Building Equivalent Fractions, the Least Common Denominator, and

295

PRE-ACTIVITY

PREPARATION

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 confi gurations for the same whole unit or group will expand your ability to look for patterns within sets of numbers.

• 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.

Building Equivalent Fractions, the Least Common Denominator,

and Ordering Fractions

LLEARNINGEARNING OOBJECTIVESBJECTIVES

TTERMINOLOGYERMINOLOGY

NEW TERMS TO LEARN

building up

common denominator

common multiple

higher terms

Least Common Denominator (LCD)

Least Common Multiple (LCM)

PREVIOUSLY USED

cross-multiply

cross-product

denominator

equivalent fraction

factors

multiple

multiplier

numerator

prime factorization

prime factors

primes

Section 3.5

296 Chapter 3 — Fractions

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

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

fact that the number 1 can take the form of

you can write an infi nite number of fractions equivalent to a given fraction.

any number

the same number,

Example : 2

5, a fraction whose value is equiv× = •

•=3

3

2 3

5 3

6

15aalent to .

2

5

2

5 shaded

6

15 shaded

VISUALIZEThe 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

5,

2

5,

2

5, and s× = × = × =2

2

4

10

4

4

8

20

13

13

26

65oo on.

Now suppose that you want to take a fraction and build an equivalent fraction with a specifi c denominator. As long as the new denominator is a multiple of the original denominator, you can use the following technique.

Building Equivalent Fractions

297Section 3.5 — Building Equivalent Fractions, the Least Common Denominator, and Ordering Fractions

TTECHNIQUEECHNIQUE

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.

Building Up an Equivalent Fraction with a Specifi ed Denominator

►►A ►►BStep 1 42 ÷ 7 = 6

Step 2 5 × 6 = 30

MMODELSODELS

57 42= ? 3

13 39= ?

Step 1 39 ÷ 13 = 3

Step 2 3 × 3 = 957

57 42

66

66

× = ••= 30 3

13 3933× = 9

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).

313

939

3 39 9 13117 117

=

× = ×=

?

?

57

3042

5 42 7 30210 210

=

× = ×=

?

?

►►C 48

= ?

Before Step 1, write the whole number as a fraction: 441 8

= = ?

Step 1 8 ÷ 1 = 8

Step 2 4 × 8 = 32

41 8

88× = 32

In this case, to validate you know that , an improper fraction, is equal to 4.328


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 infi nitely 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 ineffi cient and prone to computational errors when fi nding the multiples.

There are more effi cient 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.


MMETHODOLOGIESETHODOLOGIES

Determining the Least Common Multiple (LCM) of a Set of Numbers by Prime Factorization

Steps in the Methodology Example 1 Example 2

Step 1

Prime factor each number.

Determine the prime factorization of each number.

3 93 3 1

9 = 3 × 3

12 = 2 × 2 × 3

15 = 3 × 5

Step 2

Identify primes.

Identify all the primes that are factors in the prime factorizations. 2, 3, 5

Step 3

Choose necessary factors.

Use each prime as a factor of the LCM the greatest number of times it appears in any one of the prime factorizations.

9 = 3 × 3

12 = 2 × 2 × 3

15 = 3 × 5

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

Example 1: 9, 12, and 15


►►

►► Try It!

? ? ?Why do you do this?

Find the LCM of each set of numbers:

THINK

3 155 5 1

2 122 63 3 1Largest number is divisible

by every other number (see page 300, Model 1)

Special Case:

All are prime numbers (see page 300, Model 2)

Special Case:

Readily apparent that the numbers share no common factors (see page 301,Model 3)

Special Case:


? ? ? 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.

2 × 2 × 3 × 3 × 5

MMODELSODELS

Model 1 Special Case: Largest Number is Divisible by Every Other Number

Model 2 Special Case: All are Prime Numbers

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.

} }}9

1512

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).

2, 7, and 13 are all prime.

The LCM = 2 × 7 × 13 = 182.

If the numbers are all distinct prime numbers, there are no common factors. The LCM is the product of the prime numbers.

Determine the LCM of 2, 3, 6, and 12.

Determine the LCM of 2, 7, and 13.


Model 3 Special Case: Readily Apparent that the Numbers Share No Common Factors

7 is prime

8 is only divisible by 2 no common factors; the LCM = 7 × 8 × 9 = 504

9 is only divisible by 3

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

}Determining the Least Common Multiple (LCM) of a Set of Numbers

by “Pulling Out Primes”



►►

►► Try It!

This methodology presents another effi cient yet more compact process for determining the LCM when it is not readily apparent. It condenses the fi rst 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:


Step 1

Write the numbers.

Set up the numbers in a row with enough space below for many divisions.

9 12 15

See Special Cases in Models 1, 2, 3 (see pages 300 & 301).

Determine the LCM of 7, 8, and 9.



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.

Divide by 2

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.

Dividerow by

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.

Dividerow by

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

THINK

THINK

THINK

2 9 12 15

9 6 15

2 9 12 15

2 9 6 15

9 3 15

2 9 12 15

2 9 6 15

3 9 3 15

3 3 1 5

5 1 1 5

1 1 1


MMODELODEL

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 2 divides into 24, 36, and 60, but not 75

12 18 30 75

Step 3 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 4 LCM = 2 × 2 × 2 × 3 × 3 × 5 × 5

= 8 × 9 × 25

= 72 × 25

= 1800

THINK

THINK

In order to compare, add, and subtract fractions, you will most often fi nd 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 fi rst 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 specifi ed 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 effi cient 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.

The Least Common Denominator


MMETHODOLOGYETHODOLOGY

Building Equivalent Fractions for a Given Set of Fractions


Step 1

Find the LCD.

Determine the LCD for the given denominators.

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.

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.

Step 4

Present the answer.

Present your answer.

Example 1:

Example 2: Try It!

Determine the LCD and build equivalent fractions for:

►►

►►

2

5,

5

7, and

4

15.

5

6,

2

9, and

7

15.

3 5 7 15

5 5 7 5

7 1 7 1

1 1 1

25

= 42105

415

=28

10557

=75

105

, ,

25

42105

2121× =

57

75105

1515× =

415

28105

77× =

)5 10510

55

21

−

−

)7 10573535

15

−

−

)15 105105

73

−

multiplier for 5

multiplier for 7

multiplier for 15

Shortcut:

Using prime factors of the LCD to determine the multiplier (see pages 305 & 306, Models 1, 2, & 3)



Step 5

Validate your answer.

You can validate each equivalent fraction by applying the Equality Test for Fractions (cross-multiplying).

25

42105

2 105 5 42210 210

=

× = ×=

57

75105

5 105 7 75525 525

=

× = ×=

415

28105

4 105 15 28420 420

=

× = ×=

?

?

?

?

?

?

MMODELSODELS

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 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

42, , .

9

70 and

4

352 42 70 35

3 21 35 35

5 7 35 35

7 7 7 7

1 1 1 LCD = 2 × 3 × 5 × 7 = 210

)42 210210

51

−)70 210210

3

−)35 210210

63

−

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


Steps 3 & 4

Step 5 Validate:

Model 2

142

55

970

33

435

66

× = × = × =5210

27210

24210

142

5210

1 210 5 42210 210

=

× = ×=

970

27210

9 210 27 701890 1890

=

× = ×=

435

24210

4 210 24 35840 840

=

× = ×=

?

?

?

?

?

?

Change the following fractions to equivalent fractions, using the LCD:

Step 1

Step 2 Use the shortcut to fi nding 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 Validate:

5

18

5

21

3

14, . , and

2 18 21 14

3 9 21 7

3 3 7 7

7 1 7 7

1 1 1 LCD = 2 × 3 × 3 × 7 = 126

518

77

521

66

314

99

× = × = × =35126

30126

27126

518

5 126 35 18630 630

=

× = ×=

35126

521

5 126 30 21630 630

=

× = ×=

30126

3

143 126 27 14

378 378

=

× = ×=

27126

?

?

?

?

?

?

Model 3

Write equivalent fractions, using the LCD of the factions:

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

2

3

5

7

1

11, . , and

Note: shortcut to fi nding multipliers used

OR

18 = 2 × 3 × 3

21 = 3 × 7

14 = 2 × 7


MMETHODOLOGYETHODOLOGY

Ordering Fractions


Step 1

Identify the order.

Identify the order requested—smallest to largest, or largest to smallest.

smallest to largest

Step 2

Find the LCD and multipliers.

Determine the LCD of the fractions and identify the multipliers.

LCD = 2×2×2×5 =40

Identify the multipliers—

for 5: 40 ÷ 5 = 8

for 20: 40 ÷ 20 = 2

for 8: 40 ÷ 8 = 5

Example 1:

Example 2: Try It!

Put the following sets of fractions in order from smallest to largest:

►►

►►

Steps 3 & 4

Step 5 Validate:

23

7777

57

3333

111

2121

× = × = × =154231

165231

21231

23

2 231 154 3462 462

=

× = ×=

154231

57

5 231 165 71155 1155

=

× = ×=

165231

111

1 231 21 11231 231

=

× = ×=

21231

?

?

?

?

?

?

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.

3

5

13

20

5

8, , and

2

3

7

12

5

8, , and

2 5 20 8

2 5 10 4

2 5 5 2

5 5 5 1

1 1 1

Ordering Fractions



Step 3

Build up fractions.

Change the equivalent fractions with the LCD as the new denominator and validate by cross-multiplying.

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 identifi ed in Step 1.

Verfi y the ranking.

smallest to largest

Rank

1

3

2

Verify: 24 < 25 < 26

Step 4

Present the answer.

Present your answer with the original fractions in the proper order.

? ? ? Why can do you do this?

? ? ? 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 specifi ed how many parts are in one whole.

For example, for the whole consists of 40 equal parts. Therefore, when you look at how

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.

35

88

2440

× =

1320

22

2640

× =

58

55

2540

× =

35

88

2440

× =

1320

22

2640

× =

58

55

2540

× =

35

, 58

, 1320

24

40

26

40

25

40, , and ,


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

Issue Incorrect Process Resolution Correct

Process

Not changing both the numerator and the denominator when building up equivalent fractions

Change 3/5 to fi fteenths. Use the Identity Property of Multiplication by multiplying both numerator and denominator by the same factor.

Validate that the built-up fraction is equivalent to the original fraction.

Change 3/5 to fi fteenths.

Validate:

Guessing the order of fractions without fi nding a common denominator

Rank from smallest to largest:

Since 3 < 4 < 5 and

11 < 15 < 18, then

Compare fractions by rewriting in equivalent forms with a common denominator.

Rank from smallest to largest:

PPREPARATION REPARATION IINVENTORYNVENTORY

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

35

3315

× =

35

3 35 3

= ••= 9

15

35

915

3 15 9 545 45

=

× = ×=

?

?

415

518

3, ,

11

415

518

3, ,

11

311

415

518

< <

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 6615 66

264990

••

=

5 5518 55

275990

••

=

3 9011 90

270990

••

=

Rank

1

3

2

Answer: 4

15311

518

, ,

35 13

3×3

31

<15

<

310

ACTIVITY

Building Equivalent Fractions, the Least Common Denominator,

and Ordering Fractions

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

• Building equivalent fractions to a given denominator

• Ordering a set of fractions – identifi cation of the Least Common Denominator of the set – correctly built-up equivalent fractions, each with the LCD – correct order as specifi ed

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What is the most important difference between a common factor of a set of numbers and a common multipleof 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?

Section 3.5


4. What is the relationship between fi nding the LCM and fi nding 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?


TTIPS FOR IPS FOR SSUCCESSUCCESS

DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

1. Supply the missing numerator for each pair of equivalent fractions.

a) b) c)

2. For each of the following fractions, write three equivalent fractions.

a)

b)

c)

3. a) Determine the LCD of

b) Write their equivalent fractions and order them from smallest to largest.

• When building up fractions, use effective notation:

• 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.

original numerator

original denominator

multiplier

multiplie×

rr

⎛⎝⎜⎜⎜⎜

⎞⎠⎟⎟⎟⎟

4

9 63=

3

5 60= 2

17 51=

1

8

3

4

5

6

6

35

4

25

2

15, , and .


4. a) Determine the LCD of

b) Write their equivalent fractions and order them from largest to smallest.

5. Use the Methodology for Ordering Fractions to put the fractions in order from smallest to largest.

5

18

7

30

4

15

3

10, , , and .

5

8

2

3

7

12, , and


TEAM EXERCISESTEAM EXERCISES

1. In the grids below, fi ll in the correct numbers of rectangles to represent the following fractions.(Hint: use your knowledge of equivalent fractions.)

a) Use a pencil. c) Use a highlighter.

b) Use a pen. d) Use a different color highlighter.

2. Find two fractions between and (greater than and less than ).

17

1001

25

3

10

12

5

1

31

3

1

2

1

2

Worked SolutionWhat is Wrong Here? Identify the Errors Correct Process

1) Determine the LCD of 144 is a common denominator, but not the least (smallest) common denominator of 8 and 18.

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

78

718

and .

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.

2 8 18

2 4 9

2 2 9

3 1 9

3 1 3

1 1

LCD = × × × ×= × =

2 2 2 3 38 9 72

Answer: 72



2) Order these fractions from largest to smallest:

3) Put these fractions in order from smallest to largest:

23

57

711

, , .

23

58

710

, , .



4) Determine the LCD of

ADDITIONAL EXERCISESADDITIONAL EXERCISES

1. Supply the missing numerator:

a) b) c)

2. Order the fractions from smallest to largest:

3. Order the fractions from largest to smallest:

4. Order the fractions from smallest to largest:

2

9 108=

7

8 72= 11

14 42=

17

25

3

5

5

8, , and

7

15

3

4

7

12, , and

2

3

13

16

15

24

7

12, , , and

59

418

724

, , and .

Documents

Building Equivalent Fractions, the Least Common Denominator, and