69

2.3 REDUCING FRACTIONS

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Bringit on!

• the terminology and notation used when reducing fractions

• the meaning of equivalent fractions

• what it means to reduce (simplify) a fraction to its lowest terms

• why you can cancel common factors when reducing

• the validation of the fi nal reduced fraction

• Reducing a fraction to its lowest terms

– correct reducing techniques– validation of the fi nal answer

John didn’t win.Do the math.

Reduce 5521380 . Is it the same as 2

3 ?

Chelsea is the newest reporter for the campus newspaper. The day after the election of the new student body president, John Smith, she found an unsigned note on her desk. It read:

Chelsea realized that if there was any truth to this, it would be an incredible story. In the course of her investigation, she spoke with one of the students on the committee responsible for counting and reporting the votes.

That student told her that of 1380 students who voted, “two-thirds, a clear majority,” voted for John. She then asked the student how many votes John received.

The answer was: “552”.

Chelsea wrote down the fi gures and knew she

needed to verify that 552 is two-thirds of 1380. Is it? Is

there a story here?

25

, They are not the same.

70

Chapter 2 — Fractions

Steps in the Methodology Example 1 Example 2

Step 1

Prime factor— numerator

Determine the prime factorization of the numerator.

Quick reduction(see Model 2)

Shortcut:

2 30

3 15

5 5

1

Step 2

Prime factor— denominator

Determine the prime factorization of the denominator.

2 36

2 18

3 9

3 3

1

Step 3

Write as prime factorization.

Re-write the fraction using the prime factorizations.

2 3 52 2 3 3

× ×× × ×

Example 1: Reduce to its lowest terms.

Example 2: Reduce to its lowest terms. Try It!

This methodology breaks down the numerator and denominator of a fraction to their prime factorizations, in order to easily see their common factors. It is particularly useful to use when the common factors of the original numerator and denominator are not readily apparent to you. While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.

303642

140

Equality Test for Fractions

If two fractions are equal, their cross-products will be equal.

Are the fractions 520780 and

23

equivalent?

Calculate the cross-products (cross-multiply):520

37802

=?Does 520 × 3 equal 2 × 780?Yes, 1,560 = 1,560The two fractions are equivalent.

71

Activity 2.3 — Reducing Fractions

Steps in the Methodology Example 1 Example 2

Step 4

Cancel.

Cancel each common numerator factor with its matching denominator factor.

Recall the Special Property of Division that states that any number divided by itself equals 1. For any fraction, then, if a factor in the numerator is equal to a factor in the denominator, you can apply this property and replace the two factors with the number 1 (or 1/1), a procedure called canceling.

1

1

1

1

2 3 52 2 3 3

× ×× × ×

Step 5

Multiply remaining factors.

Multiply the remaining numerator factors to get the new numerator and the remaining denominator factors to get the new denominator.

1 1 51 2 1 3

56

× ×× × ×

=

Step 6

Present the answer.

Present your answer. 56

Step 7

Validate your answer.

Validate by using the Equality Test for Fractions. Compare the cross-products of the original fraction and the reduced fraction. The cross products must be equal.

Also, there should be no common factors between the numerator and denominator of the fi nal answer.

56

52 3

=×

3036

56

=?

no common factors

?30×6 = 5×36

180 = 180

Model 1

Reduce to lowest terms:

Step 1 Step 2 Step 3 Step 4 Steps 5 & 6

Step 7 Validate:

84 × 19 = 12 × 133

1,596 = 1,596

84133

2 84

2 42

3 21

7 7

1

7 133

19 19

1

2 2 3 77 19

× × ××

2 2 3 77 19

1

1

× × ××

4 319×

=1219

84133

1219

=?

1219

2 2 319

=× ×

no common factors

?

Answer: 1219

72

Chapter 2 — Fractions

Model 2

Reduce to lowest terms:

Use the shortcut to divide out the common factors.

Validate:?

48 × 4 = 3 × 64 192 = 192

4864

34

=? no common factors34

32 2

=×

48 864 8

68

6 28 2

÷÷

= ⇒÷÷

= 34

Answer

4864

Simplify: 440

1870Before Steps 1 and 2, divide out the

factor(s) you recognize as being common to both the numerator and denominator.

Original Methodboth divisible

by 10440 10

1870 1044

187÷÷

=

Step 1 2 440

2 220

2 110

5 55

11 11

1

2 44

2 22

11 11

1

Step 2 2 1870

5 935

11 187

17 17

1

11 187

17 17

1

Step 3 2 2 2 5 112 5 11 17× × × ×× × ×

2 2 1111 17× ×

×

Step 4 1 1 1

1 1 1

2 2 2 5 11

2 5 11 17

× × × ×

× × ×

2 2 11

11 17

1

1

× ×

×

Steps 5 & 6 Answer: 4

17

Step 7 4401870

417

=?4

172 217

=× no common factors

?440 × 17 = 4 × 1870 7,480 = 7,480

Shortcut: Quick Reduction

73

Activity 2.3 — Reducing Fractions

Make Your Own Model

Problem: _________________________________________________________________________

Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of.

1. What is a fully reduced fraction?

2. How do you validate that fractions are equivalent?

3. When reducing to lowest terms, what is the result when all the factors in the numerator cancel out?

Answers will vary.

Fractions are in its lowest terms if there are no factors common to both the numerator and denominator.

Find the cross-products by multiplying the numerator of the fi rst fraction times denominator of the second fraction, then, denominator of the fi rst fraction times numerator of the other fraction, then, comparing to make sure the

The result is that the numerator (or denominator) becomes 1.

74

Chapter 2 — Fractions

4. When reducing to lowest terms, what is the result when all the factors in the denominator cancel out?

5. How can you determine with certainty that a fraction is in lowest terms?

6. How can you be sure that your reduced fraction answer is correct?

7. What aspect of the model you created is the most diffi cult to explain to someone else? Explain why.

Fraction Factorization Reduced Fraction Validation

1)

2496

Reduce each of the following to lowest terms. If improper, write as a mixed number with its fraction in lowest terms.

Answers will vary.

Set the original fraction equal to the reduced answer. If the products of the cross multiplication are the same, then the reduced answer is correct.

The result will be the product of the factors left in the numerator over 1. But we know that by the Division Property of One that any number divided by one is equal to the beginning number.

For example: 2 3 3

32 3

161

6× ×=

×= =

You can tell a fraction is in lowest terms, if you cannot fi nd a common factor that will divide into both the numerator and the denominator. If the numerator is larger than the denominator, you must change to a mixed number.

75

Activity 2.3 — Reducing Fractions

Fraction Factorization Reduced Fraction Validation

2)

2842

3)

6450

4)

7801820

5)

68102

Reduce to lowest terms and validate your answers.

1)

42108

2)

54007500

3)

75165

4)

2756

5) 120162

718

1825

511

2756

2027

already reduced

76

Chapter 2 — Fractions

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) Reduce to lowest terms:

25100

25100

= 14

25 x 4 = 1 x 100100 = 10014

= 12 x 2

fully reducedCorrect

?

?

2) Reduce to lowest terms:

130260

Can be reduced further.

3) Reduce to lowest terms:

20150

Cannot cancel the 5 on the bottom (denominator) twice.