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Lesson 26  Section 2

How to divide fractions

Section 1:  How to multiply fractions

WE ARE ABOUT to present an alternative to the method of  "Invert and multiply." (But see below.) It is based on a techniqe the student already knows, namely finding a common denominator.

For, in division, the dividend and divisor must be units of the same kind. They must have the same name. We can only divide dollars by dollars, hours by hours, yards by yards.

15 yards ÷ 3 yards = 5

-- because 5 times 3 yards = 15 yards. (Lesson 11.)

(We cannot divide 15 yards by 3 feet -- not until we change yards to feet.)

With fractions, the units are named by the denominator. (Lesson 21.) Therefore:

6
7
  ÷   2
7
 =  3.

"6 sevenths ÷ 2 sevenths = 3"

-- because 3 times 2 sevenths = 6 sevenths.

3 is how many times 2 sevenths are contained in 6 sevenths -- which is the answer to the question that division asks.

Here is the rule:

To divide fractions, the denominators must be the same.
The quotient will be the quotient of the numerators.

  Example 1.     14
20
  ÷   15
20
  =   14  ÷ 15   =   14
15
.

See Lesson 11, Example 16.

  Example 2.   Prove:     14
20
  ÷   15
20
  =   14
15
.
  Solution.   We must show that the quotient,  14
15
, times the divisor,  15
20
, will
  equal the dividend,  14
20
. (Lesson 11.)

And on canceling the 15's --

14
15
  ×   15
20
  =   14
20
.

-- it does.  (Section 1, Example 9.)

Therefore when the denominators are the same, the quotient will be the quotient of the numerators.

  Example 3.     5
8
  ÷   7
8
  =   5
7
.
  Example 4.     7
8
  ÷   5
8
  =   7
5
  =  1 2
5
.

Different denominators

When the denominators are not the same --

5
8
  ÷   2
3

-- we can make a common denominator in the same way that we add fractions:

5
8
  ÷   2
3
  =   15
24
  ÷   16
24
  =   15
16
.

The common denominator here is  8 × 3 = 24.

  Example 5.     2
5
  ÷   3
4
  =    8 
20
  ÷   15
20
  =    8 
15
.
  Example 6.    1 1
4
  ÷  2 1
2
  =   5
4
  ÷   5
2
 
    =   5
4
  ÷   10
 4
 
    =    5 
10
 
    =   1
2
.

As in multiplication, we must change mixed numbers to improper fractions.  The common denominator is this example is 4.

  Example 7.     3
5
  ÷  2   =   3
5
  ÷   10
 5
  =    3 
10
.

To change a whole number into a fraction, multiply the whole number by the denominator.

2  =   10
 5

That product will be the numerator. (Lesson 21.)

Example 8.    A bottle of medicine contains 15 oz.  Each dose of the medicine is 2½ oz.  How many doses are there in the bottle?

 Solution.   This is a division problem (Lesson 11) -- how many times can we subtract 2½ oz from 15 oz?

15 ÷ 2½  =  15 ÷   5
2
  =   30
 2
  ÷   5
2
  =  30 ÷ 5 = 6.

In that bottle there are 6 doses.

   Example 9.   On a map,  3
4
 of an inch represents 60 miles.  How many

miles does 2 inches represent?

  Solution.   For every   3
4
 of an inch there are 60 miles.  How many  3
4
 of an

inch, then, are there in 2 inches?

That number times 60 will produce the answer.

2 ÷  3
4
 =  8
4
 ÷  3
4
 =  8
3
 = 2 2
3
.
8
3
 × 60  = 8 × 20 = 160 miles.

Or:

"Two and two thirds times 60  =  Two times 60 + two thirds of 60
Lesson 16
   =  120 + 40
 
   =  160 miles."

For a solution based on proportions, see Lesson 23, Example 5.

"Invert and multiply"

A method often taught is:  "Invert the divisor and multiply."

5
8
  ÷   2
3
  =   5
8
  ×   3
2
  =   15
16
.

As with many written methods, this is a trick that gives the right answer.  It is based on the principle of equal denominators.  Because
-- equivalent to inverting -- if we cross-multiplied:

5
8
  ÷   2
3
  =   15   ÷   16   =   15
16

-- we get the numerators if we changed those fractions to a common denominator.

Invert and multiply is merely a rule, and therefore it is not very educational.  Nevertheless, for certain problems it can be skillful, especially when the dividend is a whole number.
 Example 9.     40   ÷   4
5
  =   40   ×   5
4
  =   10 × 5   =   50.

Invert the divisor -- the number after the division sign ÷ .  Divide 4 into 40, then multiply.

When we invert a fraction, the number we obtain is called its

  reciprocal.  The reciprocal of   4
5
  is  5
4
.  And the reciprocal of   5
4
  is  4
5
.

Reciprocals come in pairs.

Thus the reciprocal of   1
2
 is  2
1
, or 2.  And the reciprocal of 2 is  1
2
.

See Lesson 29, Examples 6 - 8.

The method of common denominators, however, is to be preferred. It uses a skill the student has already learned.  And what is more, it emphasizes a basic property of division, namely:  The units -- the names of what we are dividing -- must be the same.

In algebra, which in any event is just rules, it is skillful to divide by multiplying by the reciprocal. Skill in arithmetic, however, requires understanding.

In summary:


 5.   How do we divide fractions?
 
 
  The denominators must be the same. The quotient will be the quotient of the numerators.
 

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

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