Section 3.6
PRE-ACTIVITY
PREPARATION
Adding and Subtracting Fractions
and Mixed Numbers
In our U.S. measuring system, numbers are often presented as fractions—
cooking and carpentry come readily to mind—so your ability to work with
them is a computational skill not to be overlooked. Moreover, your further
study of mathematics and other quantitative courses in a variety of fields
will assume your competency and comfort with fractions.
If you have already built an 18¼ by 20⅝ feet deck and now wish to add
a railing along three sides, how do you determine the linear feet of rail
you will need? How do you account for the two 4½ feet wide entrances to
the deck that must remain open? This practical application requires basic
addition and subtraction of mixed numbers.
LEARNING OBJECTIVES
•
Master the addition of fractions and mixed numbers.
•
Master the subtraction of fractions and mixed numbers.
•
Gain an understanding of borrowing with mixed numbers.
TERMINOLOGY
PREVIOUSLY USED
addend
mixed number
borrowing
multiplier
build up
numerator
common denominator
proper fraction
equivalent fraction
reduce
improper fraction
Least Common Denominator (LCD)
minuend
319
Chapter 3 — Fractions
320
METHODOLOGIES
The methodologies for addition and subtraction are based upon the concept introduced in the previous
section— in order to add or subtract fractions, they must share a common denominator.
The first methodology presents the simple process to use when the denominators are the same. It is followed
by a methodology for adding or subtracting proper or improper fractions when the denominators are
different.
Adding or Subtracting Proper or Improper Fractions with the Same Denominator
4
3
and .
5
5
5
7
from .
8
8
►
Example 1: Find the sum of
►
Example 2: Subtract
Steps in the Methodology
Step 1
Write the
problem.
Step 2
Add or
subtract the
numerators.
Step 3
Convert to
mixed number
if necessary.
Write the problem with the
correct operation sign.
Add (or subtract) the
numerators and place the
sum (or difference) over the
fractions’ common denominator.
Example 1
4 3
+
5 5
VISUALIZE
Example 2
7 5
−
8 8
VISUALIZE
X X X
X X
Convert an improper fraction
answer to a mixed number.
4
3
shaded +
shaded
5
5
7
5
shaded −
(X'd)
8
8
4+3 7
=
5
5
7−5 2
=
8
8
7
2
=1
5
5
VISUALIZE
1
2
5
2
is a proper fraction
8
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology
321
Example 1
Reduce the fraction to lowest
Step 4
2÷2
1
=
8÷2
4
terms.
Reduce.
Example 2
2
is reduced
5
VISUALIZE
¼
Step 5
Present your answer.
1
Present the
answer.
Step 6
Validate your
answer.
Validate your answer with the
opposite operation.
2
5
1
4
2
7
=
5
5
3
3
−
−
5
5
1
Begin with your answer, and
match the result to the original
addend or minuend.
Note: If your answer is a
mixed number or if it has a new
(reduced) denominator, refer to
the following Methodologies.
4
9
5
MODEL
Add:
7 4 1
+ +
9 9 9
Steps 1 & 2
7 4 1
7 + 4 + 1 12
+ + =
=
9
9
9 9 9
Steps 3 & 4
12
3
=1
9
9
Step 5
Step 6
⇒
1
3÷3
1
=1
9÷3
3
1
3
1
3
Validate: 1
= 1
3
9
1
1
−
= −
9
9
Answer: 1
1
2
9
2
9
4
−
9
1
X X X
X X
11
9
4
= −
9
=
7
9
9
1 2
2
×
=
4 2
8
5
5
+
+
8
8
7
9
8
Chapter 3 — Fractions
322
Adding or Subtracting Proper or Improper Fractions with Different Denominators
To add or subtract fractions whose denominators are not the same, you must first convert them to equivalent
fractions with a common denominator (the LCD, for example). This methodology includes this necessary
step.
►
►
2
3
and .
3
4
5
5
−
Example 2: Subtract:
9 36
Example 1: Find the sum of
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Determine
the LCD.
Step 3
Build
equivalent
fractions.
Set up the problem
vertically.
Determine the LCD of
the fractions and identify
the multipliers needed
to build up equivalent
fractions with the LCD.
Build equivalent fractions
using the LCD and set up the
problem with the equivalent
fractions.
???
Example 1
Example 2
2
3
3
+
4
5
9
5
−
36
LCD = 12
by inspection
LCD = 36
by inspection
Multiplier for 3 is 4
Multiplier for 9 is 4
Multiplier for 4 is 3
2 4
8
× =
3 4
12
3 3
9
+ × =+
4 3
12
5 4
20
× =
9 4
36
5
5
−
=−
36
36
Why do you do this?
Step 4
Add or
subtract the
numerators.
Add (or subtract) the
numerators and place the
sum (or difference) over the
common denominator.
2 4
8
× =
3 4
12
3 3
9
+ × =+
4 3
12
5 4
20
× =
9 4
36
5
5
−
=−
36
36
17
12
15
36
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology
Step 5
Convert
to mixed
number if
necessary.
Step 6
Convert an improper fraction
answer to a mixed number.
Reduce.
Reduce the fraction to lowest
terms.
Step 7
Present your answer.
Step 8
Validate your
answer.
Example 1
Example 2
17
5
=1
12
12
15
is proper
36
5
is reduced
12
15 ÷ 3
5
=
36 ÷ 3 12
1
Present the
answer.
Validate your answer with
the opposite operation.
Begin with your answer
and match the result to the
original addend or minuend.
Note: If your answer is a
mixed number, refer to the
following methodology.
323
5
12
5
12
5
5
17
= 1
=
12
12
12
3 3
9
9
− × = −
=−
4 3
12
12
5 3
15
× =
12 3
36
5
5
+
=+
36
36
8
12
20
36
1
8÷4
2
= 9
12 ÷ 4
3
20 ÷ 4
5
=
9
36 ÷ 4
9
???
Why do you do Step 3?
You can add or subtract fractional parts of a whole and come up with an accurate description of the result only
if the parts are based upon the same number of parts in a whole—that is, the same denominator.
Visualize two small pan pizzas, each partially eaten so that one third (1/3) of a pizza remains on one pan and
one fourth (1/4) remains on the other.
1
3
If you were to combine them onto one pan, how much pizza remains?
1
4
1 1
+ =?
3 4
You cannot simply add the 1 and the 1 in the numerators because they represent different sized parts. And what
would you use as a denominator? Only when you use their fraction equivalents can you describe the sum of
these fractional parts.
Chapter 3 — Fractions
324
Now, instead of the one whole pizza being divided into 3 equal slices and the other pizza divided into 4 equal
slices, visualize each pizza having been cut into 12 equally-sized slices so that 1/3 pizza is the same amount
as 4 of 12 slices, and 1/4 pizza is the same amount as 3 of 12 slices.
1
4
=
3 12
+
1
3
=
4 12
3
=
4
2
5
6
7
1
Now you can reassemble/combine/add 4 equally-sized slices and 3 equally-sized slices to make 7 slices (parts)
of a whole 12-slice pizza—that is, 7/12 of a whole pizza.
1 1
4
3
7
+ =
+
=
3 4 12 12 12
Similarly, you cannot subtract fractions unless they have the same denominator.
1 1
4
3
1
For example,
− =
−
= .
3 4 12 12 12
MODEL
Add: 9 + 11 + 1
14 21 6
Step 1
9
14
11
21
1
+
6
Steps 3 & 4
14 = 2 × 7
21 = 3 × 7
6 = 2×3
LCD = 2 × 3 × 7 = 42
Step 2
9 3
27
× =
14 3
42
11 2
22
× =
21 2
42
1 7
7
+ × =+
6 7
42
Multipliers: 42 ÷ 14 = 3
42 ÷ 21 = 2
42 ÷ 6 = 7
Step 5 & 6
56
14 ÷ 14
1
=1
=1
42
42 ÷ 14
3
Step 7
Answer : 1
1
3
56
42
Step 8
Validate:
1 2
2
1 × = 1
3 2
6
1
1
−
=−
6
6
1
1
6
1
7
7
×
= 1
6
7
42
11 2
22
−
×
=−
21 2
42
1
49
42
22
=−
42
27 ÷ 3
9
9
=
42 ÷ 3 14
=
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
325
METHODOLOGIES
The following two methodologies address how to add and how to subtract mixed numbers.
Adding Mixed Numbers
►
Example 1:
►
Example 2:
7
13
+3
12
15
5
3
4 +2
6
8
8
Try It!
Steps in the Methodology
Step 1
Set up the
problem.
Example 1
Set up the problem. Stack the
problem vertically.
Example 2
7
12
13
+3
15
8
For ease of calculation when
adding mixed numbers, align
the whole numbers and align
the fractions.
???
Why do you do this?
Step 2
Determine
the LCD.
If the denominators are the
same, skip to Step 4.
If the denominators are
different, determine the LCD
of the fractions and identify
the multipliers needed to
build up equivalent fractions
with the LCD.
2
12
15
2
6
15
3
3
15
5
1
5
1
1
LCD = 2×2×3×5 = 60
Identify the multipliers:
60 ÷ 12 = 5
60 ÷ 15 = 4
Step 3
Build
equivalent
fractions.
Step 4
Add.
Build equivalent fractions
using the LCD and set up the
problem with the equivalent
fractions.
7
= 8
12
13
+3
= +3
15
7 5
35
×
= 8
12 5
60
13 4
52
× = +3
15 4
60
Add the whole numbers
separately from the fractional
components.
7
= 8
12
13
+3
= +3
15
7 5
35
×
= 8
12 5
60
13 4
52
× = +3
15 4
60
Note: Refer to the
methodology for adding
fractions with the same
denominator.
8
8
11
87
60
Chapter 3 — Fractions
326
Steps in the Methodology
Step 5
Convert
improper
fractions.
Step 6
Example 1
In the answer, convert
an improper fractional
component to a mixed
number and add the whole
number parts.
Reduce.
Reduce the fractional
component to lowest terms.
Step 7
Present your answer.
11
12
Step 8
Validate
your
answer.
87
27
= 11 + 1
60
60
27
= 12
60
27 ÷ 3
9
= 12
60 ÷ 3
20
12
Present
the
answer.
Validate your answer by
subtraction, using the
Methodology for Subtracting
Mixed Numbers.
Begin with your final answer,
use the original fraction
and/or mixed numbers in the
validation, and match the
result to the original addend.
Example 2
9
20
9 3
× = 12
20 3
13 4
−3
× = −3
15 4
12
60+
27 11
27
= 12
= 11
60
60
52
52
= −3
=−3
60
60
87
60
52
60
35
60
7
35
7
= 8 12
=8
9
12
60
8
Note: You will learn subtraction
of mixed numbers in the next
methodology (see page 328).
???
Why do you do Step 1?
Since a mixed number is simply the addition of a whole number plus a fraction, the example problem can be
rewritten as 8 +
7
13
+3+ .
12
15
7
13
The Commutative Property of Addition allows you to rearrange the terms: 8 + 3 +
+ , and arrive at the
12 15
same answer.
The Associative Property of Addition allows you to add as follows:
same answer.
⎛7
(8 + 3) + ⎜⎜⎜
⎝12
+
13 ⎞⎟
⎟ , and arrive at the
15 ⎟⎠
That is, to add the mixed numbers, you can add the whole numbers and separately add the fractions; then
combine the results.
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
327
MODELS
Model 1
Add: 4
Step 1
5 3
6
+ +1
6 4
7
5
6
3
4
6
+1
7
Step 2
4
Steps 3 & 4
5 14
×
= 4
6 14
3 21
×
=
4 21
6 12
+1 ×
= +1
7 12
4
5
Steps 5 & 6
Step 7
Step 8
5
2
6
4
7
2
3
2
7
3
3
1
7
7
1
1
7
84 ÷ 4 = 21
1
1
1
84 ÷ 7 = 12
Validate:
Multipliers:
84 ÷ 6 = 14
70
84
63
84
72
84
205
84
205
37
37
=5+2
=7
84
84
84
Answer : 7
LCD = 2 × 2 × 3 × 7 =84
37
reduced
2•2•3•7
37
84
84+
6
37
37
= 7
= 6
84
84
6 12
72
−1 ×
= −1
= −1
7 12
84
7
5
121
84
72
84
49 ÷ 7
7
=5
84 ÷ 7
12
12+
4
7
7
= 5
=4
12
12
3 3
9
−
× = −
=−
4 3
12
5
19
12
9
12
5
4
6
10
12
=4
5
9
6
Chapter 3 — Fractions
328
Model 2
Add: 12
8
3
+ 13 + 15
9
4
8
9
Step 2
Step 1
12
13
LCD is 36, by inspection
Steps 3 & 4
3
+15
4
8 4
× = 12
9 4
13
= 13
3 9
+15 × = +15
4 9
12
40
Step 5 40
Step 6
32
36
Note: Keep the whole number as it is.
27
36
59
36
59
23
23
= 40 + 1
= 41
36
36
36
23 is prime and is not a factor of 36.
Step 7 Answer : 41
Step 8 Validate:
23
is fully reduced.
36
23
36
36+
40
23
23
=
41
= 40
36
36
3 9
27
−15 × = −15
= −15
4 9
36
41
59
36
27
36
32
8
25
= 25
36
9
25
8
9
−13
12
8
9
9
Subtracting Mixed Numbers
►
►
8
2
−3
21
3
1
2
Example 2: 3 −1
8
3
Example 1: 6
Steps in the Methodology
Step 1
Set up the
problem.
Set up the problem. Stack the
problem vertically.
For ease of calculation when
subtracting mixed numbers, align
the whole numbers and align the
fractions.
Try It!
Example 1
8
21
2
−3
3
6
Example 2
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology
Step 2
Determine
the LCD.
Step 3
Build
equivalent
fractions.
Step 4
Borrow if
necessary.
Example 1
If the denominators are the
same, skip to Step 4.
Example 2
21 is divisible by 3.
21 is the LCD.
If the denominators are
different, determine the LCD of
the fractions and identify the
multipliers needed to build up
equivalent fractions with the
LCD.
Build equivalent fractions using
the LCD (refer to Section 3.5)
and set up the problem using the
equivalent fractions.
329
21 ÷ 3 = 7
8
= 6
21
2
−3
= −3
3
8
= 6
21
2 7
× = −3
3 7
6
8
21
14
21
Determine if borrowing from
8 < 14
the whole number part of the top
Borrowing is necessary.
number is necessary. Borrowing
is necessary when the numerator
8
8
of the first fraction is less than
6
=5+ 1 +
the numerator of the second
21
21
fraction.
21
29
8
Borrowing with Fractions
=5+
21
+
21
=5
21
To borrow using fractions:
• Reduce the ones digit in the
whole number by one (1).
OR
Use this notation:
• Rewrite the borrowed 1 as a
fraction, using the common
denominator.
5
???
6
21+
8
29
=5
21
21
Why do you do this?
Subtracting a mixed
Special number from a whole
Case: number (see page 332,
Model 1)
Step 5
Subtract.
Subtract the whole numbers
separately from the fractional
numbers.
Note: Refer to the Methodology
for Subtracting Fractions with the
Same Denominator.
21+
8
=5
21
14
−3
= −3
21
5
6
29
21
14
21
15
2
21
continued on the next page
Chapter 3 — Fractions
330
Steps in the Methodology
Step 6
Reduce.
Reduce the fractional component
to lowest terms.
Step 7
Present your answer.
Example 1
1
15
3 •5
5
2
=21
=2
21
7
3 •7
Present
the
answer.
Step 8
Validate
your
answer.
2
Validate your answer by addition,
using the Methodology for
Adding Fractions and Mixed
Numbers.
Begin with your final answer, use
the original fractions and/or
mixed numbers in the validation,
and match the result to the
original first number.
Example 2
5
7
5
5 3
= 2 × = 2
7
7 3
2
2 7
+3 = +3 × = +3
3
3 7
15
21
14
21
5
29
21
2
5
29
8
8
= 5 +1
=6
9
21
21
21
???
Why do you do Step 4?
The borrowing process for a mixed number subtraction problem focuses on the common denominator of the
fractions.
A way to understand this borrowing process might be to think of it in terms of a familiar example. Imagine
yourself as a baker selling whole sheet cakes and individual servings that you form by slicing a whole cake
into 21 equal portions (think denominator).
For Example 1, visualize the cakes you have on hand today:
6 whole cakes and 8 individual servings
6 whole cakes and 8 of another, or 6 8 cakes
21
21
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
331
Although you can easily sell 3 whole cakes from the 6, you cannot serve 14 pieces from the 8 cut pieces
available. However, the solution is to slice one of the 6 whole cakes into 21 pieces.
Why 21? —to match the already determined size of your portions (think LCD determined in Step 2). This
borrowing results in a rearrangement of the mixed number, giving you 5 whole cakes + 1 whole cake cut
into 21 pieces + the original 8 pieces on hand.
That is, 6
8
8
21
8
29
= 5 +1+
=5+
+
=5
21
21
21
21 21
This enables you to take 14 pieces from the 29 pieces and 3 whole cakes from the 5 still-uncut whole cakes
(X’d out below), leaving you with 2 15 cakes, or 2 whole cakes and 15 of another (shaded below).
21
21
X
X
X X X
X
X X X
X X
X X
X X
X X
Chapter 3 — Fractions
332
MODELS
Special Subtracting a Mixed Number
Case: from a Whole Number
Model 1
Subtract: 13 − 5
Step 1
3
8
Subtract: 18
Step 1
13
−5
Model 2
3
8
18
4
− 12
5
4
5
−12
Steps 2 & 3 There is only one fraction.
It remains the same. Skip to Step 4.
Step 2
There is only one fraction, 4 .
5
Skip to Step 4.
Step 4
Step 4
Borrowing is not necessary in
4
this case because
is the
5
top fraction, from which no
When subtracting a mixed number from
a whole number, there is no fractional
component to subtract from. Borrowing
is necessary. Use the denominator of the
bottom fraction as the LCD.
13 = 12 +
8
8
or 12
8
8
fraction is being subtracted.
Step 5
18
4
5
−12
8
8
3
3
−5
= −5
8
8
Steps 5 & 6 13
7
Step 7
6
= 12
Answer : 7
5
8
Step 6
4
is reduced.
5
Step 7
Answer : 6
Step 8
Validate:
reduced
5
8
4
5
4
5
6
4
5
+12
Step 8
5
Validate: 7
8
3
+5
8
12
8
= 12 + 1 = 13 9
8
18
4
9
5
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
333
Model 3
Subtract 3
Step 1
5
4
from 7 .
6
9
4
9
5
−3
6
Step 2
7
Steps 3, 4 & 5
9=3×3
6=2×3
LCD = 2 × 3 × 3 = 18
4•2
= 7
9•2
5•3
−3
= −3
6•3
7
18
8
26
8
= 6+
+
=6
18
18 18
18
15
15
=
−3
18
18
3
or use the notation:
6
7
−3
18+
8
18
15
18
26
18
15
= −3
18
=
6
3
Step 6
11
11
=
reduced
18 2 • 3 • 3
Step 7
Answer : 3
Step 8
Validate:
11
18
11
18
11
18
11
11
=3
18
18
5 3
15
+3 × = 3
6 3
18
3
4
26
8
8
4
6
= 6 +1
=79
=7 9
18
18
9
18
Chapter 3 — Fractions
334
METHODOLOGY
The following is an alternate methodology for adding or subtracting mixed numbers. It avoids the process
of borrowing by first converting the mixed numbers to improper fractions. However, for many problems,
the number of calculations combined with the size of the numbers becomes cumbersome and prone to
computational errors. If you do decide to use this methodology, keep in mind that you must present the final
answer as a mixed number, not as an improper fraction.
Adding and Subtracting Mixed Numbers by Conversion to Improper Fractions
optional, alternate methodology
►
1
2
Example 1: 3 − 1
8
3
►
Example 2:
4
5
11
−1
12
8
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Write as
improper
fractions.
Step 3
Build up
fractions.
Step 5
Add or
subtract
numerators.
Example 1
Set up the problem. Stack the
problem vertically.
1
8
2
−1
3
Change the mixed numbers to
improper fractions.
1
25
=
8
8
2
5
−1
=−
3
3
Determine the LCD.
LCD = 24,
Determine
the LCD.
Step 4
Try It!
Example 2
3
3
by inspection
Using the LCD, build equivalent
fractions. Set up the problem
with the equivalent fractions.
25
25 • 3
75
=
=
8
8•3
24
5
5•8
40
− =−
=−
3
3•8
24
Add or subtract the numerators
as indicated in the problem.
25
25 • 3
75
=
=
8
8•3
24
5
5•8
40
− =−
=−
3
3•8
24
35
24
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology
Step 6
Convert
to mixed
number.
Step 7
Example 1
Convert the answer to a mixed
number.
Reduce.
Reduce the fractional
component to lowest terms.
Step 8
Present your answer.
Validate
your
answer.
Example 2
35
11
=1
24
24
11 is prime, no common
factors with 24.
11
is reduced.
24
1
Present the
answer.
Step 9
335
Validate your final answer
with the opposite operation.
Begin with your answer, use
the original fractions or mixed
numbers and match the result
to the original term.
11
24
11
35
35
=
=
24
24
24
2
5 8
40
+1
=+ × =+
3
3 8
24
1
75
24
1
75
3
1
=38
=3 9
24
8
24
MODEL
Model of Alternate Methodolgy
Add: 12
8
3
+ 13 + 15
9
4
Steps 1 & 2
Step 3
8
116
=
9
9
13
+13
=
1
3
63
+15
=+
4
4
12
LCD is 36, by inspection
Chapter 3 — Fractions
336
Steps 4 & 5
116
4
464
×
=
9
4
36
13 36
468
×
=
1 36
36
63
9
567
+
×
=+
4
9
36
116
× 4
36
× 13
464
108
360
468
567
1499
36
Step 6
41
36 1499
−144
59
−36
)
41
63
× 9
464
468
+567
1499
23
36
23
Step 7
Step 8
23 is prime and not a factor of 36.
23
is reduced.
36
Answer : 41
36
×41
23
36
36
1440
1476
+ 23
Step 9
Validate:
23
1499
1499
=
=
36
36
36
3
63 9
567
−15
=−
× =−
4
4 9
36
1499
41
63
× 9
567
932
36
1499
−567
932
36
× 13
932
932
=
36
36
13 36
468
−
×
=−
1 36
36
464
= 12
36
12
36 464
−36
)
108
360
104
−72
468
32
8
9
32
36
= 12
8
9
9
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
337
How Estimation Can Help
As was the case with whole numbers and decimals, estimating sums and differences of fractions and mixed
numbers requires mental math skills.
It is easiest to estimate by rounding the mixed numbers to the nearest whole number, based upon how the
fractional part of the mixed number compares to ½. Recall that a fraction is equal to ½ when its numerator is
half its denominator.
•
If the fraction is < ½, round the mixed number down to the whole number part.
•
If the fraction is > ½, round up to the next higher whole number.
•
If the fraction = ½, retain the ½.
Occasionally you might get a better estimate if you can tell if the fraction is “very close” to ½,
in which case you might round to the ½ (see the fourth example below).
Example:
THINK
8
3
4
+ 2
3 1
>
4 2
Actual answer: 11
38
Estimate:
Actual answer:
+
1 1
<
3 2
9 + 2 +
Estimate:
Example:
1
3
1
2
1 1
=
2 2
1
1
= 11
2
2
31
1
, a bit larger than 11
2
48
7
13
+ 2
9
15
39
+
3
= 42
29
41 ,
45
a bit closer to 42 than to 41
Example:
THINK
Estimate:
Actual answer:
23
5
2
− 15
8
3
5 1
>
8 2
2 1
>
3 2
24 − 16 = 8
7
23
, close to 8
24
18
1
− 1
35
10
Example:
4
Estimate #1:
5
– 1 = 4
Estimate #2:
4
1
1
− 1 = 3
2
2
Actual answer: 3 29 , closer to 3 1 than to 4
70
2
As these examples remind you, estimation is not meant to be precise. However, it will give you a number
against which you can determine if your answer is reasonable.
Go back and estimate the answers to the first and second Examples of the Methodologies for Adding and for
Subtracting Fractions and Mixed Numbers. Was each answer reasonable as compared to its estimate?
Chapter 3 — Fractions
338
ADDRESSING COMMON ERRORS
Incorrect
Process
Issue
4 4
2
Incorrectly
5
= 5
= 5
3
6
identifying
5
5
the number of
−2
= −2
= −2
6
6
parts in a whole
when borrowing
2
from the whole
number for
subtraction
9
3
1
2 =3 =3
6
6
2
Not reducing
the final answer
to lowest terms
1
4
6
5
6
1 3
× =8
4 3
7
+5
= +5
12
8
Correct
Process
Resolution
3
12
7
12
10
13
12
9
6
When borrowing,
the borrowed “1”
must be in fraction
5
form, using the
denominator
−2
of the given
fractions.
4
6 4
= 4+ + = 4
6
6 6
5
5
= −2
= −2
6
6
2
Do a prime
factorization
of your final
answer to assure
that there are
no remaining
common factors to
cancel.
continues on the next page
8
1
=
40 5
5
6
1 3
× =8
4 3
7
+5
= +5
12
3
12
7
12
10
13
12
8
5
6
5
+2
6
2
4
10
4
= 4 +1
6
6
2
4
6
=5
2
9
3
5 2
× = 13
6 2
7
−5
= −5
12
1
Use the
Methodology
for Building
Equivalent
Fractions (see
Section 3.5).
Apply the Identity
Property of
Multiplication.
3
10
12
7
12
3
8
12
13
10
2 •5
13
= 13 1
12
2 •2•2
= 13
3
3
×8 =
5
40
4
5
5
+ ×5 = +
8
40
4
10
6
5
6
=5
10
Answer:
swer: 13
3
12
Not adjusting
the numerator
to balance the
change in the
denominator
when writing
equivalent
fractions
Validation
8
3
1
=8 9
12
4
5
6
3 8
24
× =
5 8
40
5 5
25
+ × =+
8 5
40
49
9
=1
40
40
0 40+ 9
9
49
= 1
=
40
40
40
5 5
25
25
− × = −
=−
8 5
40
40
1
24
40
24
=
40
3
5
24
40
=
3
5
9
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Issue
Incorrect
Process
Forgetting to
include the
whole number
parts of mixed
numbers
3 3
9
× =
12
4 3
2 4
8
+5 × = +
3 4
12
2
17
12
17
5
=1
12
12
Not borrowing
when
subtracting
from a whole
number
7
−4
2
5
3
2
5
339
Correct
Process
Resolution
For mixed number
addition and
2
subtraction,
vertically align the +5
whole numbers
and align the
fractions.
When you need
to rewrite the
fractional parts of
mixed numbers,
always bring along
the whole number
parts as well,
before adding or
subtracting.
When you subtract
a mixed number
from a whole
number, you must
borrow from the
whole number in
order to subtract
the fractional
part of the mixed
number.
7
Validation
3 3
× = 2
4 3
2 4
× = +5
3 4
9
12
8
12
12+
7
5
5
= 8
=7
12
12
2 4
8
−5 × = −5
= −5
3 4
12
17
12
8
12
7
17
12
2
9
12
8
17
5
= 7 +1
12
12
5
=8
12
3
2
5
5
2
2
−4
= −4
5
5
7
= 6
2
4
9
12
=2
3
9
4
3
5
2
+4
5
3
5
2
6
5
= 6 +1
5
=7 9
Use the
denominator
of the bottom
fraction as the
LCD when you
borrow.
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with adding and subtracting fractions and mixed numbers
why you need a common denominator to add or subtract fractions and mixed numbers
what it means to borrow from the whole number in order to subtract fractional parts
how to validate the answer to an addition or subtraction problem when the terms are fractions or mixed
numbers
Section 3.6
ACTIVITY
Adding and Subtracting Fractions
and Mixed Numbers
PERFORMANCE CRITERIA
• Adding any combination of fractions and mixed numbers correctly
– final answer in mixed number, fully reduced form
– validation of the final answer
• Subtracting any combination of fractions and mixed numbers correctly
– final answer in mixed number, fully reduced form
– validation of the final answer
CRITICAL THINKING QUESTIONS
1. What is the best way to set up the addition or subtraction of mixed numbers?
2. Why do you need a common denominator to add or subtract fractions?
3. Why is it to your advantage to use the Least Common Denominator when adding or subtracting fractions?
4. Where is the Identity Property of Multiplication used in the addition and subtraction of fractions?
340
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
341
5. What is the meaning of borrowing within mixed number subtraction?
6. What are the advantages and disadvantages of the alternate Methodology for Subtracting Mixed Numbers?
7. Why is it important to use terms from the original problem when you validate your presented answer?
TIPS
FOR
SUCCESS
• For ease of computation and for clarity when presenting an answer, use a horizontal fraction bar rather than
a slash. ⎛⎜ 2 not 2/3⎞⎟⎟
⎜⎜
⎝3
⎟⎠
• If alignment is a problem, use a vertical line between the whole numbers and the fractional parts of mixed
numbers as shown in the Models.
• The LCD is the easiest common denominator to use.
• Use effective notation for borrowing.
• Always validate your final answer using the original fraction or mixed number terms. Otherwise you may
not detect the interim errors that may have been made in building up or reducing.
Chapter 3 — Fractions
342
DEMONSTRATE YOUR UNDERSTANDING
Problem
1)
5
7
3
+
+
12 16 4
2) 28
5
7
− 10
6
12
3) 6 − 2
4
9
Worked Solution
Validate
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Problem
4) 3
2
4
+ 2 +1
9
5
5) 5
7
−2
12
6)
11
7
3
+2
+7
15
10
4
Worked Solution
343
Validate
Chapter 3 — Fractions
344
Problem
7)
9
Worked Solution
Validate
1
6
−5
3
7
8) Subtract
3
1
12 from 15
8
7
TEAM EXERCISE
One third (1/3) of the monthly income for my family is used to pay the rent, one twelfth (1/12) of it is used to
pay the utilities, one fourth (1/4) of it is used to pay for food, and one eighth (1/8) of the monthly income is
used to make the car payment. What part of my family’s monthly income is left for other things?
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
IDENTIFY
AND
CORRECT
THE
345
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) 6 − 4
1
3
Identify Errors
or Validate
Just broght down
the 1/3.
Did not borrow
from the whole
number 6 to
subtract the 1/3.
Correct Process
3
3
1
1
−4 = −4
3
3
2
1
3
6
=
5
Answer: 1
2) 3
3)
1
5
−1
4
8
1
3
7
−2
8
5
2
3
Validation
2
3
1
+4
3
3
5 = 5 +1
3
=6 9
1
Chapter 3 — Fractions
346
Worked Solution
What is Wrong Here?
4)
7
5) 2
3
7
+1
5
10
1
3
7
+1 + 2
4
8
12
Identify Errors
or Validate
Correct Process
Validation
Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
ADDITIONAL EXERCISES
Solve each of the following and validate your answers.
1.
2 5 7
+ +
3 6 9
2.
5
3.
14
5
−9
12
4.
15
1
11
7
+ 7 + 40
3
15
9
5.
16
1
3
− 11
4
5
6.
5
3
5
+1 + 4
21
14
18
7.
25 − 5
2
1
−2
5
4
5
8
8. Subtract 4
11
2
from 16
15
5
347