Dividing fractions -- A complete course in arithmetic

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A R I T H M E T I C

Lesson 25 Section 2

Dividing fractions

IN DIVISION, the dividend and divisor be units of the same kind. 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 10.)

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

When we divide pure numbers --

6 ÷ 2 = 3

-- we mean

6 units ÷ 2 units = 3.

For there is no "6" apart from 6 units, even though we do not say the word "units."

With fractions, the units are named by the denominator. (Lesson 20.) Therefore:

6
7

2
7

= 3.

"6 sevenths ÷ 2 sevenths = 3"

-- because 3 times 2 sevenths = 6 sevenths.

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

14 ÷ 15 is equal to

14
15

because we any quotient of whole numbers

a ÷ b is equal to

a
b

3 ÷ 4

3
4

because

3
4

× 4 = 3.

Compare Lesson 10, Example 14.

Example 2.

5
8

7
8

5
7

Example 3.

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 usual way:

5
8

2
3

15
24

16
24

15
16

The common denominator in this case is 8 × 3 = 24.

Example 4.

2
5

3
4

8
20

15
20

8
15

Example 5. 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 then 4.

Example 6.

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 20.)

Example 7. 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 10) -- 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.

"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. But it is based on the principle of equal denominators -- because it gives the numerators, 15 and 16, if we were to make the denominators the same!

5
8

2
3

15
24

16
24

15
16

(We see that we could also obtain the numerators by cross-multiplying.)

Invert and multiply is merely a rule, and therefore it is not very educational. Nevertheless, for certain problems it can be skillful, particularly when the dividend is a whole number.

Example 8. 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

5
4

. And the reciprocal of

5
4

4
5

Reciprocals always come in pairs.

Thus the reciprocal of

1
2

2
1

, or 2. And the reciprocal of 2 is

1
2

See Lesson 28, Examples 6 - 8.

In general, however, the method of common denominators 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 must be the same.

In summary:

4.	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.

Continue on to the next Lesson.

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