section 3.6 adding and subtracting fractions pre … · section 3.6 — adding and subtracting...
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319
PRE-ACTIVITY
PREPARATION
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 fi elds 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.
• Master the addition of fractions and mixed numbers.
• Master the subtraction of fractions and mixed numbers.
• Gain an understanding of borrowing with mixed numbers.
Adding and Subtracting Fractionsand Mixed Numbers
LLEARNINGEARNING OOBJECTIVESBJECTIVES
TTERMINOLOGYERMINOLOGY
PREVIOUSLY USED
addend mixed number
borrowing multiplier
build up numerator
common denominator proper fraction
equivalent fraction reduce
improper fraction Least Common Denominator (LCD)
minuend
Section 3.6
320 Chapter 3 — Fractions
Steps in the Methodology Example 1 Example 2
Step 1
Write the problem.
Write the problem with the correct operation sign.
Step 2
Add or subtract the numerators.
Add (or subtract) the numerators and place the sum (or difference) over the fractions’ common denominator.
Step 3
Convert to mixed number if necessary.
Convert an improper fraction answer to a mixed number.
Adding or Subtracting Proper or Improper Fractions with the Same Denominator
Example 1: Find the sum of
Example 2: Subtract
►►
►►
4
5 and
3
5.
5
8
7
8 from .
45
35
+ 78
58
−
VISUALIZE
78
58
shaded (X'd)−
7 58
28
− =
X X XX X
VISUALIZE
45
35
shaded + shaded
4 35
75
+ =
28
is a proper fraction
75
125
=
VISUALIZE
125
MMETHODOLOGIESETHODOLOGIES
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 fi rst 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.
321Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology Example 1 Example 2
Step 4
Reduce.
Reduce the fraction to lowest terms.
Step 5
Present the answer.
Present your answer.
Step 6
Validate your 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 or if it has a new (reduced) denominator, refer to the following Methodologies.
125
75
35
3545
=
− −
14
22
28
58
5878
× =
+ +
MMODELODEL
Add:
Steps 1 & 2
Steps 3 & 4
Step 5 Answer:
Step 6 Validate:
7
9+
4
9+
1
9
79
49
19
+ + = + + =7 4 19
129
129
139
113
= ⇒ ÷÷= 1
3 39 3
125
14
25
is reduced
2 28 2
14
÷÷=
VISUALIZE
X X XX X
¼
113
113
139
19
19
129
=
− = −
129
119
49
4979
=
− = −
322 Chapter 3 — Fractions
Steps in the Methodology Example 1 Example 2
Step 1
Set up the problem.
Set up the problem vertically.
Step 2
Determine the LCD.
Determine the LCD of the fractions and identify the multipliers needed to build up equivalent fractions with the LCD.
LCD = 12 by inspection
Multiplier for 3 is 4
Multiplier for 4 is 3
LCD = 36 by inspection
Multiplier for 9 is 4
Step 3
Build equivalent fractions.
Build equivalent fractions using the LCD and set up the problem with the equivalent fractions.
Step 4
Add or subtract the numerators.
Add (or subtract) the numerators and place the sum (or difference) over the common denominator.
Adding or Subtracting Proper or Improper Fractions with Different Denominators
Example 1: Find the sum of
Example 2: Subtract:
►►
►►
2
3 and
3
4.
5
9− 5
36
To add or subtract fractions whose denominators are not the same, you must fi rst convert them to equivalent fractions with a common denominator (the LCD, for example). This methodology includes this necessary step.
? ? ? Why do you do this?
23
+ 34
59
− 536
23
44
34
33
8129
12
× =
+ × = +
59
44
536
2036536
× =
− = −
23
44
812
34
33
9121712
× =
+ × = +
59
44
2036
536
5361536
× =
− = −
323Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology Example 1 Example 2
Step 5
Convert to mixed number if necessary.
Convert an improper fraction answer to a mixed number.
Step 6
Reduce.
Reduce the fraction to lowest terms.
Step 7
Present the answer.
Present your answer.
Step 8
Validate your 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.
1712
1512
= 1536
is proper
512
is reduced15 336 3
512
÷÷=
512
1512
1512
1512
1712
34
33
912
9128
12
= =
− × = − =−
512
33
1536
536
5362036
× =
+ =+
8 412 4
23
÷÷
= 20 436 4
59
÷÷
=
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.
If you were to combine them onto one pan, how much pizza remains?
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.
1
3?+ =1
4
1
41
3
? ? ? Why do you do Step 3?
324 Chapter 3 — Fractions
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.
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.
Similarly, you cannot subtract fractions unless they have the same denominator.
For example,
1
3
4
12=
1
4
3
12=
+ = 1
2
34 5
67
1
3+ = + =1
4
4
12
3
12
7
12
1
3− = − =1
4
4
12
3
12
1
12.
MMODELODEL
Add:
Step 1 Step 2
Steps 3 & 4 Step 5 & 6
Step 7
Step 8 Validate:
9
14+ +11
21
1
6
914112116
+
Multipliers: 42 14 342 21 242 6 7
÷ =÷ =÷ =
914
33
2742
1121
22
2242
16
77
7425642
× =
× =
+ × = +
5642
114 1442 14
113
= ÷÷
=
Answer : 113
116
77
1742
1121
22
2242
× =
− × = −
113
22
126
16
16
116
× =
− =−
=
= −
÷÷=
4942224227 342 3
914
14 2 721 3 76 2 3
2 3 7 42
= ×= ×= ×= × × =LCD
325Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Adding Mixed Numbers
Steps in the Methodology Example 1 Example 2
Step 1
Set up the problem.
Set up the problem. Stack the problem vertically.
For ease of calculation when adding mixed numbers, align the whole numbers and align the fractions.
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.
LCD = 2×2×3×5 = 60Identify the multipliers: 60 ÷ 12 = 5 60 ÷ 15 = 4
Step 3
Build equivalent fractions.
Build equivalent fractions using the LCD and set up the problem with the equivalent fractions.
Step 4
Add.
Add the whole numbers separately from the fractional components.
Note: Refer to the methodology for adding fractions with the same denominator.
Example 1:
Example 2:
►►
►►
? ? ? Why do you do this?
8 37
12
13
15+
4 25
6
3
8+
8712
31315
+
2 12 15
2 6 15
3 3 15
5 1 5
1 1
Try It!
8712
8712
55
8
31315
31315
44
3
35605260
= × =
+ = + × = +
8712
8712
55
83560
31315
31315
44
35260
118760
= × =
+ = + × = +
MMETHODOLOGIESETHODOLOGIES
The following two methodologies address how to add and how to subtract mixed numbers.
326 Chapter 3 — Fractions
Steps in the Methodology Example 1 Example 2
Step 5
Convert improper fractions.
In the answer, convert an improper fractional component to a mixed number and add the whole number parts.
Step 6
Reduce.
Reduce the fractional component to lowest terms.
Step 7
Present the answer.
Present your answer.
Step 8
Validate your answer.
Validate your answer by subtraction, using the Methodology for Subtracting Mixed Numbers.
Begin with your fi nal answer, use the original fraction and/or mixed numbers in the validation, and match the result to the original addend.
? ? ? 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
The Commutative Property of Addition allows you to rearrange the terms: and arrive at the same answer.
The Associative Property of Addition allows you to add as follows: and arrive at the same answer.
That is, to add the mixed numbers, you can add the whole numbers and separately add the fractions; then combine the results.
87
123
13
15+ + + .
8 37
12
13
15+ + + ,
8 37
12
13
15+( ) + +
⎛⎝⎜⎜⎜
⎞⎠⎟⎟⎟ ,
11 11
122760
8760
12760
= +
=
12920
1227 360 3
12920
÷÷=
1315
60+
12920
33
122760
1227
6011
8760
344
35260
11× = = =
− × = − = − =−35260
35260
83560
= =835
608
712
7
12
Note: You will learn subtraction of mixed numbers in the next methodology (see page 328).
327Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
MMODELSODELS
Add:
Step 1 Step 2
Steps 3 & 4
Steps 5 & 6
Step 7
Step 8 Validate:
45
6
3
41
6
7+ +
2 6 4 7
2 3 2 7
3 3 1 7
7 1 1 7
1 1 1
LCD = 2 × 2 × 3 × 7 =84
Multipliers:
84 ÷ 6 = 14
84 ÷ 4 = 21
84 ÷ 7 = 12
45634
167
+
Model 1
520584
5 23784
73784
= + = 372 2 3 7• • •
reduced
Answer : 73784
456
1414
47084
34
2121
6384
167
1212
17284
520584
× =
× =
+ × = +
73784
737
846
167
1212
6 84
12184
= =
− × =
+
−− = −
÷÷=
17284
17284
549 784 7
5712
5712
57
124
1912
34
33
912
912
4
4 12
= =
− × = − = −
+
5
6
10
124
56
=
328 Chapter 3 — Fractions
Model 2
Add:
Step 1 Step 2 LCD is 36, by inspection
Step 5
Step 6 23 is prime and is not a factor of 36.
Step 7
Step 8 Validate:
128
913 15
3
4+ +
1289
13
1534
+
405936
40 12336
412336
= + =
Answer : 412336
Steps 3 & 4
2336
is fully reduced.
412336
4123
3640
1534
99
152736
15
40 36
5936
= =
− × = − = −
+
22736
253236
2589
=
2589
13
1289
−
1289
44
123236
13 13
1534
99
152736
405936
× =
=
+ × = +
Note: Keep the whole number as it is.
Steps in the Methodology Example 1 Example 2
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.
Example 1:
Example 2:
Subtracting Mixed Numbers
►►
►►
68
213
2
3−
31
81
2
3− Try It!
6821
323
−
329Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
continued on the next page
Steps in the Methodology Example 1 Example 2
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.
21 is divisible by 3.
21 is the LCD.
21 ÷ 3 = 7
Step 3
Build equivalent fractions.
Build equivalent fractions using the LCD (refer to Section 3.5) and set up the problem using the equivalent fractions.
Step 4
Borrow if necessary.
Determine if borrowing from the whole number part of the top number is necessary. Borrowing is necessary when the numerator of the fi rst fraction is less than the numerator of the second fraction.
Borrowing with Fractions
To borrow using fractions:
• Reduce the ones digit in the whole number by one (1).
• Rewrite the borrowed 1 as a fraction, using the common denominator.
8 < 14Borrowing is necessary.
OR
Use this notation:
Step 5
Subtract.
Subtract the whole numbers separately from the fractional numbers.
Note: Refer to the Methodology for Subtracting Fractions with the Same Denominator.
? ? ? Why do you do this?
5 21
68
215
2921
+=
5 21
68
215
2921
31421
31421
21521
+=
− = −
6821
5 1821
521
821
52921
21
= + +
= + + =
Special Case:
Subtracting a mixed number from a whole number (see page 332, Model 1)
6821
6821
6821
323
323
77
31421
= =
− = − × = −
330 Chapter 3 — Fractions
? ? ? 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 of another, or cakes8
216
8
21
Steps in the Methodology Example 1 Example 2
Step 6
Reduce.
Reduce the fractional component to lowest terms.
Step 7
Present the answer.
Present your answer.
Step 8
Validate your answer.
Validate your answer by addition, using the Methodology for Adding Fractions and Mixed Numbers.
Begin with your fi nal answer, use the original fractions and/or mixed numbers in the validation, and match the result to the original fi rst number.
257
257
33
21521
323
323
77
31421
52921
= × =
+ = + × = +
52921
5 1821
6821
= + =
21521
23 5
3 72
57
1
1= •
•=
257
331Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
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, 68
215 1
8
215
21
21
8
215
29
21= + + = + + =
XX X
X X
X X
X X
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 cakes, or 2 whole cakes and of another (shaded below).215
21
15
21
X X X
X X X
X X
332 Chapter 3 — Fractions
MMODELSODELS
Model 1
Subtract: 13 53
8−
Step 1
Steps 2 & 3 There is only one fraction. It remains the same. Skip to Step 4.
Step 4
Steps 5 & 6
Step 7
Step 8 Validate:
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 1288
= + or 1288
13
538
−
13 1288
538
538
758
=
− = −
reduced
Answer : 758
758
538
1288
12 1 13
+
= + =
Subtracting a Mixed Number from a Whole Number
Model 2
Subtract: 184
512−
Step 1
Step 2 There is only one fraction, . Skip to Step 4.
Step 4 Borrowing is not necessary in
this case because is the
top fraction, from which no
fraction is being subtracted.
Step 5
Step 6 is reduced.
Step 7
Step 8 Validate:
1845
12
−
1845
12
645
−
45
Answer : 645
45
45
645
+12
1845
Special Case:
333Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Model 3
Step 1 Step 2 9 = 3 × 3
6 = 2 × 3
LCD = 2 × 3 × 3 = 18
Steps 3, 4 & 5
or use the notation:
Step 6
Step 7
Step 8 Validate:
Subtract from 35
67
4
9.
749
356
−
74 29 2
78
186
1818
818
62618
35 36 3
31518
••= = + + =
− ••= − =
−31518
31118
6 187
818
31518
+
−
=
= −
62618
31518
31118
1118
112 3 3
=• •
reduced
Answer : 31118
31118
31118
3
=
+ 556
33
31518
62618
6 18
187
8
187
49
4
9
× =
= + = =
334 Chapter 3 — Fractions
Steps in the Methodology Example 1 Example 2
Step 1
Set up the problem.
Set up the problem. Stack the problem vertically.
Step 2
Write as improper fractions.
Change the mixed numbers to improper fractions.
Step 3
Determine the LCD.
Determine the LCD. LCD = 24,
by inspection
Step 4
Build up fractions.
Using the LCD, build equivalent fractions. Set up the problem with the equivalent fractions.
Step 5
Add or subtract numerators.
Add or subtract the numerators as indicated in the problem.
Example 1:
Example 2:
Adding and Subtracting Mixed Numbers by Conversion to Improper Fractions
optional, alternate methodology
►►
►► Try It!
31
81
2
3−
45
81
11
12−
318
123
−
318
258
123
53
=
− = −
258
25 38 3
7524
53
5 83 8
4024
= ••=
− = − ••= −
MMETHODOLOGYETHODOLOGY
258
25 38 3
7524
53
5 83 8
40243524
= ••=
− = − ••= −
The following is an alternate methodology for adding or subtracting mixed numbers. It avoids the process of borrowing by fi rst 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 fi nal answer as a mixed number, not as an improper fraction.
335Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Steps in the Methodology Example 1 Example 2
Step 6
Convert to mixed number.
Convert the answer to a mixed number.
Step 7
Reduce.
Reduce the fractional component to lowest terms.
11 is prime, no common factors with 24.
is reduced.
Step 8
Present the answer.
Present your answer.
Step 9
Validate your answer.
Validate your fi nal answer with the opposite operation. Begin with your answer, use the original fractions or mixed numbers and match the result to the original term.
11124
11124
3524
3524
123
53
88
4024
7524
7524
33
243
18
1
8
= =
+ = + × = +
= =
1124
MMODELODEL
Model of Alternate Methodolgy
Steps 1 & 2
Add: 128
9+ +13 15
3
4
1289
1169
13131
1534
634
=
+ =
+ = +
Step 3 LCD is 36, by inspection
3524
11124
=
336 Chapter 3 — Fractions
Steps 4 & 5
Step 6 )36 1499144
593623
4141
2336
−
−
Answer : 412336
Step 8
Step 7 23 is prime and not a factor of 36.
is reduced.2336
Step 9 Validate:
93236
93236
131
3636
4683
=
− × = −
66
46436
1232
3612
89
8
9= =
412336
149936
149936
1534
634
99
5673693236
= =
− = − × = −
116
4
464
×36
13
108
360
468
×
63
9
567
×464
468
567
1499
+
36
41
36
1440
1476
23
1499
×
+
63
9
567
×
1499
567
932
−
36
13
108
360
468
× )36 464
36
104
72
32
12
−
−
1169
44
46436
131
3636
46836
634
99
56736
149936
× =
× =
+ × = +
337Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
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: Example:
Estimate: Estimate:
Actual answer: Actual answer:
Example: Example:
Estimate: Estimate #1: 5 – 1 = 4
Estimate #2:
Actual answer:
How Estimation Can Help
THINK
235
8
2
3 15−
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 fi rst and second Examples of the Methodologies for Adding and for Subtracting Fractions and Mixed Numbers. Was each answer reasonable as compared to its estimate?
THINK3
4>
1
2
1
3<
1
2
1
2=
1
2
83
42
1
3
1
2 + +
5
8>
1
2
2
3>
1
2
9 21
2 11
1
2+ + = 24 16 8− =
1131
48, a bit larger than 11
1
27
23
24, close to 8
387
92
13
15 + 4
18
35
1
10 1−
39 42 + 3 =
4129
45,
a bit closer to 42 than to 413
29
70, closer to 3
1
2 than to 4
Actual answer: 41
2
1
2 − =1 3
338 Chapter 3 — Fractions
AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS
Issue Incorrect Process Resolution Correct
Process Validation
Incorrectly identifying the number of parts in a whole when borrowing from the whole number for subtraction
When borrowing, the borrowed “1” must be in fraction form, using the denominator of the given fractions.
Not reducing the fi nal answer to lowest terms
Do a prime factorization of your fi nal answer to assure that there are no remaining common factors to cancel.
Not adjusting the numerator to balance the change in the denominator when writing equivalent fractions
Use the Methodology for Building Equivalent Fractions (see Section 3.5). Apply the Identity Property of Multiplication.
523
546
546
256
256
256
296
4 1
= =
− = − = −
296
336
312
= =
546
466
46
4106
256
256
256
256
= + + =
− = − = −
256
256
4106
4 146
54
65
23
2
3
+
= +
= =
35
8340
58
5540
840
15
× =
+ × = +
=
35
88
2440
58
55
2540
4940
1940
× =
+ × = +
=
1940
19
404940
58
55
2540
2540
2440
0 40= =
− × = − = −
+
2440
24
40
35
3
5= =
814
33
83
12
5712
5712
131012
× =
+ = +
814
33
83
12
5712
5712
131012
× =
+ = +
Answer: 131012
131012
132 5
2 2 2
1356
1
1= •
• •
=
1356
22
131012
5712
5712
83
12
× =
− = −
83
128
14
=
continues on the next page
=6
54 =
=56
=
sw 3er:
5 +
45
=88
4
×8
+
339Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
PPREPARATION REPARATION IINVENTORYNVENTORY
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
Issue Incorrect Process Resolution Correct
Process Validation
Forgetting to include the whole number parts of mixed numbers
For mixed number addition and subtraction, vertically align the 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.
Not borrowing when subtracting from a whole number
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.
Use the denominator of the bottom fraction as the LCD when you borrow.
234
33
912
523
44
812
1712
1712
1512
× =
+ × = +
=
234
33
29
12
523
44
58
12
71712
71712
7 1512
8512
× =
+ × = +
= +
=
8512
85
127
1712
523
44
58
125
812
29
12
7 12
= =
− × = − = −
+
29
122
34
3
4 =
42
4×
4
× =
=
34 3
4
×
=
7
425
325
−
7
45
7 655
425
425
235
=
− = −
235
425
655
6 1
7
+
= +
=
340
ACTIVITY
PPERFORMANCE ERFORMANCE CCRITERIARITERIA
• Adding any combination of fractions and mixed numbers correctly – fi nal answer in mixed number, fully reduced form – validation of the fi nal answer
• Subtracting any combination of fractions and mixed numbers correctly – fi nal answer in mixed number, fully reduced form – validation of the fi nal answer
CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS
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?
Adding and Subtracting Fractionsand Mixed Numbers
Section 3.6
341Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
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?
TTIPS FOR IPS FOR SSUCCESSUCCESS
• For ease of computation and for clarity when presenting an answer, use a horizontal fraction bar rather than a slash.
• 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 fi nal 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.
2
3 not 2/3
⎛⎝⎜⎜⎜
⎞⎠⎟⎟⎟
342 Chapter 3 — Fractions
DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING
Problem Worked Solution Validate
1)
2)
3)
512
716
34
+ +
2856
10712
−
6 249
−
343Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
Problem Worked Solution Validate
4)
5)
6)
329
2 145
+ +
5712
2−
1115
2710
734
+ +
344 Chapter 3 — Fractions
Problem Worked Solution Validate
7)
8)
913
567
−
Subtract
from 1238
1517
TEAM EXERCISETEAM 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?
345Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS
In the second column, identify the error(s) you fi nd 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 SolutionWhat is Wrong Here?
Identify Errors or Validate Correct Process Validation
1) Just broght down the 1/3.
Did not borrow from the whole number 6 to subtract the 1/3.
2)
3)
6 413
−
314
158
−
513
278
−
6 5 33
4 13
4 13
123
=
− = −
Answer: 123
123
4 13
5 33
5 1
6
+
= +
=
346 Chapter 3 — Fractions
Worked SolutionWhat is Wrong Here?
Identify Errors or Validate Correct Process Validation
4)
5)
735
1710
+
214
138
2712
+ +
347Section 3.6 — Adding and Subtracting Fractions and Mixed Numbers
ADDITIONAL EXERCISESADDITIONAL EXERCISES
Solve each of the following and validate your answers.
1.
2.
3.
4.
5.
6.
7.
8. Subtract
2
3
5
6
7
9+ +
52
52
1
4−
145
129−
151
37
11
1540
7
9+ +
161
511
3
4−
5
211
3
144
5
18+ +
25 55
8−
411
15
2
5 from 16