Section 3.3
PRE-ACTIVITY
PREPARATION
Reducing Fractions
You must often use numbers to communicate information to
others. When the message includes a fraction whose components
are large, it may not be easily understood. In that case, you
might use a simplified form of the fraction to most effectively
convey the same information. The following example illustrates
the practicality of the mathematical skill of reducing fractions to
their simplest form.
Consider, for an example, the five hundred twenty out of seven
hundred eighty third graders in a certain school district who ride
the bus to school (520/780) . The same fraction in its simplest
form is 2/3—two thirds of the third graders ride the bus to school.
This simplest form more efficiently communicates the same
information to prospective school parents.
LEARNING OBJECTIVES
•
Reduce fractions to lowest terms by dividing or canceling out common factors.
•
Use the meaning of equivalent fractions for validation.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
denominator
cancel
divisibility tests
common factor
factor
cross-multiply
numerator
cross-product
prime factorization
does not equal sign ≠
equivalent fraction
lowest terms
reduce
simplify
259
Chapter 3 — Fractions
260
BUILDING MATHEMATICAL LANGUAGE
Fractions are equivalent (equal) when they represent the same part of a whole or group, the same
division, or the same ratio.
1
2
For example,
is equivalent to
2
4
VISUALIZE
One way to check whether two given fractions are equivalent is to apply the following test.
Equality Test for Fractions
If two fractions are equal, their cross-products will be equal.
A cross-product is found by multiplying the numerator of one fraction by the denominator of
the other.
520
2
For example, to determine whether the fractions in the Introduction,
and , are equivalent fractions,
780
3
you could calculate the cross-products (cross-multiply):
520 ?
=
780
2
3
Does 520 × 3 equal 2 × 780?
Yes, 1,560 = 1,560
The two fractions are equivalent.
2
4
and .
5
9
These two fractions are not equivalent because they do not pass the Test for Equality.
Now consider
That is, 2 × 9 ≠ 4 × 5
18
≠
20
Read, “2 times 9 does not equal 4 times 5”
Read, “18 does not equal 20.”
A common factor of two or more numbers is a factor that they share. That is, it divides evenly into
each of the numbers. For example, 3 and 6 are the two common factors of 12, 18, and 36.
To reduce a fraction is to rewrite it as an equivalent fraction with a smaller numerator and smaller
denominator. To reduce a fraction to its lowest terms or to simplify a fraction is to write the equivalent
fraction whose numerator and denominator have no common factors.
2 1
Example:
= in reduced form.
4 2
Section 3.3 — Reducing Fractions
261
In addition to the divisibility tests for the numbers 2, 3, and 5 which you used to determine the prime
factorization of a number, there are several other divisibility tests which you might use to determine
common factors:
Additional Divisibility Tests
•
If the final two digits of a number form a number divisible by 4, then the original number is
divisible by 4.
•
If a number is divisible by both 2 and 3, then it is divisible by 6.
•
If the sum of the digits of a number is divisible by 9, then the original number is divisible by 9.
•
If the number ends in 0, then it is divisible by 10; if it ends in 00, it is divisible by 100;
in 000, it is divisible by 1000, and so on.
METHODOLOGY
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.
Be sure to note its shortcut options!
Reducing a Fraction
►
►
30
to its lowest terms.
36
42
Example 2: Reduce
to its lowest terms.
140
Example 1: Reduce
Steps in the Methodology
Step 1
Prime factor—
numerator
Step 2
Prime factor—
denominator
Determine the prime factorization of the
numerator.
Shortcut:
Quick reduction
(see page 263, Model 2)
Determine the prime factorization of the
denominator.
Try It!
Example 1
2
30
3
15
5
5
1
2
36
2
18
3
9
3
3
1
Example 2
Chapter 3 — Fractions
262
Steps in the Methodology
Step 3
Write as prime
factorization.
Step 4
Example 1
Re-write the fraction using the prime
factorizations.
Cancel each common numerator factor with
its matching denominator factor.
Cancel.
???
Example 2
2 ×3×5
2×2×3×3
1
2 × 13 × 5
1
2 × 2 × 13 × 3
Why can you do this?
Step 5
Multiply
remaining
factors.
Multliply the remaining numerator factors to
get the new numerator and the remaining
denominator factors to get the new
denominator.
Step 6
Present your answer.
5
6
Present the
answer.
Step 7
Validate your
answer.
1×1× 5
5
=
1× 2 ×1× 3 6
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 final answer.
30 ? 5
=
36
6
?
30×6 = 5×36
180 = 180 9
5
5
=
6
2×3
no common
factors 9
???
Why can you do Step 4?
Recall the Special Property of Division that states that any number divided by itself equals 1.
any number
=1
that same number
1
2
3
= 1,
= 1,,
= 1, and so on.
1
2
3
That is,
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
6
3× 2
3
3
Example:
=
or
× 1 or, by the Identity Property of Multiplication, simply
1
10
5
5
5× 2
You will get the same result if you divide both the numerator and denominator by the same common factor.
6
6÷2
3
=
= OR, with canceling notation,
10 10 ÷ 2
5
3
5
6
10
=
3
.
5
Section 3.3 — Reducing Fractions
263
MODELS
Model 1
84
133
Reduce to lowest terms:
Step 1
2
84
2
42
3
21
7
Step 2
Step 3
133
2×2×3×7
7 × 19
7
19
19
Step 4
Steps 5 & 6
2 × 2 × 3 ×1 7
1
7 × 19
4 × 3 12
=
19
19
1
Answer:
Step 7
7
Validate:
1
84
133
? 12
=
19
?
84 × 19 = 12 × 133
12
19
12 2 × 2 × 3
=
19
19
1,596 = 1,596 9
no common factors 9
Shortcut:
Quick Reduction
Model 2
A
►
440
1870
Simplify:
Shortcut: Before you do Steps 1 and 2, first divide
out the factor(s) you readily recognize as being
common to both the numerator and denominator.
Shortcut Version (optional)
Step 1
Step 2
2
440
2
1870
2
220
5
935
2
110
11
187
5
55
17
11
11
THINK
both divisible by 10
440 ÷ 10
44
=
1870 ÷ 10 187
Step 1
Step 2
17
2
44
11 187
1
2
22
17
1
11 11
1
17
1
2 × 2 × 2 × 5 × 11
2 × 5 × 11 × 17
Step 3
1
Step 4
1
Steps 5 & 6
Step 7
1
1
2 × 5 × 11 × 17
Answer:
Validate:
4
17
440 ? 4
=
1870
17
2 × 2 × 11
11 × 17
1
1
2 × 2 × 2 × 5 × 11
1
Step 3
Step 4
2 × 2 × 11
1
11 × 17
Steps 5 & 6
?
440 × 17 = 4 × 1870
7,480 = 7,480 9
Answer:
4
17
4
2×2
=
17
17
no common factors 9
Chapter 3 — Fractions
264
B
►
48
64
Reduce to lowest terms:
48 ÷ 8
6
=
⇒
64 ÷ 8
8
Use the shortcut to divide out the common factors.
48 ? 3
=
64
4
Validate:
?
48 × 4 = 3 × 64
192 = 192 9
6÷2
3 Answer
=
8÷2
4
3
3
no common factors 9
=
4 2×2
ADDRESSING COMMON ERRORS
Incorrect
Process
Issue
150
Not reducing Simplify:
240
all the way
to lowest
terms when 150 ÷ 10
15
=
simplifying a 240 ÷ 1
24
10
fraction
Correct
Process
Resolution
Simplify:
For this issue, validating
by cross-multiplying
alone does not catch
the error since both
fractions are, in fact,
equivalent.
?
Reduce:
60
=
126
1
=
=
1
1
60
126
1
2 × 2 × 3 ×5
1
1
2 × 3 × 3 ×7
5
7
150 ? 5
=
240
8
1
?
150 × 8 = 5 × 240
1200 = 1200 9
15
3 ×5
=
1
24
2×2×2× 3
5
=
8
Reduce:
Use effective notation.
Factors must be
canceled in pairs: one
numerator factor with
only one denominator
factor, with each pair
equaling one (1).
150
240
150 ÷ 10
15
=
240 ÷ 10
24
150 × 24
15 × 240 Always do a prime
4=1
factorization of your
3600 = 3600
final answer to assure
that there are no
remaining common
factors to cancel.
Mismatching
the factors
when
canceling
Validation
60
=
126
1
1
60
126
5
5
=
8
2×2×2
no common
factors 9
?
60 × 21 = 10 × 26
1260 = 1260 9
1
2 ×2 × 3 ×5
1
2 × 3 ×3×7
10
=
21
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with 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 final reduced fraction
10
2 ×5
=
21 3 × 7
no common
factors 9
Section 3.3
ACTIVITY
Reducing Fractions
PERFORMANCE CRITERIA
• Reducing a fraction to its lowest terms
– correct reducing techniques
– validation of the final answer
CRITICAL THINKING QUESTIONS
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?
265
Chapter 3 — Fractions
266
4. When reducing to lowest terms, what is the result when all the factors in the denominator cancel out?
5. How can you tell if a fraction is in lowest terms?
6. How can you be sure that your reduced fraction answer is correct?
TIPS
FOR
SUCCESS
• Know and use Divisibility Tests to quickly cancel common factors.
• When you cannot readily recognize common factors, use the prime factorization of the numerator and
denominator to reduce (by using the Methodology for Reducing a Fraction).
• Cross multiply to test the equivalency of your reduced fraction.
• Even when you do quick reduction by common factors, always prime factor your final answer to assure that
there are no remaining common factors.
Section 3.3 — Reducing Fractions
267
DEMONSTRATE YOUR UNDERSTANDING
Reduce each of the following to lowest terms. If improper, write as a mixed number with its fraction in lowest terms.
Fraction
1)
24
96
2)
28
42
3)
64
50
4)
780
1820
5)
68
102
Factorization
Reduced
Fraction
Validation
Chapter 3 — Fractions
268
IDENTIFY
AND
CORRECT
THE
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) Reduce to lowest terms:
25
100
Identify Errors
or Validate
25
100
25 × 4
100
Correct Process
Validation
? 1
=
4
= 1 × 100
= 100
?
1
1
=
4
2 ×2
fully reduced
Correct
2) Reduce to lowest terms:
130
260
3) Reduce to lowest terms:
20
150
ADDITIONAL EXERCISES
Reduce to lowest terms and validate your answers.
1)
42
108
2)
5400
7500
3)
75
165
4)
27
56
5)
120
162