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 must 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)

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.

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

14 ÷ 15 is equal to

14
15

because 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 15.

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 this example is 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. And it is based on the principle of equal denominators -- because if we were to make the denominators the same, the numerators would become 15 and 16

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, especially 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 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 -- the names of what we are dividing -- must be the same.

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.

Continue on to the next Lesson.

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