Download 19 Addition and Subtraction of Fractions

19
Addition and Subtraction of Fractions
You walked 15 of a mile to school and then 35 of a mile from school to your
friend’s house. How far did you walk altogether? How much farther was the
second walk than the first?
To solve the first problem, you would add 51 and 35 . To solve the second problem, you would subtract 15 from 35 . Adding and subtracting fractions are the
subjects of this section.
Addition of Fractions
Pictures and manipulatives help understand the process of finding the sum
of two fractions. Let’s consider the question of finding 51 + 35 . This sum is
illustrated in two ways in Figure 19.1. The colored region and the numberline models both show that 51 + 35 = 45 .
Figure 19.1
The above models suggest the following definition of fractions with like denominators:
The sum of two fractions with like denominators is found by adding the numerators and dividing the result by the common denominator. Symbolically
a b
a+b
+ =
.
c c
c
Example 19.1
Find the value of the sum
3
8
+
1
8
and write the answer in simplest form.
1
Solution.
By the definition above we have
3 1
3+1
4
4·1
1
+ =
= =
= .
8 8
8
8
4·2
2
Next we consider adding two fractions with unlike denominators. This is
done by rewriting the fractions with a common denominator. The common
denomintor is referred to as the least common denominator and is nothing
else then the least common multiple of both denominators. For example, the
least common denominator for 41 and 32 is the LCM(4,3), which is 12. Thus,
1
4
+
2
3
=
=
3
12
1·3
4·3
+
+
8
12
2·4
3·4
=
11
12
The procedure just described can be modeled with fraction strips as shown
in Figure 19.2
Figure 19.2
The general rule for adding fractions with unlike denominators is as follows.
To add ab + dc where b 6= 0, d 6= 0 and b 6= d :
(1) Rename each fraction with an equivalent fraction using the least common
denominator, that is, LCM(a,b).
(2) Add the new fractions using the addition rule for like denominators.
Example 19.2
Find 43 + 56 + 23 .
2
Solution.
The least common denominator is LCM (4, 6, 3) = 4 · 3 = 12. Thus,
3
4
+ 65 +
2
3
=
=
=
3·3
+ 5·2
+ 2·4
4·3
6·2
3·4
9
8
+ 10
+ 12
= 9+10+8
12
12
12
9·3
9
27
=
=
12
4·3
4
Example 19.3
The sum of a whole number and a fraction is most often written as a mixed
number. For example, 2 + 34 = 2 34 .
(a) Express 3 25 as a fraction.
(b) Express 36
as a mixed number.
7
Solution.
(a) We have: 3 25 = 3 + 25 = 15
+ 25 = 17
.
5
5
36
35
1
1
1
(b) We have 7 = 7 + 7 = 5 + 7 = 5 7 .
Example 19.4
Compute 2 43 + 4 25 .
Solution.
The least common denominator is LCM (4, 5) = 4 · 5 = 20. Thus,
2 43 + 4 25 =
=
2 + 4 + 43 + 25
15
8
6 + 20
+ 20
= 6+
23
20
=6+
20
20
+
3
20
3
= 7 20
Properties of Addition of Fractions
The following properties of addition can be used to simplify computations.
Theorem 19.1
(a) Closure: The sum of two fractions is a fraction.
(b) Commutativity: Let ab and dc be any fractions. Then
a c
c a
+ = + .
b d
d b
(c) Associativity: Let ab , dc and
e
f
be any fractions. Then
a c e
a
c
e
+
+ = +
+
.
b d
f
b
d f
3
(d) Additive identity: 0 is the additive identity since for any fraction
have
a
a
+0= .
b
b
a
b
we
Proof.
We will prove parts (a) and (b) and leave the rest for the reader.
(a) This follows from ab + dc = ad+bc
and the fact that both the numerator
bd
and denominator are whole numbers.
(b) By the definition of addition of fractions and the fact that addition and
multiplication of whole numbers are commutative we have
a
b
+
c
d
=
=
=
ad+bc
bd
bc+ad
cb+da
=
bd
db
c
a
+
d
b
Practice Problems
Problem 19.1
If one of your students wrote
that this is incorrect?
1
4
+ 23 = 37 , how would you convince him or her
Problem 19.2
Use the colored region model to illustrate these sums, with the unit given by
a circular disc.
(a) 25 + 65 (b) 23 + 14 .
Problem 19.3
Find 61 + 14 using fraction strips.
Problem 19.4
Represent each of these sums with a number-line diagram.
(a) 18 + 38 (b) 23 + 12 .
Problem 19.5
Perform the following additions. Express each answer in simplest form.
213
(a) 72 + 37 (b) 38 + 11
(c) 450
+ 12
.
24
50
Problem 19.6
A child thinks 12 +
the answer.
1
8
=
2
.
10
Use fraction strips to explain why
4
2
10
cannot be
Problem 19.7
Compute the following without a calculator.
5
+ 38 (b) a1 + 2b .
(a) 12
Problem 19.8
Compute the following without a calculator.
1
3
4
2
+ 21
(b) 2n
+ 5n
.
(a) 15
Problem 19.9
Compute 5 43 + 2 58 .
Problem 19.10
Solve mentally:
(a) 81 +? = 85
(b) 4 81 + x = 10 38 .
Problem 19.11
Name the property of addition that is used to justify each of the following
equations.
(a) 73 + 27 = 27 + 37
4
4
(b) 15
+ 0 = 15
2
3
(c) 5 + 5 + 47 = 52 + 53 + 47
(d) 25 + 73 is a fraction.
Problem 19.12
Find the following sums and express your answer in simplest form.
(a) 37 + 73
1
3
+ 16
(b) 89 + 12
8
4
(c) 31 + 51
143
759
+ 100,000
.
(d) 1000
Problem 19.13
Change the following mixed numbers to fractions.
(a) 3 56 (b) 2 78 (c) 7 19 .
Problem 19.14
Use the properties of fraction addition to calculate each of the following sums
mentally. (a) 73 + 19 + 47
9
4
(b) 1 13
+ 56 + 13
2
3
(c) 2 5 + 3 8 + 1 54 + 2 83
5
Problem 19.15
Change the following fractions to mixed numbers.
(b) 49
(c) 17
(a) 35
3
6
5
Problem 19.16
(1) Change each of the following to mixed numbers: (a) 56
3
(2) Change each of the following to a fraction of the form
are whole numbers: (a) 6 43 (b) 7 21 .
(b) 293
.
100
a
where
a and b
b
Problem 19.17
Place the numbers 2, 5, 6, and 8 in the following boxes to make the equation
true:
Problem 19.18
A clerk sold three pieces of one type of ribbon to different customers. One
piece was 13 yard long, another 2 34 yd long, and the third was 3 12 yd. What
was the total length of that type of ribbon sold?
Problem 19.19
Karl wants to fertilize his 6 acres. If it takes 8 32 bags of fertilizer for each
acre, how much fertilizer does he need to buy?
Subtraction of Fractions
You walked 36 of a mile to school and then 76 of a mile from school to your
friend’s house. How much farther was the second walk than the first? This
problem requires subtracting fractions with like denominators, i.e. 76 − 36 .
Figure 19.3 shows how the take-away, measurement, and missing-addend
models of the subtraction operation can be illustrated with colored regions,
the number line, and fraction strips. In each case we see that 67 − 36 = 46 .
6
Figure 19.3
In general, subtraction of fractions with like denominators is determined as
follows:
a−c
a c
− =
, a ≥ c.
b b
b
Now, to subtract fractions with unlike denominators we proceed as follows:
Suppose that ab ≥ dc , i.e. ad ≥ bc. Then
a
b
−
c
d
ad
bc
− bd
bd
ad−bc
bd
=
=
We summarize this result in the following theorem.
Theorem 19.2
If ab and dc are any fractions with
a
b
≥
c
d
Example 19.5
Find each difference.
(a) 85 − 14 (b) 5 13 − 2 43 .
7
then
a
b
−
c
d
=
ad−bc
.
bd
Solution.
(a) Since LCM(4,8) = 8 we must have
5 1
5 2
3
− = − = .
8 4
8 8
8
(b) Since LCM(3,4) = 12 we must have
(5 + 13 ) − (2 + 34 )
5 13 − 2 34 =
4
9
= ( 70
+ 12
) − ( 24
+ 12
)
12
12
33
31
7
74
= 12 − 12 = 12 = 2 12
Practice Problems
Problem 19.20
Use fraction strips, colored regions, and number-line models to illustrate
2
− 14 .
3
Problem 19.21
Compute these differences, expressing each answer in simplest form.
(a) 85 − 28 (b) 35 − 42 (c) 2 23 − 1 31 .
Problem 19.22
(a) Find the least common denominator of
3
7
− 28
.
(b) Compute 20
7
20
Problem 19.23
Compute the following without a calculator.
5
1
5
3
(a) 12
− 20
(b) 6c
− 4c
.
Problem 19.24
Compute 10 16 − 5 32 .
Problem 19.25
Compute 90 13 − 32 97 .
Problem 19.26
9
3
Solve mentally: 3 10
−? = 1 10
.
8
and
3
.
28
Problem 19.27
Fill in each square with either a + sign or a - sign to complete each equation
correctly.
Problem 19.28
On a number-line, demonstrate the following differences using the take-away
approach.
1
5
− 12
(b) 23 − 41 .
(a) 12
Problem 19.29
Perform the following subtractions.
9
5
7
(a) 11
− 11
(b) 45 − 43 (c) 21
− 39
.
51
Problem 19.30
Which of the following properties hold for fraction subtraction?
(a) Closure (b) Commutative (c) Associative (d) Identity
Problem 19.31
Rafael ate one-fourth of a pizza and Rocco ate one-third of it. What fraction
of the pizza did they eat?
Problem 19.32
You planned to work on a project for about 4 12 hours today. If you have been
working on it for 1 43 hours, how much more time will it take?
Problem 19.33
Martin bought 8 34 yd of fabric. He wants to make a skirt using 1 78 yd, pants
using 2 38 yd, and a vest using 1 23 yd. How much fabric will be left over?
Problem 19.34
A recipe requires 3 12 c of milk. Don put in 1 43 c and emptied the container.
How much more milk does he need to put in?
Problem 19.35
A class consists of 25 freshmen,
of the class is seniors?
1
4
sophomore, and
9
1
10
juniors. What fraction
Problem 19.36
Sally, her brother, and another partner own a pizza restaurant. If Sally owns
1
and her brother owns 14 of the restaurant, what part does the third partner
3
own?
10