DIVISION OF VULGAR FRACTIONS.

RULE. Having reduced compound fractions to simple. ones, and mixed numbers to improper fractions, multiply the numerator of the divisor by the denominator of the dividend, the product will be the denominator of the quotient; then, multiply the denominator of the divisor by the numerator of the dividend, the product will be the numerator of the quotient, which being reduced to its lowest terms will be the answer; or, invert the divisor, and then proceed as in multiplication of vulgar fractions.

4 is ; then, 4)19(95 quot. and 15 is 515 Ans.

16

2.

Divide

of by of 4.

3. Divide

of 17 by 3 of 3

by

5. Divide by 7.

6. Divide 145 by 3 of

DECIMAL FRACTIONS.

DECIMAL FRACTIONS are wrought in the same manner as whole numbers, and are distinguished from whole numbers, by a comma* placed at the left hand; thus,,25,75 ,258,7853 which are read twenty-five hundredths, seventy five hundredths, two hundred and fifty-eight thous andths, &c.

The comma, in decimals fractions, or the point which separates a decimal fraction from a whole number, is called a Separatrix.

The denominator of a decimal fraction is always understood, it being 1, with so many ciphers at the right hand as there are figures in the numerator, that is, if the numerator be one figure only, it is so many tenths as there are units in the figure; if the numerator consist of two places, it is so many hundredths; if three, so many thousandths, &c. Thus,2,25 ,258 if expressed in the manner of vulgar fractions would be

25 258

780.100.1000.

When there are more ciphers in the denominator, than there are places of figures in the numerator, the deficiency must be supplied by_prefixing* a cipher, or ciphers, to the numerator; thus, 85, must be written, ,03, ,025, &c.

To enumerate any decimal, begin at the left hand figure and proceed to the right. The first figure at the left hand, or the figure next to the comma, is called tens; the second, hundreds; the third, thousands, &c. ; each figure, from left to right, decreasing in the some proportion that whole numbers increase from right to left, as is shown in the following

TABLE.

Tens, or tenth parts.

Hund. of thous. or hund. thousandth parts.
Tens of thous, or ten thousandth parts.
Hundreds, or hundredth parts.
Thousands, or thousandth parts.
Millions, or millionth parts.

Ten of millions, or ten millionth parts.
Hund. of mill. or hund. millionth parts.

NOTE. The first, second, third, fourth, &c. places of decimals, reckoning from the left hand to the right, are called primes, seconds, thirds, fourths, fifths,&c.

From the above table it appears plain, that every figure

counting from left to right, decreases in a tenfold proportion ; that is, the value of a figure in the place of hundreds, is ten times less than the value of the same figure, in the place of tens ; and the value of a figure in the place of thousands, is ten times less than that of the same figure in the place of hundreds, &c. for, an hundredth part is ten times less than a tenth part, and a thousandth part is ten times less than a hundredth part, &c.

* When ciphers are to be placed at the left hand of any number, they are said to be prefixed, but when they are to be placed at the right band, they are said to be annexed.

The annexing of ciphers to a decimal fraction does neither increase nor diminish its value; for,5,50 and,500, that is, are all equivalent, being equal to 4.

50 500 TO 100 1000'

But by prefixing ciphers to a decimal fraction, the value is decreased; for every cipher which is prefixed renders the value of the fraction ten times less than it was before the cipher was prefixed; thus,,5,05 and,005 are of different values, as may be seen by expressing them in the manner of vulgar. fractions; thus,,5 is;,05 is 150,005 is 1000.

ADDITION OF DECIMALS.

RULE. Whether the sums to be added are mixed, or altogether decimals, place them in such manner, as that each figure of every sum may stand directly under those of the same name; then proceed in every respect as in Simple Addition. Point off so many places from the total sum, for decimals, as there are decimals in the greatest number added.

6. What is the sum of 85,385-848,25,3085-28,75---3,4867 and ,835?

967,0152 Ans.

856,22776 Ans.

7. What is the sum of 850-3,587—,5873-1,255 and ,79846 ? 8. What is the sum of,8536—7,75—81,113—583—,684 and 3,333 ?

9. What is the sum of 4,25-5,075 and 7,0025 ?

676,7336 Ans.

16,3275 Ans.

SUBTRACTION OF DECIMALS.

RULE. Place the numbers as directed in addition of decimals; then subtract as in whole numbers, and from the remainder point off so many places for decimals, as there are decimals in the greatest number.

MULTIPLICATION OF DECIMALS.

RULE. Proceed as in Multiplication of whole numbers; then point off so many places of the product for decimals, as there are decimals in both the multiplicand and multiplier ; but, if the product does not consist of so many places, the deficiency must be supplied by prefixing ciphers.

5072 17752. 7608

,00943392

In this example, the number of places in the product being less than the number of decimals in the multiplicand and multiplier, the defect is supplied by prefixing two ciphers.

DIVISION OF DECIMALS.

RULE. Proceed as in Division of whole numbers; then point off so many places of the quotient for decimals, as the dividend has decimal places more than the divisor.

Note 1. If there happen to be not so many piaces in the quotient as are required, prefix a sufficient number of ciphers to make up the defect.

Note 2. When the decimal places of the divisor are more than those of the dividend, this defect must be supplied by annexing ciphers.*

Note 3. When there is a remainder, ciphers may be annexed to it, which render it capable of being further divided, and the succeeding figures in the quotient are decimals; which by annexing a cipher, or ciphers, to every succeeding remainder, may be continued at pleasure.

Note 4. When the dividend is a whole number, and the divisor a decimal, annex so many ciphers to the dividend as there are decimal places in the divisor, the quotient figures will be whole numbers till all the annexed ciphers are brought down and divided; then, if there be a remainder, annex a cipher or ciphers to it and divide; the remaining figures in the quotient will be decimals.

N. B. Four or five places of decimals are generally sufficient. Note 5. When the dividend is a decimal, and the divisor a whole number, divide as in whole numbers till every figure of the dividend is brought down and divided; and if there are not so many places in the quotient as there are decimals in the dividend, supply the defect by prefixing a cipher or ciphers; and if there be a remainder, ciphers may be annexed, and the quotient carried on still further.

Note 6. When the dividend is a decimal and the divisor a whole number, if the divisor is not contained in the dividend place a cipher in the quotient in the first place; then annex a cipher to the dividend, and if the divisor is not contained in the dividend after one cipher is annexed, place another cipher in the quotient and annex another to the dividend; thus proceed till the dividend can be divided, and if there be a remainder, a cipher or ciphers may be annexed, and the quotient carried on still further,

When the dividend consists of an integer only, or when it is a mixed number, if the decima! places of the divisor be more than those of the dividend, supply the defect by annexing ciphers to the integer, or decimal; the quotient figures will be whole numbers, till all the annexed ciphers are brought down and divided; then, if there be a remainder, annex a cipher, or ciphers, and divide; the remaining figures in the quotient will be decimals. G