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We are now ready to look at the operation of division on fractions. First we will need a newconcept, the reciprocal of a fraction.
We invert, or turn over, a fraction to write its reciprocal.
Rules and Properties: The Reciprocal of a Fraction
The product of any number and its reciprocal is 1. (Every number except zerohas a reciprocal.)
Rules and Properties: Reciprocal Products
NOTE In general, the
reciprocal of the fraction is .
Neither a nor b can be 0.
ba
ab
Example 1
Finding the Reciprocal of a Fraction
Find the reciprocal of (a) , (b) 5, and (c) .
(a) The reciprocal of is .
(b) The reciprocal of 5, or , is .
(c) The reciprocal of , or , is .3
5
5
31
2
3
1
5
5
1
Just invert, or turn over, thefraction.
4
3
3
4
12
3
3
4
C H E C K Y O U R S E L F 1
Find the reciprocal of (a) and (b) .314
58
An important property relating a number and its reciprocal follows.
Write 5 as and then turn
over the fraction.
51
Write as , then invert.53
123
We are now ready to use the reciprocal to find a rule for dividing fractions. Recall thatwe can represent the operation of division in several ways. We used the symbol � earlier.Remember that a fraction also indicates division. For instance,
3 � 5 �In this statement, 5 is called thedivisor. It follows the division sign �and is written below the fraction bar.
A complex fraction is written by placing thedividend in the numerator and the divisor inthe denominator.
2
3
4
5
23
2
3�
4
5
Example 2
C H E C K Y O U R S E L F 2
Write � as a complex fraction.34
25
Let’s continue with the same division problem.
Dividing Two Fractions
(1)
(2)
We see from lines (1) and (2) that
2
5�
3
4�
2
5�
4
3
Recall that a number divided by 1 is justthat number.
�2
5�
4
3
The denominator becomes 1.�
2
5�
4
3
1
�
2
5�
4
3
3
4�
4
3
Write the original quotient as acomplex fraction.
�
2
5
3
4
2
5�
3
4
Multiply the numerator and denominator
by , the reciprocal of the denominator.
This does not change the value of thefraction.
43
Example 3
Using this information, we can write a statement involving fractions and division as acomplex fraction, which has a fraction as both its numerator and denominator, as Example 2illustrates.
We would certainly like to be able to divide fractions easily without all the work of thelast example. Look carefully at the example. The following rule is suggested.
To divide one fraction by another, invert the divisor (the fraction after thedivision sign) and multiply.
Rules and Properties: To Divide Fractions
C H E C K Y O U R S E L F 3
Write � as a multiplication problem.78
35
C H E C K Y O U R S E L F 4
Divide.
2
5�
3
4
Example 4
Dividing Two Fractions
Divide.
We invert the divisor, , then multiply.
�1 � 7
3 � 4�
7
12
47
�1
3�
7
4
1
3�
4
7
Let’s look at another similar example.
Example 4 applies the rule for dividing fractions.
Dividing Two Fractions
Divide.
Write the quotient as a mixednumber if necessary.
�5
8�
5
3�
5 � 5
8 � 3�
25
24� 1
1
24
5
8�
3
5
Example 5
NOTE Remember, the numberinverted is the divisor. It followsthe division sign.
Simplifying will also be useful in dividing fractions. Consider the next example.
C H E C K Y O U R S E L F 6
Divide.
4
9�
8
15
Dividing Two Fractions
Divide.
1
2
�3 � 7
5 � 6�
7
10
Invert the divisor first! Thenyou can divide by thecommon factor of 3.
�3
5�
7
6
3
5�
6
7
When mixed or whole numbers are involved, the process is similar. Simply change themixed or whole numbers to improper fractions as the first step. Then proceed with thedivision rule. Example 7 illustrates this approach.
Dividing Two Mixed Numbers
Divide.
1
2
�19
14� 1
5
14
Invert the divisor and multiplyas before.
�19
8�
4
7
Write the mixed numbers asimproper fractions.
�19
8�
7
42
3
8� 1
3
4
Example 6
Example 7
NOTE Be careful! We mustinvert the divisor before anysimplification.
Example 8 illustrates the division process when a whole number is involved.
Dividing a Mixed Number and a Whole Number
Divide and simplify.
3
2
�3
10
Invert the divisor, then divideby the common factor of 3.
�9
5�
1
6
�9
5�
6
11
4
5� 6
Example 8
C H E C K Y O U R S E L F 7
Divide.
31
5� 2
2
5
C H E C K Y O U R S E L F 8
Divide.
8 � 44
5
NOTE Write the whole
number 6 as .61
Units AnalysisWhen dividing by denominate numbers that have fractional units, we multiplyby the reciprocal of the number and its units.
Examples
500 mi � � 500 mi � � 20 gal
$24,000 � � 24,000 dol � � 24,000 dol � � 60 yr
(As always, note that in each case, the arithmetic of the units produces the finalunits.)
1 yr400 dol
400 dol1 yr
$4001 yr
1 gal25 mi
25 mi1 gal
As was the case with multiplication, our work with the division of fractions will be usedin the solution of a variety of applications. The steps of the problem-solving process remainthe same.
49. Quantity. A butcher wants to wrap lb packages of ground beef from a cut of meat
weighing lb. How many packages can be prepared?
50. Quantity. A manufacturer has yards (yd) of imported cotton fabric. A shirt
pattern uses yd. How many shirts can be made?
51. Number of pieces. A stack of in. thick plywood is 48 in. high. How many sheets
of plywood are in the stack?
52. Area. A landfill occupies land that measures mi by mi. If there are 144 cellsin the landfill, what is the area of each cell?
53. Manuel has yd of cloth. He wants to cut it into strips yd long. How many strips
will he have? How much cloth remains, if any?
54. Evette has ft of string. She wants to cut it into pieces ft long. How many
pieces of string will she have? How much string remains, if any?
55. In squeezing oranges for fresh juice, three oranges yield about of a cup.
(a) How much juice could you expect to obtain from a bag containing 24 oranges?
(b) If you needed 8 cups of orange juice, how many bags of oranges should you buy?
56. A farmer died and left 17 cows to be divided among three workers. The first worker
was to receive of the cows, the second worker was to receive of the cows, and the
third worker was to receive of the cows. The executor of the farmer’s estate realized
that 17 cows could not be divided into halves, thirds, or ninths and so added aneighbor’s cow to the farmer’s. With 18 cows, the executor gave 9 cows to the firstworker, 6 cows to the second worker, and 2 cows to the third worker. This accountedfor the 17 cows, so the executor returned the borrowed cow to the neighbor. Explainwhy this works.