Download Building Equivalent Fractions, the Least Common Denominator, and

Section 3.5
PRE-ACTIVITY
PREPARATION
Building Equivalent Fractions,
the Least Common Denominator,
and Ordering Fractions
You have learned that a fraction might be written in an equivalent
form by reducing it to its lowest terms. In this section, you will
explore the techniques of how to write an equivalent form of the
fraction in another manner—to build up the fraction with a larger
numerator and a larger denominator, yet still retain its value.
The skill of rewriting fractions in higher terms is valuable when
comparing, ordering, adding, and subtracting fractions. Additionally,
looking at different configurations for the same whole unit or group
will expand your ability to look for patterns within sets of numbers.
LEARNING OBJECTIVES
•
Use a methodology to determine the Least Common Multiple (LCM) of a set of numbers and the Least
Common Denominator (LCD) of a set of fractions.
•
Build up equivalent fractions.
•
Use the LCD to put a set of fractions in order from smallest to largest or largest to smallest.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
cross-multiply
building up
cross-product
common denominator
denominator
common multiple
equivalent fraction
higher terms
factors
Least Common Denominator (LCD)
multiple
Least Common Multiple (LCM)
multiplier
numerator
prime factorization
prime factors
primes
295
Chapter 3 — Fractions
296
BUILDING MATHEMATICAL LANGUAGE
Building Equivalent Fractions
Recall that when you reduce a fraction, you divide out common factors from the numerator and denominator,
the result being an equivalent fraction in lower terms. Building up an equivalent fraction to
higher terms is the opposite process. To build up a fraction, you multiply both numerator and
denominator by the same number, resulting in a higher number for both.
By the Identity Property of Multiplication (multiplying a number by 1 does not change its value) and the
any number
fact that the number 1 can take the form of
,
the same number
you can write an infinite number of fractions equivalent to a given fraction.
Example :
2 3 2•3 6
2
× =
= , a fraction whose value is equivalent to .
5 3 5 • 3 15
5
6
shaded
15
2
shaded
5
VISUALIZE
The whole rectangle is now broken up into 15 parts, and it takes 6 of them to
equal the original 2 out of 5 parts.
In fact, you can choose any fraction form of the number 1 to build up an equivalent fraction.
For the same example,
2 2
4
× = ,
5 2 10
2 4
8
× = ,
5 4 20
2 13 26
× = ,
5 13 65
and so on.
Now suppose that you want to take a fraction and build an equivalent fraction with a specific
denominator. As long as the new denominator is a multiple of the original denominator, you can use
the following technique.
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
297
TECHNIQUE
Building Up an Equivalent Fraction with a Specified Denominator
Technique
Step 1:
Divide the larger denominator by the denominator of the given fraction to
determine the multiplier.
Step 2:
Multiply the numerator of the given fraction by that same number.
MODELS
A
►
5
?
=
7
42
3
?
=
13 39
B
►
Step 1
42 ÷ 7 = 6
Step 2
5 × 6 = 30
5 6
5•6
30
× =
=
7 6
7•6
42
Step 1
39 ÷ 13 = 3
Step 2
3×3=9
3 3
9
× =
13 3 39
You can validate that your answer and the original fraction are equivalent by applying the
Equality Test for Fractions (comparing the cross-products which should be equal).
3 ? 9
=
13 39
?
3 × 39 = 9 × 13
117 = 117 9
5 ? 30
=
7
42
?
5 × 42 = 7 × 30
210 = 210 9
C
►
4=
?
8
Before Step 1, write the whole number as a fraction:
Step 1
8÷1=8
Step 2
4 × 8 = 32
4=
4
?
=
1
8
4 8
32
× =
1 8
8
In this case, to validate you know that
32
, an improper fraction, is equal to 4. 9
8
Chapter 3 — Fractions
298
Common Multiples and the Least Common Multiple
A common multiple of two or more numbers is a multiple of each of them. That is, each of the
numbers will divide evenly into their common multiple.
For example, 90 is a common multiple of the numbers 5, 6, and 9 because it is a multiple of 5 (18 × 5 = 90),
of 6 (15 × 6 = 90), and of 9 (10 × 9 = 90).
Another common multiple of 5, 6, and 9 is 180, because 5, 6, and 9 each divide evenly into 180. In fact,
there are infinitely more common multiples of 5, 6, and 9, among them 270, 360, 450, and so on.
The smallest multiple that two or more numbers have in common is called their Least Common
Multiple (LCM).
For the previous example, the Least Common Multiple (LCM) of the numbers 5, 6, and 9 is 90, the
smallest number that all three numbers can divide into evenly.
Determining the Least Common Multiple
For two or more given numbers, how can you determine their LCM?
Example: Find the LCM of 9, 12, and 15.
You could list the multiples of each number and pick out the smallest they have in common—
The multiples of 9, 12, and 15 (which you would have to compute) are:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, …
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, …
15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, …
Their LCM is 180.
However, as the example demonstrates, this approach to determining an LCM is inefficient and prone to
computational errors when finding the multiples.
There are more efficient methods for determining the LCM. When the Least Common Multiple is not readily
apparent to you, use either of the following two methodologies to determine the LCM.
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
299
METHODOLOGIES
Determining the Least Common Multiple (LCM) of a Set of Numbers
by Prime Factorization
Find the LCM of each set of numbers:
►
Example 1: 9, 12, and 15
►
Example 2: 12, 14, and 15
Try It!
Steps in the Methodology
Step 1
Prime factor
each number.
Determine the prime factorization
of each number.
Largest number is divisible
Special
by every other number
Case:
(see page 300, Model 1)
Special
Case:
Example 1
2
2
3
3
3
9
3
1
3
5
15
5
1
Example 2
12
6
3
1
All are prime numbers
(see page 300, Model 2)
9=3×3
Readily apparent that the
Special
numbers share no common
Case:
factors (see page 301,Model 3)
Step 2
Identify
primes.
Step 3
Choose
necessary
factors.
Identify all the primes that are
factors in the prime factorizations.
Use each prime as a factor of the
LCM the greatest number of times
it appears in any one of the prime
factorizations.
12 = 2 × 2 × 3
15 = 3 × 5
2, 3, 5
9=3×3
12 = 2 × 2 × 3
15 = 3 × 5
THINK
???
Why do you do this?
two 2’s and one 5
are needed. 3 is a
factor of 9 twice and
of 12 and 15 once
each—need two 3’s.
LCM=2×2×3×3×5
Step 4
Multiply the
factors.
Multiply these prime factors.
The result is the LCM of the original
numbers.
= 180
Chapter 3 — Fractions
300
???
Why do you do Step 3?
There must be the correct number of each of the prime factors in the LCM to make it divisible by each of the
numbers in the set. In the worked example, two 2’s are needed as the factors of the LCM (180). If you would
choose, for example, only one 2 as a factor of the LCM, the number 12 (which is 2 × 2 × 3) would not divide
into it.
}
}
At the same time, there are no extra factors in the Least Common Multiple—only those necessary to make it
divisible by all three numbers in the set. The following illustrates how all necessary factors are included in the
LCM of Example 1.
12
15
}
2×2×3×3×5
9
MODELS
Model 1
Special Case: Largest Number is Divisible by Every Other Number
Determine the LCM of 2, 3, 6, and 12.
If, by inspection and or/application of the Divisibility Tests, you can
readily determine that the largest number is divisible by all other
numbers in the set, it is the LCM (no need to do Steps 2-4).
12, the largest number, is divisible by 2, by 3, by 6, and, of course, by itself.
Therefore, the Least Common Multiple (LCM) is 12.
Model 2
Special Case: All are Prime Numbers
Determine the LCM of 2, 7, and 13.
If the numbers are all distinct prime numbers, there are no common factors.
The LCM is the product of the prime numbers.
2, 7, and 13 are all prime.
The LCM = 2 × 7 × 13 = 182.
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
Model 3
301
Special Case: Readily Apparent that the Numbers Share No Common Factors
Determine the LCM of 7, 8, and 9.
If, by inspection and or/application of the Divisibility Tests, you can
readily determine that the numbers share no common factors, the
LCM is simply the product of the numbers.
THINK
7 is prime
8 is only divisible by 2
9 is only divisible by 3
}
no common factors; the LCM = 7 × 8 × 9 = 504
Determining the Least Common Multiple (LCM) of a Set of Numbers
by “Pulling Out Primes”
This methodology presents another efficient yet more compact process for determining the LCM when it is
not readily apparent. It condenses the first methodology by pulling out only the necessary prime factors of the
numbers from smallest to largest. It may remind you of the “factor ladder” process of prime factoring.
Find the LCM of each set of numbers:
►
Example 1: 9, 12, and 15
►
Example 2: 12, 14, and 15
Steps in the Methodology
Step 1
Write the
numbers.
Set up the numbers in a row with
enough space below for many
divisions.
See Special Cases
in Models 1, 2, 3
(see pages 300 & 301).
Try It!
Example 1
9
12
Example 2
15
Chapter 3 — Fractions
302
Steps in the Methodology
Step 2
Divide by the
smallest prime
factor.
Divide by the smallest prime factor
of any of the numbers.
If the chosen factor does not divide
into a number evenly, bring down
that number into the next row,
indicating this with an arrow.
Example 1
Example 2
Divide by 2
2
9
12
15
9
6
15
THINK
9 is not divisible by 2.
12 is divisible by 2.
15 is not divisible by
2.
Step 3
Divide the
next row by
the smallest
prime factor.
Look at the numbers in the second
row.
Divide by the same prime number
if it is still a factor of any of the
numbers in the row, or by the next
higher prime number that is a factor
of any of the numbers in the row.
Bring down the numbers not divisible
by the prime.
Divide
row by
2
9
12
15
2
9
6
15
9
3
15
THINK
6 is divisible by 2.
9 and 15 are not.
Step 4
Divide until
the quotients
are all 1’s.
Continue this process with the third
row and so on until you have only 1’s
remaining.
Divide
row by
2
9
12
15
2
9
6
15
3
9
3
15
3
3
1
5
5
1
1
5
1
1
1
THINK
9, 3, and 15 are all
divisible by 3.
3 is divisible by 3.
1 and 5 are not.
5 is divisible by 5.
Step 5
Multiply the
factors.
Collect all of the factors on the
outside and multiply. The product
is the LCM of the original set of
numbers.
LCM = 2×2×3×3×5
= 180
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
303
MODEL
Find the LCM of 24, 36, 60, and 75 by “pulling out primes.”
Step 1
24
36
60
75
Step 2
2 24
36
60
75
12
18
30
75
2
24
36
60
75
2
12
18
30
75
2 divides into 12, 18, and 30, not 75
2
6
9
15
75
2 divides into 6
3
3
9
15
75
3 divides into 3, 9, 15, and 75
3
1
3
5
25
3 divides into 3
5
1
1
5
25
5 divides into 5 and 25
5
1
1
1
5
5 divides into 5
1
1
1
1
all prime factors found
Step 3
Step 4
THINK
2 divides into 24, 36, and 60, but not 75
THINK
LCM = 2 × 2 × 2 × 3 × 3 × 5 × 5
=
8
=
72
=
1800
×
9
×
25
×
25
The Least Common Denominator
In order to compare, add, and subtract fractions, you will most often find it necessary to build them up to
equivalent fractions with the same denominator. This is because the rewrite allows you to easily compare,
add, or subtract parts (the numerators) when you represent the same number of equal parts in a whole (the
denominators) by the same number for each fraction.
The first step, therefore, is to determine which denominator is suitable to use for your entire set of fractions.
Recall that you can build up a fraction to a specified denominator only if the new denominator is a multiple
of the original one. This new common denominator, therefore, must be a multiple of each of the given
denominator numbers—their common multiple.
To avoid working with larger than necessary numbers, it is most efficient to use the smallest, or Least
Common Denominator (LCD); that is, the LCM of the denominators.
The following methodology presents the steps necessary to rewrite a set of fractions, using their Least Common
Denominator.
Chapter 3 — Fractions
304
METHODOLOGY
Building Equivalent Fractions for a Given Set of Fractions
Determine the LCD and build equivalent fractions for:
►
►
2
,
5
5
,
Example 2:
6
Example 1:
5
, and
7
2
, and
9
4
.
15
7
.
15
Try It!
Steps in the Methodology
Step 1
Find the LCD.
Example 1
Determine the LCD for the
given denominators.
Example 2
3
5
7
15
5
5
7
5
7
1
7
1
1
1
1
LCD = 3×5×7=105
Step 2
Identify
multipliers.
Identify the multipliers for the
numerators and denominators
of each fraction by dividing the
LCD by each denominator.
Shortcut:
Using prime
factors of the LCD
to determine the
multiplier
(see pages 305 &
306,
Models 1, 2, & 3)
21
5 105
−10
multiplier
for 5
)
5
−5
15
7 105
−7
35
−35
)
7
3
15 105
−1 05
)
Step 3
Build up
fractions with
LCD.
Build each fraction to have the
LCD as its new denominator.
Use the multipliers determined
in Step 2 and apply the Identity
Property of Multiplication for
the building up process.
multiplier
for 7
multiplier
for 15
2 21
42
×
=
5 21 105
5 15
75
×
=
7 15 105
4 7
28
× =
15 7 105
Step 4
Present the
answer.
Present your answer.
2
42
,
=
5 105
5
75
=
7 105
4
28
=
,
15 105
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
Steps in the Methodology
Step 5
Example 1
You can validate each
equivalent fraction by applying
the Equality Test for Fractions
(cross-multiplying).
Validate your
answer.
305
Example 2
2 ? 42
=
5
105
?
2 × 105 = 5 × 42
210 = 210 9
5 ? 75
=
7
105
?
5 × 105 = 7 × 75
525 = 525 9
4 ? 28
=
15
105
? 15 × 28
4 × 105 =
420 = 420 9
MODELS
Model 1
Shortcut: Using the Prime Factors of the LCD to Determine the Multipliers
Rewrite the equivalent fractions, using the LCD, for
Step 1
Step 2
2
42
70
35
3
21
35
35
5
7
35
35
7
7
7
7
1
1
1
for 42: 210 ÷ 42 = 5
for 70: 210 ÷ 70 = 3
for 35: 210 ÷ 35 = 6
Shortcut (optional):
Instead of doing the
divisions, use the
prime factors of the
LCD to determine
the multipliers.
1
9
4
,
, and
.
42 70
35
LCD = 2 × 3 × 5 × 7 = 210
5
4 2 210
−210
1
)
3
70 210
−210
)
6
35 210
−210
3
)
210 = 2 × 3 × 5 × 7 and 42 = 2 × 3 × 7
so 210 ÷ 42 = 5, the remaining factor
210 = 2 × 3 × 5 × 7 and 70 = 2 × 5 × 7
so 210 ÷ 70 = 3, the remaining factor
210 = 2 × 3 × 5 × 7 and 35 = 5 × 7
so 210 ÷ 35 = 2 × 3 = 6, the product of the remaining factors
Chapter 3 — Fractions
306
1
5
5
× =
42 5 210
Steps 3 & 4
Step 5
Validate:
9 3
27
× =
70 3 210
1 ?
5
=
42
210
?
1 × 210 = 5 × 42
210 = 210 9
4 6
24
× =
35 6 210
9 ? 27
=
70
210
?
9 × 210 = 27 × 70
1890 = 1890 9
4 ? 24
=
35
210
?
4 × 210 = 24 × 35
840 = 840 9
Model 2
Change the following fractions to equivalent fractions, using the LCD:
Step 1
Step 2
2
18
21
14
3
9
21
7
3
3
7
7
7
1
7
7
1
1
1
5 5
3
,
, and
.
18 21
14
18 = 2 × 3 × 3
21 = 3 × 7
OR
14 = 2 × 7
LCD = 2 × 3 × 3 × 7 = 126
Use the shortcut to finding multipliers:
for 18: 126 ÷ 18 = 126 ÷ (2 × 3 × 3) = 7
for 21: 126 ÷ 21 = 126 ÷ (3 × 7) = 2 × 3 = 6
for 14: 126 ÷ 14 = 126 ÷ (2 × 7) = 3 × 3 = 9
Steps 3 & 4
Step 5
5 7
35
× =
18 7 126
Validate:
5 6
30
× =
21 6 126
5 ? 35
=
18
126
?
5 × 126 = 35 × 18
630 = 630 9
3 9
27
× =
14 9 126
5 ? 30
=
21
126
?
5 × 126 = 30 × 21
630 = 630 9
3 ? 27
=
14
126
?
3 × 126 = 27 × 14
378 = 378 9
Model 3
Write equivalent fractions, using the LCD of the factions:
2 5
1
, , and .
3 7
11
Step 1
The denominators 3, 7, and 11 are all prime. The LCD = 3 × 7 × 11 = 231
Step 2
for 3: 231 ÷ 3 = 7 × 11 = 77
for 7: 231 ÷ 7 = 3 × 11 = 33
for 11: 231 ÷ 11 = 3 × 7 = 21
Note: shortcut to finding multipliers used
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
2 77 154
×
=
3 77 231
Steps 3 & 4
Step 5
Validate:
5 33 165
×
=
7 33 231
2 ? 154
=
3
231
?
2 × 231 = 154 × 3
462 = 462 9
307
1 21
21
×
=
11 21 231
5 ? 165
=
7
231
?
5 × 231 = 165 × 7
1155 = 1155 9
1 ? 21
=
11
231
?
1 × 231 = 21 × 11
231 = 231 9
Ordering Fractions
The most reliable way to order a set of fractions is to determine the LCD of the fractions, build equivalent
fractions using the LCD, and then easily compare the numerators, as in the following methodology.
METHODOLOGY
Ordering Fractions
Put the following sets of fractions in order from smallest to largest:
►
Example 1:
3 13
5
,
, and
5 20
8
►
Example 2:
2 7
5
,
, and
3 12
8
Try It!
Steps in the Methodology
Step 1
Identify the
order.
Step 2
Find the
LCD and
multipliers.
Identify the order requested—
smallest to largest, or largest
to smallest.
Determine the LCD of the
fractions and identify the
multipliers.
Example 1
Example 2
smallest to largest
2
5
20
8
2
5
10
4
2
5
5
2
5
5
5
1
1
1
1
LCD = 2×2×2×5 =40
Identify the multipliers—
for 5: 40 ÷ 5 = 8
for 20: 40 ÷ 20 = 2
for 8: 40 ÷ 8 = 5
Chapter 3 — Fractions
308
Steps in the Methodology
Step 3
Build up
fractions.
Change the equivalent
fractions with the LCD as
the new denominator and
validate by cross-multiplying.
Example 1
Example 2
3 8
24
× =
5 8
40
13 2
26
× =
20 2
40
5 5
25
× =
8 5
40
Validate:
3×40=120 and 24×5=120
13×40=520 and 26×20=520
5×40=200 and 25×8=200
Step 4
Order
numerators.
Compare the numerators of
the new equivalent fractions
and rank them according to
the order identified in Step 1.
Verfiy the ranking.
???
Why can do you do this?
smallest to largest
3 8
24
× =
5 8
40
Rank
1
13 2
26
× =
20 2
40
3
5 5
25
× =
8 5
40
2
Verify: 24 < 25 < 26
Step 4
Present the
answer.
Present your answer with
the original fractions in the
proper order.
3 5 13
, ,
5 8 20
???
Why can do you do Step 4?
Fractions with different numerators and denominators can best be compared when the same common
denominator is the basis for comparison. Once a common denominator has been determined as the basis for
comparison, you have specified how many parts are in one whole.
24 26
25
For example, for
,
, and
, the whole consists of 40 equal parts. Therefore, when you look at how
40 40
40
many parts out of the whole to consider (the numerators), you can easily determine the smaller the numerator,
the smaller the fractional part of the whole that fraction represents.
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
309
ADDRESSING COMMON ERRORS
Issue
Incorrect
Process
Not changing
both the
numerator and
the denominator
when building
up equivalent
fractions
Change 3/5 to fifteenths.
Guessing
the order
of fractions
without finding
a common
denominator
Rank from smallest to
largest:
4
5
3
,
,
15 18 11
3
3
×3 =
5
15
1
Correct
Process
Resolution
Use the Identity
Property of Multiplication
by multiplying both
numerator and
denominator by the
same factor.
Change 3/5 to fifteenths.
3 3•3
9
=
=
5 5 • 3 15
Validate:
3 ? 9
=
5
15
?
3 × 15 = 9 × 5
45 = 45 9
Validate that the built-up
fraction is equivalent to
the original fraction.
Since 3 < 4 < 5 and
11 < 15 < 18, then
3
4
5
<
<
11
1 15 18
Compare fractions by
rewriting in equivalent
forms with a common
denominator.
Rank from smallest to
largest: 4
5
3
,
,
15 18 11
Rank
2
15
18
11
3
15
9
11
3
5
3
11
5
5
1
11
11
1
1
11
1
1
1
LCD = 2×3×3×5×11
= 990
4 • 66
264
=
15 • 66
990
1
5 • 55
275
=
18 • 55
990
3
3 • 90
270
=
11 • 90
990
2
Answer:
4
3
5
,
,
15 11 18
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with building equivalent fractions and ordering fractions
how to apply the Identity Property of Multiplication when building fractions
how to use the Methodology for Finding the LCM of a Set of Numbers to determine the LCD
the reliable way to order fractions
Section 3.5
ACTIVITY
Building Equivalent Fractions,
the Least Common Denominator,
and Ordering Fractions
PERFORMANCE CRITERIA
• Building equivalent fractions to a given denominator
• Ordering a set of fractions
– identification of the Least Common Denominator of the set
– correctly built-up equivalent fractions, each with the LCD
– correct order as specified
CRITICAL THINKING QUESTIONS
1. What is the most important difference between a common factor of a set of numbers and a common multiple
of the numbers?
2. What are three characteristics of a Least Common Denominator?
•
•
•
3. Even though you can use any common denominator for comparing fractions, what are the advantages of
using the lowest common denominator?
310
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
311
4. What is the relationship between finding the LCM and finding the LCD?
5. How do you determine what factors to multiply the numerator and denominator by in order to build up an
equivalent fraction?
6. Why should you use a common denominator to compare two or more fractions?
7. Why must the numerator change when building up a fraction?
Chapter 3 — Fractions
312
TIPS
FOR
SUCCESS
⎛ original numerator
multiplier ⎞⎟
• When building up fractions, use effective notation: ⎜⎜
×
⎟
⎜⎝ original denominator multiplier ⎟⎟⎠
• Use the Least Common Denominator to easily compare the size of fractions rather than trying a “visualize
and guess” approach.
• Use cross-products to validate equivalent fractions.
DEMONSTRATE YOUR UNDERSTANDING
1. Supply the missing numerator for each pair of equivalent fractions.
a)
4
=
9
63
b)
3
=
5
60
c)
2. For each of the following fractions, write three equivalent fractions.
a) 1
8
b)
3
4
c) 5
6
3. a) Determine the LCD of
6
4
2
,
, and
.
35 25
15
b) Write their equivalent fractions and order them from smallest to largest.
2
=
17
51
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
4. a) Determine the LCD of
313
5
7
4
3
,
,
, and
.
18 30 15
10
b) Write their equivalent fractions and order them from largest to smallest.
5 2
7
, and
5. Use the Methodology for Ordering Fractions to put the fractions ,
in order from smallest
8 3
12
to largest.
Chapter 3 — Fractions
314
TEAM EXERCISES
1. In the grids below, fill in the correct numbers of rectangles to represent the following fractions.
(Hint: use your knowledge of equivalent fractions.)
17
100
1
b)
25
a)
Use a pencil.
c)
Use a pen.
d)
2. Find two fractions between
IDENTIFY
AND
3
10
2
1
5
Use a highlighter.
Use a different color highlighter.
1
1
1
and
(greater than 1 and less than ).
3
2
2
3
CORRECT
THE
ERRORS
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column.
Worked Solution
What is Wrong Here?
1) Determine the LCD of
7
8
and
7
.
18
Identify the Errors
144 is a common
denominator, but not
the least (smallest)
common denominator
of 8 and 18.
Correct Process
2
8 18
2
4
9
2
2
9
3
1
9
3
1
3
1
1
Answer:
72
LCD = 2 × 2 × 2 × 3 × 3
= 8 ×9 = 72
Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions
Worked Solution
What is Wrong Here?
2) Order these fractions from
largest to smallest:
2 5 7
, ,
.
3 7 11
3) Put these fractions in order
from smallest to largest:
2 5 7
, ,
.
3 8 10
Identify the Errors
Correct Process
315
Chapter 3 — Fractions
316
Worked Solution
What is Wrong Here?
Identify the Errors
Correct Process
4) Determine the LCD of
5
4
7
,
, and
.
9 18
24
ADDITIONAL EXERCISES
1. Supply the missing numerator:
a)
2
=
9 108
b)
7
=
8
72
c)
11
=
14
42
2. Order the fractions from smallest to largest:
17 3
5
,
, and
25 5
8
3. Order the fractions from largest to smallest:
7
3
7
,
, and
15 4
12
4. Order the fractions from smallest to largest:
2 13 15
7
,
,
, and
3 16 24
12