in the original fraction. A fraction is changed into an
equal fraction with higher terms by multiplying both the numerator and denominator by the came
whole number greater than 1.

I 7
 5

EXAMPLE 3: Change the terms of each fraction so that its new denominator is 64.
7
a)
16

7
b)
8

Solution:

a) The 16 denominator must be multiplied by 4 to make a new denominator of 64.

7 (7)(4)
16 (16)(4)

28
64
Solution:

b) The 8 denominator needa to be multiplied by 8 to get 64 in the denominator.

7 (7)(8)
8 (8)(8)

56
64
Solution:

c) The 4 denominator divides into 64 exactly 16 times. The 4 must be multiplied

by 16 to get 64.
1 (1)(16)
4 (4)(16)

16
64

PRACTICE PROBLEMS: Reduce each fraction to its lowest terms.

15 30 12
11. 12. 13.
2C 48 128

6 60 c 38
14. izt 15.
4
I 0.
320 64

24 270
17. T-41 18. -6-6- 19.
480

36
20.
72

8
 6
I Change each of the given fraction into sixteenths (denominators of 16).

I 21. 3
4
92. 5
8
23 12

I 24. i 7
25.
1
71

I Change each fraction into a fraction with denominator of 32.

I 26. 3
8 27'
3
16
2 8. 34

I Fractions in a measurement are frequently encountered when the English measurement syst9rn is
being used. When the distance or length dimensions for a machine part are given using the English
inch measurement unit, these measurements can be given either in fraction form, like 3 inch, or in
I decimal form, 0.375 inch. Most simple measurement instruments, which record length in the fraction-
8

al inch scale, will allow size to be determined to the nearest 64th (sixty-fourth) inch. Table 1, at the
I end of Module 7, contains the decimal equivalent for evory proper fraction of denominator 64. For
each number of 64ths, Table 1 has its decimal equivalent and all the possible reductions of sixth-
fourths into a fraction of smaller denominator. Table 1 is used to read fraction and decimal
I equivalents.

EXAMPLE 4: Use Table 1 to give the reduced fraction and decimal equivalent of the given
I fractions.
18
a)
I 64

h 56

I 1'1 64

24
c) T-2
I
Solution:
I a) 9 is the reduced fraction. 0.28125 is the equivalent decimal.
32

I Solution:
b) 7 is the reduced fraction. 0.875 is the equivalent decimal.
8
I
Solution:

1 c) 3 is the reduced fraction. 0.75 is the equivalent decimal.

4

I A MIXED NUMBER is the combination of a whole number and a proper fraction. Examples of
mixed numbers include 2i
1 5
3-8 and 5
' ' 32
I 9
 7

A mixed number which contains a fraction in 64ths, or a reduction of

64ths, can be changed into a
decimal using Table 1. The whole number part of the mixed
number is repeated in the whole number
part of the decimal. The fraction of the mixed number is converted
to a decimal by Table 1 and at-
tached to the whole number.

EXAMPLE 5: Use Table 1 to give the decimal equivalent of the mixed numbers.
a) 4-3
16

23
b) 7 Ts:

c)
8

Solution.

a) Working with the fraction and then whole number:

0.1875
16

Attaching whole number 4 to decimal 0.1875 gives 4?-6 = 4.1875

Solution:

b) The fraction of the mixed number is 23 = 0.71875

32
23
With whole number 7, 7 = 7.71875
32

Solation:

c) Since 1 = 0.375 , then 6 2 6.375

8 8

PRACTICE PROBLEMS: Use Table 1 to determine the decimal value of these proper fractions
and mixed numbers.
7
30 31. 39
29' T6- 32 64

26
32.
32
33. 1 3 34 . 2-5
4 8

35. 7 11
16
36. 5-31
64
37. 29
3
32

38. 4 732 39' 6 16:

8
40. 41648
10
 8

EXAMPLE 6: Use Table 1 to find the reduced fraction or mixed number value fcr the decimals.
a) 0.65625

b) 0.328125

c) 4.03125

d) 9.375

Solution:
a) 0.65625 . 21
32

Solution:
21
b) 0.328125 . -6-4-t.

Solution:
1
c) 0.03125 . yi, so

4.03125 . 4 1
32

Solution:
d) 0.375 .
8
9.375 . 93
8

Fractions in which the whole number in the numerator (s greater than or equal to the whole num-
ber in the denominator are called IMPROPER FRACTIONS. Don't let the name of th;s kind of fraction
influence your opinion about the usefulness of such fractions. If a family names their first son Pierre,
that doesn't make him French. Pierre may choose to name his pet hamster Goliath. That won't make
the hamster 7 foot tall and weigh-in at 300 pounds. Most names do not carry any special meaning
about its owner. IMPROPER FRACTIONS are not IMPROPER. Improper fractions make good
answers.

An improper fraction is often changed into a mixed number. To change an improper fraction into a
mixed number using a calculator and Table 1:

1) Divide the numerator by the denominator. The whole number part of the
decimal is the whole number part of the mixed number.

2) The decimal part is converted to a fraction using Table 1.

3) The whole number and the fraction are joined to form the mixed number.
 9

EXAMPLE 7: Change the improper fraction 51 into a mixed number.

8
Solution:

Directions K;..y strokes Display

Enter numerator
rl 51.

Divide 51

Enter denominator 8.

End divide 6.375

3
The whole number part of the mixed number is 6. Table 1 g:ves 0.375
8
51 3
Therefore, T3- = 6 .

EXAMPLE 8: 307
Change the improper fraction into a mixed number.
32
Solution:

Directions Key strokes Display

i
Enter numerator 3 7
307.

Divide 307

2
Enter denominator 32.

End divide 9.59375

19
The whole number part of the mixed number is 9. Table 1 gives 0.59375 = --2- .
307 19
Therefore, --5-2--- - 9 -1.

12
 10

PRAC -ICE PROBLEMS: Use Table 1 to find the reduced fraction or mixed number value for
the decimals.

41. 0.8125 42. 3.5625 43. 4.125

44. 2.828125 45. 5.21875 46. 6.03125

47. 12.75 46. 7.875

Use a calculator and Table 1 to change each improper fraction into a mixed number.

47 19
49. -31 50. 51.
8 16 4

105 413
52. -71 53. 54.
16 32 32

281 745
55. 56.
64 64

When a given decimal is not exactly oqual to any of the decimals provided in Table 1, the table
can be used to determine an approximate fraction to the nearest 64th. The given decimal will fall be-
tween two consecutive table decimal entries. The entry just smaller will correspond to the next
smaller fraction. The entry just larger will correspond to the next larger fraction. Subtraction of
decimals will show which fraction is closest to the given decimal.

EXMAPLE 9: For each given decimal, use Table 1 to find the fractions in 64ths which are the:

(1) next smaller fraction (mixed number)

(2) next larger fraction (mixed number)
(3) closest fraction (mixed number)

a) 0.6800
b) 3.1750
c) 5.6

Solution:
a) The decimal 0.6800 lies between 0.671875 and 0.6875

(1) The next smaller fraction is 0.671875 = g

44 11
(2) The next larger fraction is 0.6875 = -64
16
(3) The differences are
0.6800 - 0.671875 = 0.008125 and 0.6875 0.68 = 0.0075
The smaller difference is 0.0075
11
Closest fraction 0.6875 = -44
64 16

1 '3
 11
Solution:
b) The decimal in 3.1750 is 0.1750 which lies between 0.171875 and 0.1675
11
(1) The next smaller fraction is 0.171875 = -6--4- .

The next smaller mixed number is 3 -11

64
12 3
(2) The next larger fraction is 0.1875 = -64 = -16
3
The next larger mixed number is 3
16
(3) Computing differences:
0.1750 - 0.171875 = 0.003125 and
0.1875 - 0.1750 = 0.0125
The smaller difference is 0.003125.
11 11
The closest fraction to 0.1750 = -64 The closest mixed number to 3.1750 is 3 -64.

Solution:
c) The decimal part 0.6 lies between 0.59375 and 0.609375
38 19
(1) The next smaller fraction is 0.59375 = -64 = -32*

The r: xt smaller mixed number is 5-19

32
(2) The next larger fraction is 0.609375 = -39
64
The next larger mixed number is 5 -39
64
(3) Computing differences:
0.6 0.59375 = 0.00625 and 0.609375 - 0.6 = 0.009375
The smaller difference is 0.00625
9 19
The closest fractiun is 0.593/5 - -1 and the closest mixed number is 5 -32*
32
PRACTICE PROBLEMS: For each decimal, use Table 1 to find the fractions in 64ths
which are the:
(a) next smaller fraction,
(b) next larger fraction, and
(c) closest fraction.

57. 0.235 58. 0.872 59. 0.0825

60. 0.94375 61. 0.6050 62. 0.40055

For each decimal, use Table 1 to find the mixed numbers in 64ths
which are the:
(a) next smaller mixed number,
(b) next larger mixed number, and
(c) closest mixed number.

63. 3.27875 64. 2.8945 65. 1.07625

66. 4.32750 67. 7.5675 68. 5.6025

14
 12

Table 1 -FRACTIONt AND DECIMAL EQUIVALENTS

I Fractions
1/32
1/64
2/64
Decimals
0.015625
0.03125
3/64 0.046875
1/16 2/32 4/64 0.062
5/64 0.078125
3/32 6/64 0.0937

I
7/64 0.109375
1/8 2/16 4/32 8/64 0.125
9 /64 0.140625
5/32 10/64 0.15625

I 3/16 6/32

7/32
11P34
12/64
1 3/64
14/64
0.171875
0.1875
0.203125
0.21875

I 1/4 2/8 4/16 8/32

15/64
16/64
17/64
0.234375
0.25
0.265625
9/32 18/64 0.28125

I 5/16 10/32
19/64
20/64
21/64
0.296875
0.3125
0.328125
11/32 22/64 0.34375

I 3/8 6/16 12/32

23/64
24/64
25/64
0.359375
0.375
0.390625
13/32 26/64 0.40625

I 7/16 14/32
27/64
28/64
29/64
0.421875
0.4375
0.453125
15/32 30/64 0.46875

I 102 2/4 4/8 8/16 1C./32

31/64
32/64
33/64
0.484375
0.5
0.515625
17/32 34/64

I
0.53125
35/64 0.546875
9/16 18/32 36/64 0.5625
37/64 0.578125

i 5/8 10/16
19/32

20/32
38/64
39/64
40/64
41/64
0.59375
0.609375
0.625
0.640625

I 11/16
.:1/32

22/32
42/64
43/64
44/64
0.65625
0.671875
0.6875
45/64 0.703125

I 3/4 6/8 12/16

23/32

24/32
46/64
47/64
48/64
0.71875
0.734375
0.75
49/64 0.765625

I 13/16
25/32

26/32
50/64
51/64
52/64
0.78125
0.796875
0.8125
53/64 0.828125

I 7/8 14/16
27/32

28/32
54/64
55/64
56/64
0.84375
0.859375
0.875
57/64

I
0.890625
29/32 58/64 0.90625
59/64 0.921875
15/16 30/32 60/64 0.9375

I
61/64 0.953125
31/32 62/64 0.96875
63/64 0.984375
2/2 4/4 8/8 16/16 32/32 64/64 1.000000

15
 13
SOLUTIONS TO PRACTICE PROBLEMS-Module 7
1. < 2. > 3. < 4. = 5. = 6. <
7. > 8. < 9. = 3
10. > 11. 12. -5
4 8

3 1 3 19
13. -di- 14. -,
15' T6 16. -ii
3 9
17.
8
18 . 1 20.
1
-2
4 19' -1-6

12 10 8
21. 22 . 24 . -14
16 16 23 -1-6 16

4 12 6 24
25. 26. 27. r2- 28. 5-2-
16

29. 0.4375 30. 0.46875 31. 0.609375 32. 0.8125

33. 1.75 34. 2.625 35. 7.6875 36. 5.484375

37. 3.90625 38. 4.21875 39. 6.5 40. 8.640625

13 9 1
41. 42. 3 43. 44. 2-53 45. 51-
16 16 64 32

7-78 7 5
1
46. 6-32 47. 12-3 48. 49. 3 50. 21--
4 8 16

7 9 9
51. 4 1.
4
52. 4
16
53. 3-
32
54. 12L
32
55. 4L5
64

41 15 55 5
56. 11-- 58. a) -6-4
64 57. a' 64 59' a) -6-4-

h 16 hl 56 hl 6
11/ 64 '1' 64 -1 64

56 5
c) -611

60 38 25 17
60. a) -6-z 61. a) -6-4- 62. a) -6--4 63. a) 3-64
1
h 61 39 hl 26
64 b) 3-18
'1' 1" 64 '1' 64 64

60 39 26 118
C' 64 C' 64 C) C' '164

A6
 14

4 20
64. a) 2-57 65. a) 66. a) 4 67. a) 7-36
64 164 o4 64

b) 2-58
64
b) 1-5
64
b) 4
64
21
b) 37647
c) 2-57
64
c) 1
64
5 A21
64
c) 7
36
64

38
68. a) 5-64

b) 5-39
64

c) 5-39
64
 END
U.S. Dept. of Education

Office of Educational
Research and Improvement (OERI)

ERIC
Date Filmed
July 17, 1991