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which may be either a mixed number less than the divisor, or a proper fraction; then dividing the two parts of the dividend by the divisor, and taking the sum of the partial quotients for the complete quotient.

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In practice, these and similar calculations may be made as follows.

1. 7)78543

1122+ rem. ÷7=*

... 78543÷7=112235.

2. 9)1327

147 +4 rem2. 41÷9=}÷9=1.

.. 1327÷9=1471.

c. The quotient of a large mixed number by a fraction may be obtained by combining the processes of Arts. 94. a. and 99. b.

For example, the quotient of 156271 by is found by multiplying 15627} by 5 and dividing the product by 4, thus,—

156271

5

4)781363

19534+rem". and ÷4==}.

.. 15627÷195341.

If the divisor is a mixed number, it may be reduced to an improper fraction, and the division made as in the preceding example.

d. These abridged processes of division are described in the following rules.

I. To divide a fraction by a whole number:-Divide the

numerator or multiply the denominator of the dividend by the divisor.

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II. To divide a whole number by a fraction : Multiply the dividend by the denominator of the divisor and divide the product by the numerator of the divisor.

III. To divide a mixed number by a whole number :-Divide the integral part of the dividend by the divisor; to the remainder, if there be one, annex the fractional part of the dividend; divide this mixed number or fraction by the divisor and annex the second partial quotient (which is a proper fraction) to the first. The mixed number thus formed is the required

quotient.

IV. To divide a mixed number by a fraction or a mixed number (having reduced the divisor, if a mixed number, to an improper fraction):- Multiply the dividend by the denominator of the divisor and divide this product by the numerator of the divisor.

100. Additional examples in Division of vulgar fractions. It is required to divide

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26. 1762 by 133.

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102. The fractional expressions hitherto considered are fractions of unity: thus,

of 1.

But a fractional part or certain fractional parts may also be taken of any number either less or greater than unity: as, 3 of ; of 3; of 43.

A fraction of unity is called a simple fraction; and a fraction of any other number, a complex or compound fraction.

If the sum or difference of any complex fractions or of complex and simple fractions is required, it becomes necessary to reduce the complex to simple fractions as a preparation for their reduction to a common denominator with each other or with simple fractions.

Now the fraction }, or ¿ of 1, signifies that 1 is divided into 8 equal parts, and that 7 of these equal parts compose the fraction. In like manner, the fraction of, or 3 of 7 of 1, signifies that of 1 are divided into 5 equal parts, and that 3 of these equal parts compose the fraction of 7.

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Whence in these instances (and the method is general) the reduction of a fraction of any other fractional expression to a

3 x 9

27

=

=

39.

7

fraction of unity is made by multiplying together the numerators of the parts of the complex fraction for the numerator of the simple fraction, and the denominators for its denominator. If the proposed complex fraction be, for example, of of 1, it is found in the same manner that

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The reduced or simple fraction, in this instance, has for numerator the product of the numerators, and for denominator the product of the denominators of the fractional expressions composing the complex fraction.

The method followed in the reduction of the preceding examples of complex fractions being general and capable of extension to cases in which the number of dependent fractional expressions is greater than three, it follows that the rule for reducing any complex fraction to an equivalent simple fraction is,

Multiply together the numerators of the complex fraction for the numerator of the simple fraction, and the denominators for its denominator.

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DECIMAL FRACTIONS.

104. A Decimal Fraction, or, more briefly, a Decimal, is a fraction which has 10 or some power of 10 for its denominator. It follows, from Art. 3., that a figure placed to the right of

another figure expresses units of the order immediately inferior to those of that other figure. Hence, a figure on the right of that which expresses the simple units of a number must express tenth parts of unity; a figure on the right of that which expresses tenth parts must express tenth parts of tenth parts, or hundredth parts of unity, and so on; the relative value of the parts of unity depending on the place of the figure, and the number of parts on the absolute value of the figure, precisely as in the case of figures representing whole numbers.

The integral part of a mixed number, expressed decimally, is separated from the fractional part by a point (*), named the decimal point. When expressed in words, the integral part is named in the usual manner (Art. 5.); the fractional part, afterwards, also in the usual manner, the name of this part being followed by a word describing the relative value of the last decimal figure.

Thus, the expression in words of 29.374 is twenty-nine, three hundred and seventy-four thousandths.

The fractional part of this number, for example, may also be expressed in words, thus, three tenths, seven hundredths, and four thousandths.

These forms are mutually convertible,

For 3 tenths=30 hundredths=300 thousandths=300 thousandths 7 hundredths = 70 thousandths = 70

the sum of which is 374 thousandths.

Conversely,

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300 thousandths=30 hundredths=3 tenths-3 tenths

70 thousandths = 7 hundredths=

7 hundredths

4 thousandths=

4 thousandths.

Whence, by the use of the decimal point, and the extension of the conventions of the decimal system to numbers less than unity, a sum of fractions such as To, and is expressed in this simple form, 374.

105. In a decimal fraction, the decimal point indicates, by its position, the magnitude of the denominator, which is 1 followed by as many zeros as there are figures on the right of the decimal point.

Hence 0 on the right of a decimal has no effect on the value of that decimal: for 6 = $; ·60== $; ·600== · · ·

.*. ·6="60=*600 ...

...

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