10. Can you divide by 5 by dividing the numerator by the integer an integral number of times?

11.5 means that of is to be taken. Since of , then, ÷ 5 = 16·

=

12.

=

?

each? Then

Can 18 be separated into 5 equal parts of +5=5. ÷5 옳. Observe that multiplying the by 5 changes the fractional unit from thirds to fifteenths and takes one of the 5 equal parts into which has been changed.

denominator of

A fraction may be divided by an integer by multiplying the denominator by the integer.

Solve the following problems by the more convenient method and give your reasons:

A fraction may be divided by an integer either by dividing the numerator or by multiplying the denominator of the fraction by the integer.

NOTE. Small mixed numbers are frequently reduced to improper fractions and then divided by the same principle as proper fractions.

NOTE. The division may be performed by changing both numbers to sevenths, but it is a much shorter process to perform the work as indicated, changing the remaining mixed number, 84, to an improper fraction, 2, and dividing it by 9. Thus, of }}.

9)12869

14280

Dividing fractions by first making them similar.

1. What is the unit of measure in each of the following:

8; 6 yd.; 5 ft.; 4 in.; 8 rd.; 5 oz.; 2 lb.; 10 mi.?

2. What kind of units may be added or subtracted?

3. Since both the dividend and the divisor of concrete numbers must represent like units, to divide 24 yd. by 3 ft. we must either change yards to feet, thus, 72 ft. ÷ 3 ft.; or feet to yards, thus, 24 yd. ÷ 1 yd.

4. In ÷ are the fractional units alike in size and kind? Then how often is contained in &?

5. Since is contained 3 times in §, how can we divide one fraction by another when the denominators are alike? A fraction may be divided by another fraction by changing both fractions to a common denominator and dividing the

By inverting the terms of the divisor and multiplying.

1. How many half inches are there in 1 in.? in 2 in.? Then how many times is contained in 1? in

3 in.? 4 in.

2? in 3? in 4?

To divide any number by a fractional unit multiply the number by the denominator; thus, 12 ÷ 1 = 12 × = 48, or 48.

how many?

19. 1÷

= how many

?

16.

=

how many

?

1+4

=

1÷季 = how many? 20. 1÷2

Observe that the number of times each of the above fractions is contained in 1 equals the number of times the numerator is contained in the denominator.

The number of times a fraction is contained in 1 is called the reciprocal of the fraction. Thus, is contained in 1, § times. Hence is the reciprocal of .

1 divided by any fraction equals the fraction inverted.

Find quotients:

Any number is divided by a fraction by inverting the terms of the divisor and multiplying.

30. Work 18+ by both methods. What principle does the second method introduce that the first does not?

NOTE.- The process should be shortened by cancellation when possible.

Written Work

Change the mixed numbers to improper fractions before dividing. Divide:

195

38.22 56. (of)+(of 3)